Nonconforming Immersed Finite Element Methods for Interface Problems

Transcription

1 Nonconforming Immersed Finite Element Metods for Interface Problems Xu Zang Dissertation submitted to te Faculty of te Virginia Polytecnic Institute and State University in partial fulfillment of te requirements for te degree of Doctor of Pilosopy in Matematics Tao Lin, Cair Slimane Adjerid Eric de Sturler Yuriko Renardy April 4, 03 Blacksburg, Virginia Keywords: Immersed Finite Element, Elliptic Interface Problems, Cartesian Mes, Nonconforming Rotated Q Finite Element, Error Analysis, Elasticity Interface Problems, Moving Interface Problems, Discontinuous Galerkin Metods Copyrigt 03, Xu Zang

2 Nonconforming Immersed Finite Element Metods for Interface Problems Xu Zang (ABSTRACT) In science and engineering, many simulations are carried out over domains consisting of multiple materials separated by curves/surfaces. If partial differential equations (PDEs) are used to model tese simulations, it usually leads to te so-called interface problems of PDEs wose coefficients are discontinuous. In tis dissertation, we consider nonconforming immersed finite element (IFE) metods and error analysis for interface problems. We first consider te second order elliptic interface problem wit a discontinuous diffusion coefficient. We propose new IFE spaces based on te nonconforming rotated Q finite elements on Cartesian meses. Te degrees of freedom of tese IFE spaces are determined by midpoint values or average integral values on edges. We investigate fundamental properties of tese IFE spaces, suc as unisolvency and partition of unity, and extend well-known trace inequalities and inverse inequalities to tese IFE functions. Troug interpolation error analysis, we prove tat tese IFE spaces ave optimal approximation capabilities. We use tese IFE spaces to develop partially penalized Galerkin (PPG) IFE scemes wose bilinear forms contain penalty terms over interface edges. Error estimation is carried out for tese IFE scemes. We prove tat te PPG scemes wit IFE spaces based on integralvalue degrees of freedom ave te optimal convergence in an energy norm. Following a similar approac, we prove tat te interior penalty discontinuous Galerkin scemes based on tese IFE functions also ave te optimal convergence. However, for te PPG scemes based on midpoint-value degrees of freedom, we prove tat tey ave at least a sub-optimal convergence. Numerical experiments are provided to demonstrate features of tese IFE metods and compare tem wit oter related numerical scemes. We extend nonconforming IFE scemes to te planar elasticity interface problem wit discontinuous Lamé parameters. Vector-valued nonconforming rotated Q IFE functions wit integral-value degrees of freedom are unisolvent wit appropriate interface jump conditions. More importantly, te Galerkin IFE sceme using tese vector-valued nonconforming rotated Q IFE functions are locking-free for nearly incompressible elastic materials. In te last part of tis dissertation, we consider potential applications of IFE metods to time dependent PDEs wit moving interfaces. Using IFE functions in te discretization in space enables te applicability of te metod of lines. Crank-Nicolson type fully discrete scemes are also developed as alternative approaces for solving moving interface problems. Tis work received support from NSF grants DMS and DMS-0633.

3 Dedication To Yi and Olivia. iii

4 Acknowledgments I would like to take tis opportunity to look over te past five years and remember all te friends and family wo elped and supported me along tis fulfilling journey. First of all, I want to express my eartfelt gratitude to my advisor, Dr. Tao Lin, wo led me into tis exciting finite element world. His constant guidance and support ave greatly elped me work troug numerous callenging problems. His integrity and entusiasm inspire me to become better as a scolar, an educator, and a person. I ope tat one day I would become as good an advisor to my students as Dr. Lin as been to me. Members of P.D. committee Dr. Slimane Adjerid, Dr. Eric de Sturler, and Dr. Yuriko Renardy deserve my sincerest tanks. I ave learned a lot from eac of tem and I am grateful for teir valuable advises and insigtful comments on my researc and dissertation. I am tankful to Dr. Oleg Roderick for offering me a wonderful summer researc experience at Argonne National Laboratory and leading me to te uncertainty quantification researc. I would also like to tank Dr. Yanping Lin, Dr. Dongwoo Seen, Dr. Xiaoming He, and Dr. Zu Wang for teir elp and collaboration in various researc projects. Special tanks to Ms. Eileen Sugart, Ms. Margaret McQuain and Dr. Donald McKeon for being my teacing mentors for te past years and saring a lot of teir valuable experiences wit me. I wis to express my sincerest gratitude to my parents. Teir constant support provides inspiration and driving force for me to pursue in te academic road. I am tankful to my wife Yi, witout wom, te dissertation could not be completed. I will always appreciate er love and support, sometimes even sacrifices to elp me focus on my work. iv

5 Contents List of Figures List of Tables vii x Introduction. Interface Problems and Teir Applications Survey of Numerical Metods for Interface Problems Survey of Previous Work for Immersed Finite Element Metods Motivations to Study Nonconforming IFE Metods Outline of te Dissertation Nonconforming IFE Spaces 6. Preliminaries and Notations Nonconforming Rotated Q Functions Nonconforming IFE Space S P (Ω) Nonconforming IFE Space S I (Ω) Properties of IFE Spaces 9 3. Fundamental Properties Approximation Capabilities Error Analysis of Interpolation on S P (Ω) Error Analysis of Interpolation on S I (Ω) Numerical Experiments v

6 4 IFE Metods and Error Estimation IFE Metods Galerkin IFE Metods Partially Penalized Galerkin IFE Metods IPDG IFE Metods Error Estimation Error Estimation for PPG IFE Solutions in S I (Ω) Error Estimation for IFE Solutions in S P (Ω) Error Estimation for IPDG IFE Solutions Discussions on Related Scemes Nonconforming IFE Metods for Elasticity Interface Problems Introduction Vector-Valued Nonconforming IFE Spaces Nonconforming IFE functions Properties of Nonconforming Rotated Q IFE Spaces Interpolation and Galerkin Metod Numerical Experiments Applications of IFEs to Moving Interface Problems 7 6. Introduction IFE Metod of Lines Semi-Discrete Scemes Crank-Nicolson IFE Fully Discrete Algoritms Implementation for Moving Interfaces Numerical Experiments Future Work 54 Bibliograpy 56 vi

7 List of Figures. A sketc of solution domain for interface problems Comparison of a body-fitting triangular mes (left) and non-body-fitting rectangular (middle) and triangular (rigt) meses One dimensional linear IFE local basis functions One dimensional quadratic IFE local basis functions One dimensional cubic IFE local basis functions One dimensional linear, quadratic, and cubic IFE global basis functions Interface edges (marked by red color) on a Cartesian mes Two dimensional linear FE (left) and IFE (rigt) local basis functions Two dimensional linear FE (left) and IFE (rigt) global basis functions Two dimensional bilinear FE/IFE local basis functions. From left to rigt: FE basis, Type I IFE basis, Type II IFE basis Two dimensional bilinear FE (left) and IFE (rigt) global basis functions Te left plot is a rectangular domain wit a circular interface. Te rigt plot is te point-wise error of a bilinear IFE solution to an elliptic interface problem define on te geometry on te left A non-interface rectangular element Type I interface rectangles: Case,,3 (midpoint-value degrees of freedom)...3 Type II interface rectangles: Case, (midpoint-value degrees of freedom)...4 Nonconforming rotated Q FE (left), Type I (middle) and Type II (rigt) IFE local basis functions wit midpoint-value degrees of freedom vii

8 .5 Nonconforming rotated Q FE (left) and IFE (rigt) global basis functions wit midpoint-value degrees of freedom Type I and Type II interface rectangles (integral-value degrees of freedom) Nonconforming rotated Q FE (left), Type I (middle) and Type II (rigt) IFE local basis functions wit integral-value degrees of freedom Nonconforming rotated Q FE (left) and IFE (rigt) global basis functions wit integral-value degrees of freedom A sketc of interface rectangle: Type I, Case Points selected to calculate te L norm on a rectangular element T Comparison of errors in different nonconforming rotated Q IFE metods wit β =, β + = Point-wise error comparison of Galerkin solution and NPPG solution u P Point-wise error comparison of Galerkin solution and NPPG solution u I Point-wise error comparison of bilinear Galerkin IFE solution and NPPG IFE solution u Comparison of errors in different Galerkin IFE metods Comparison of errors in different NPPG IFE metods Comparison of discontinuity for different IFE global basis functions wit fixed value β = and different values of β + =, 5, Te domain of planar elasticity interface problems A vector-valued nonconforming rotated Q finite element local basis function. 5.3 A vector-valued nonconforming rotated Q IFE local basis function on a Type I interface element A vector-valued nonconforming rotated Q IFE local basis function on a Type II interface element Errors of bilinear IFE solutions and nonconforming rotated Q IFE solutions u. From left to rigt: L, L, H norms A comparison of body-fitting triangular mes wit a non-body-fitting Cartesian mes for a boundary layer problem viii

9 5.7 Errors of nonconforming rotated Q IFE solutions u for te boundary layer example in different norms Solution domain of moving interface problems A body-fitting triangular mes (left) and a non-body-fitting triangular Cartesian mes (rigt) A sketc of te interface configuration in a triangle at time t Cases of interface triangle cut by two interface line segments tat intersect inside te triangle Cases of interface triangle cut by two interface line segments tat intersect outside te triangle Cases of Type I interface rectangle cut by two interface line segments tat intersect inside te rectangle Cases of Type I interface rectangle cut by two interface line segments tat intersect outside te rectangle Cases of Type II interface rectangle cut by two interface line segments tat intersect inside te rectangle Cases of Type II reference rectangle cut by two interface line segments tat intersect outside te rectangle Te left plot sows ow te radius r(t) of te interface circle Γ(t) canges; te rigt plot is for te time step sizes used by te IFE-MoL combined wit te adaptive DIRK45 ODE solver Te left plot contains curves of L norm error for tree IFE solutions generated on te same mes wit = /64. Te rigt plot is te enlarged part for time between 0.8 to ix

10 List of Tables 3. Errors of IFE interpolations I I u wit β =, β + = Errors of IFE interpolations I P u wit β =, β + = Errors of IFE interpolations I I u wit β =, β + = Errors of IFE interpolations I P u wit β =, β + = Errors of NPPG IFE solutions u u I wit β =, β + = Errors of SPPG IFE solutions u u I wit β =, β + = Errors of IPPG IFE solutions u u I wit β =, β + = Errors of NPPG IFE solutions u u I wit β =, β + = Errors of SPPG IFE solutions u u I wit β =, β + = Errors of IPPG IFE solutions u u I wit β =, β + = Errors of NPPG IFE solutions u u P wit β =, β + = Errors of SPPG IFE solutions u u P wit β =, β + = Errors of IPPG IFE solutions u u P wit β =, β + = Errors of NIPDG IFE solutions u u DG wit β =, β + = Errors of SIPDG IFE solutions u u DG wit β =, β + = Errors of IIPDG IFE solutions u u DG wit β =, β + = Errors of Galerkin IFE solutions u u P wit β =, β + = Errors of Galerkin IFE solutions u u I wit β =, β + = Errors of Galerkin IFE solutions u u I wit β =, β + = Errors of bilinear Galerkin IFE solutions u u wit β =, β + = x

11 4.7 Errors of bilinear NPPG IFE solutions u u wit β =, β + = Errors of bilinear SPPG IFE solutions u u wit β =, β + = Errors of bilinear IPPG IFE solutions u u wit β =, β + = Errors of nonconforming rotated Q IFE interpolations and Galerkin IFE solutions wit λ + = 5, λ =, µ + = 0, µ =, ν ± = Errors of nonconforming rotated Q IFE interpolations and Galerkin IFE solutions wit λ + = 00, λ =, µ + = 00, µ =, ν ± = Errors of nonconforming rotated Q IFE interpolations and Galerkin IFE solutions wit λ + =, λ = 00, µ + =, µ = 00, ν ± = Errors of bilinear and nonconforming rotated Q Galerkin IFE solutions in locking test wit λ + = 0, λ =, µ + = 0.0, µ = 0.00, ν ± Errors of bilinear and nonconforming rotated Q Galerkin IFE solutions in locking test wit λ + = 00, λ =, µ + = 0., µ = 0.00, ν ± Errors of nonconforming rotated Q Galerkin IFE solutions for problems wose interfaces are at different locations Comparison of errors of te linear FE and nonconforming rotated Q IFE solutions for te boundary layer example wit λ + =, λ =, µ + = 3, µ =, ν + = 0., ν Errors of nonconforming rotated Q IFE solutions for te boundary layer example wit λ + =, λ =, µ + = 3, µ =, ν + = 0., ν Errors of nonconforming rotated Q IFE solutions for te boundary layer example wit λ + = 000, λ = 000, µ + = 3, µ =, ν , and ν Errors of linear IFE solutions wit β = using DIRK at time t = Errors of linear IFE solutions wit β =, β + = using 4t order scemes at time t = Errors of linear IFE adaptive DIRK45 solutions wit β = 0.5, β + = at time t = Errors of linear CN-IFE solutions wit β =, β + = and τ = at time t = Errors of linear CN-IFE solutions wit β =, β + = and τ = at time 8 t = xi

12 6.6 Errors of linear CN-IFE solution wit β =, β + = 00 and τ = at time 8 t = Errors of linear IFE solutions in CN-IFE Algoritm 3 wit β = and β + = 00 at time t = Errors of linear IFE solution using CN-IFE-A wit β = and τ = at time t = xii

13 Capter Introduction In tis capter, we start wit te introduction to te typical second order elliptic interface problems and teir applications. Next we provide a brief survey of numerical metods for interface problems and a sort review of te recently developed immersed finite element (IFE) metods. Our motivations to work on nonconforming IFE metods are presented at te end of tis capter.. Interface Problems and Teir Applications We consider te classic second order elliptic interface problem tat appears in many applications: (β u) = f, in Ω, (.) u = g, on Ω, (.) were te pysical domain Ω R d, d =,, 3, is assumed to be formed by multiple materials. Witout loss of generality, we assume tat te domain Ω is separated by an interface Γ into two sub-domains Ω +, Ω, suc tat Ω = Ω + Ω Γ, see Figure. for an illustration. Eac sub-domain contains only one material. Te coefficient function β(x) is discontinuous across te interface Γ due to te cange in material properties. For simplicity, we assume β(x) is a piece-wise constant function defined by { β β(x) =, if X Ω, β +, if X Ω + (.3), were β, β + > 0. Across te interface Γ, te solution u(x, y) is assumed to satisfy te jump conditions: [u] Γ = 0, (.4) [ β u n]γ = 0, (.5)

14 Xu Zang Capter. Introduction were n is te unit normal vector of te interface Γ. Here, for every piece-wise function v defined as { v v(x) = (X), if X Ω, v + (X), if X Ω +, we let [v] Γ = v + Γ v Γ. Figure.: A sketc of solution domain for interface problems. Ω + Γ Ω Ω Te elliptic interface problem arises in many applications, one of wic is te plasma particle simulations in ion truster optics [4, 83, 84]. An ion truster is a type of electric propulsion device wic emits a ig-energy ion beam to propel a spacecraft. Te standard algoritm to model a plasma is particle-in-cell (PIC) in wic te te propellant beam ions are represented by macro particles. Te trajectory of eac ion plasma particle is determined by Newton s second law: mp i = ma i = F i (Φ), i =,,, M, (.6) were p i is te position of eac particle, M is te total number of particles, and Φ is te electric potential. Te electric field is governed by te elliptic equation: (β Φ) = f(φ). (.7) Tese two sets of equations are coupled troug F(Φ) and f(φ) due to Lorentz force and oter pertinent pysical laws [83]. Note tat te simulation domain of te electric fields is usually cosen as a tree dimensional domain containing te ion truster wit parts formed by different materials under investigation; ence, tis simulation domain consists of multiple materials and leads to te interface model problems (.) - (.5). Oter applications for te elliptic interface problems include plasma simulations in spacecraft carging in space [69, 4], te projection metods to solve te Navier-Stokes problems involving multi-pase flows [40, 77], and topology optimization of eat conduction problems [56], to name just a few.

15 Xu Zang Capter. Introduction 3. Survey of Numerical Metods for Interface Problems A large number of numerical metods ave been developed for interface problems suc as finite difference (FD) metods and finite element (FE) metods. Tese conventional metods, especially finite element metods [6, 39, 49], can be applied to solve interface problems, provided tat solution meses are tailored to fit te interface; oterwise, te convergence of te numerical solutions migt be impaired [9]. Suc meses are often called body-fitting, as illustrated in te left plot of Figure.. Geometrically, te body-fitting restriction requires eac element to be placed essentially on one side of a material interface. Pysically, it means eac element in a mes to be occupied mainly by one of te materials. Figure.: Comparison of a body-fitting triangular mes (left) and non-body-fitting rectangular (middle) and triangular (rigt) meses. On te oter and, it is usually time consuming to generate a satisfactory body-fitting mes for an interface problem in wic te interface separating te materials is geometrically complicated. Suc a difficulty becomes even more severe if te interface evolves in a simulation because a new mes as to be generated for eac of te material configurations to be considered. For many applications, it is terefore desirable to develop numerical metods tat can be used wit non-body-fitting meses, suc as te Cartesian meses in te plot in te middle and on rigt in Figure., to solve interface problems. A Cartesian mes can also be obviously advantageous in many simulations. Particle-In-Cell metod for plasma particle simulations [83, 84, 06, 07] is a typical example tat prefers te electric potential interface problem (.7) to be solved on a Cartesian mes for efficient particle tracking. Many numerical metods based on Cartesian meses ave been developed for interface problems. In te finite difference formulation, Peskin developed te immersed boundary metod [0,, 8] in 977 to study flow patterns around eart valves. Since ten, various finite difference scemes using Cartesian meses were proposed, suc as te immersed interface metod [5, 90, 9, 94], te gost fluid metod [54,, ], te matced interface and boundary metod [54, 55, 56], and te Cartesian grid metod [, 3, 9], to name just a

16 Xu Zang Capter. Introduction 4 few. In te finite element formulation, special treatments need to be taken for elements around te interface. One way is to modify te weak formulation of finite element equations near te interface. Te penalty finite element metod [9, 8] penalizes te solution jump around interface. Te unfitted finite element metod [65, 66, 48] using Nitsce s sceme, modifies te bilinear form near interface by using te weigted average flux. Discontinuous Galerkin formulation metods [0, 63] penalize te interface jump conditions to solve interface problems. Anoter approac is to modify te finite element functions around te interface. Te general finite element metod [0,, 4, 5] utilizes special sape functions to capture critical features of te unknown solution and tey can be non-polynomials for some cases. Te multi-scale finite element metod [43, 5, 78] modifies te basis functions around te interface by solving an auxiliary subgrid problem. Oter metods in tis category include te extended finite element metod [49, 80, 5, 35], te partition of unity metod [, 3, 7]. Te recently proposed immersed finite element metods also fall into tis framework. Since tey are closely related wit te researc presented in tis dissertation, we will give a more detailed introduction in te next section..3 Survey of Previous Work for Immersed Finite Element Metods Immersed finite element (IFE) metods ave been developed for over a decade since te first article [93] was publised. Te main idea for IFE metods is to adapt finite element functions instead of solution meses for interface problems. IFE metods can use non-bodyfitting meses for interface problems, suc as Cartesian meses, wic are independent of te interface. Consequently te interface are allowed to cut te interior of elements, in oter words, te interface can be immersed in some of te elements, and tis is were te name immersed originated. In an IFE metod, elements are divided into two categories: interface elements, wose interior intersects wit te interface, and non-interface elements, consisting of te rest of elements. Standard finite element functions are utilized on non-interface elements, wile special IFE functions are constructed on interface elements. According to te interface location, IFE functions are constructed in te form of piece-wise polynomials on interface elements to incorporate interface jump conditions. In tis section, we give a sort review of previous work on IFE metods for different types of interface problems.

17 y y Xu Zang Capter. Introduction 5 IFE Metods for One Dimensional Elliptic Interface Problems In 998, Li introduced a linear IFE metod for one dimensional two-point boundary value problem wit one interface point [93]: (β(x)u (x)) + q(x)u(x) = f(x), 0 < x <, (.8) Te jump condition at te interface α (0, ) is given by: u(0) = 0, u() = 0. (.9) [u(α)] = 0, [βu (α)] = 0. (.0) To capture essential features of te solution across te interface, IFE basis functions on an interface element were defined by piece-wise linear polynomials satisfying interface jump conditions (.0). To be more specific, on an interface element T = (x, x ), wit T = (x, α) and T + = (α, x ), te local linear IFE space was defined as follows: S (T ) = {φ : φ T s P (T s ), s = +,, and [φ(α)] = 0, [βφ (α)] = 0}. (.) Local linear IFE basis functions φ i, i =, were cosen from S (T ) suc tat φ i (x j ) = δ ij, i, j =,. (.) IFE spaces formed by iger degree polynomials ave been constructed. In particular, several types of quadratic IFE basis functions were introduced in [33, 99]. Te approximation capability of corresponding IFE spaces was analyzed. IFE spaces wit an arbitrary polynomial degree p were developed in []. Figure.3: One dimensional linear IFE local basis functions x x We illustrate te linear, quadratic and cubic IFE local basis functions in Figure.3, Figure.4, and Figure.5, respectively. Global IFE basis functions are illustrated in Figure.6. Tese one dimensional IFE functions are continuous; ence, tey are in H (Ω). A Galerkin sceme using any of tese IFE spaces is a conforming finite element metod. Approximation

18 y y y y y y y y y y Xu Zang Capter. Introduction 6 Figure.4: One dimensional quadratic IFE local basis functions x x x Figure.5: One dimensional cubic IFE local basis functions x x x x Figure.6: One dimensional linear, quadratic, and cubic IFE global basis functions x x x capabilities of tese IFE spaces and error estimates of related IFE solutions ave been proved to be optimal in bot L and H norms []. We recall teir results in te following two teorems. Let H p (Ω) = { u C(Ω) : u Ω s H p+ (Ω s ), s = +,, and [ βu (j) (α) ] = 0, j =,,, p }. Teorem.. Tere exists a constant C independent of interface α, suc tat for all u

19 Xu Zang Capter. Introduction 7 H p (Ω) 4p I p u u 0 + I p u u C (p )! p+ u p+. (.3) Here I p u is te p-t degree IFE interpolation of u. Teorem.. Tere exists a constant C independent of interface α, suc tat 4p u u 0 + u u C (p )! p+ u p+. (.4) Here u is te p-t degree IFE solution for D elliptic interface problems, and u H p (Ω) is te solution to te interface problem defined by (.8)-(.0). IFE Metods for Two Dimensional Elliptic Interface Problems IFE metods for two dimensional elliptic interface problems ave been extensively studied in te past decade, see for instance, [4, 4, 60, 6, 69, 70, 73, 87, 95, 96, 98, 30, 47], and te references terein. Note tat te above mentioned one dimensional IFE spaces [, 93] are conforming, i.e., eac of tese IFE function spaces is a subset of te trial function space H (Ω) of weak formulation of te elliptic interface problem. However, for iger dimensional cases, te global continuity of IFE functions cannot be guaranteed in general. In fact, locally imposing te interface jump condition on eac interface element only guarantees te continuity of IFE functions witin an element, but IFE functions are usually discontinuous across edges wic are cut by interfaces as tose edges marked by red color in Figure.7. In [96], bot conforming and nonconforming linear IFE metods ave been developed on triangular meses for te two dimensional elliptic interface problem. Te nonconforming linear IFE space is a natural extension of te one dimensional IFE spaces to two dimensional case. Te nonconforming IFE spaces ave many advantages. First, te construction of IFE functions is rater straigtforward because one only need to consider a single element and neglect te continuity between elements. Second, te resulting IFE linear system as te same structure as standard FE linear system on te same mes, wic makes efficient solvers for FEs applicable for solving interface problems. Te disadvantages are due to its nonconformity wic makes te error estimation more callenging. On te oter and, in te conforming IFE metod proposed in [96], to maintain te continuity along edges, eac conforming IFE basis as an enlarged support and takes an average or a weigted average value of nonconforming IFE basis wit te same values at nodal points. One benefit of te conforming IFEs is tat convergence analysis can be establised straigtforward via Céa s lemma [44]. However, one obvious drawback is tat te construction of its basis is more complicated tan nonconforming linear IFE functions. Also, due to a larger

20 Xu Zang Capter. Introduction 8 Figure.7: Interface edges (marked by red color) on a Cartesian mes. support of every global basis function, te global stiffness matrix is usually denser tan te matrix from a nonconforming IFE metod. Moreover, it is muc more complicated to extend tis conforming IFE metod to deal wit more complicated PDEs or using iger degree polynomials. A comparison of te nonconforming linear FE and IFE local basis functions is illustrated in Figure.8. Global FE and IFE functions are illustrated in Figure.9. Tese plots demonstrate tat linear IFE basis functions are continuous witin an interface element, but discontinuous across interface edges. Figure.8: Two dimensional linear FE (left) and IFE (rigt) local basis functions.

21 Xu Zang Capter. Introduction 9 Figure.9: Two dimensional linear FE (left) and IFE (rigt) global basis functions. Approximation capabilities of te nonconforming linear IFE spaces was studied in [95]. Instead of using te standard scaling argument, te autors extended te multi-point Taylor expansion idea [8, 35] to prove tat te Lagrange type IFE interpolation can acieve te optimal convergence orders O( ) in L norm and O() in H norm. Higer degree IFEs for two dimensional elliptic interface problems based on triangular meses were studied in [9]. Teir numerical experiments suggested tat interior penalty terms are required in te computational scemes in order to obtain te optimal convergence. For tree dimensional elliptic interface problem, nonconforming linear IFE spaces based on tetraedra meses were constructed and applied to plasma simulation based on te particle-in-cell formulation [8, 84]. Wen te solution domain is rectangular or polygonal tat can be divided into several rectangles, it is often preferable and more natural to use a Cartesian mes formed by rectangles, see rigt plot in Figure. as an illustration. A bilinear IFE space on te Cartesian rectangular meses was introduced in [98]. Note tat an interface curve may intersect a rectangle at two adjacent edges of a rectangle or at two opposite edges of te rectangle. Procedures of constructing bilinear IFE functions on tese two types of interface rectangles were discussed [70, 98]. Comparisons of local and global bilinear FE and IFE basis are illustrated in Figure.0 and Figure., respectively. Again, tese bilinear IFE spaces are nonconforming because functions in tese spaces are not continuous across interface edges as illustrated by te plot in te rigt in Figure.. Approximation capabilities of te bilinear IFE spaces were studied in [69, 70]. Teir analysis indicated tat bilinear IFE spaces retain optimal approximation capabilities in L and H norms as te standard bilinear FE spaces. Tese bilinear IFE functions ave been used in Galerkin metod [69], interior penalty discontinuous Galerkin metod [7], and finite volume metod [7] for solving elliptic interface problems. For tree dimensional elliptic interface problem, a trilinear IFE metod as been developed for solving

22 Xu Zang Capter. Introduction 0 te electroencepalograpy forward problem on parallelepiped meses [39]. For elliptic interface problems wit nonomogeneous flux jump, local IFE spaces were enriced by adding anoter piece-wise polynomial function wic vanises at all nodes but wit discontinuous flux [73]. An alternative approac to andle nonomogeneous flux jump was introduced in [6]. Figure.0: Two dimensional bilinear FE/IFE local basis functions. From left to rigt: FE basis, Type I IFE basis, Type II IFE basis. Figure.: Two dimensional bilinear FE (left) and IFE (rigt) global basis functions.

23 Xu Zang Capter. Introduction IFE Metods for Oter Interface Problems IFE metods ave been extended to oter types of interface problems involving a system of PDEs, time dependent PDEs, and iger order PDEs. For planar elasticity interface problems, a nonconforming linear IFE metod based on te triangular Cartesian mes was presented in [60, 97, 50] and te autors concluded tat tis metod as at least O() convergence in L norm. In [60, 6], te autors developed a conforming linear IFE metod were te optimal convergence rate in L norm was observed. However, in tis conforming IFE configuration, global IFE basis functions ave a rater complicated and larger support around te interface. In [08], a nonconforming bilinear IFE metod was developed for te elasticity interface problems. Te autors studied error beaviors for bot linear and bilinear IFE metods in L and H norms. More importantly, tey discovered tat linear and bilinear IFE functions for elasticity interface problems do not always ave te unisolvent property. As a result, in some configurations of interface location elasticity materials, linear and bilinear IFE functions cannot be constructed. Also in [08], te autors analyzed te unisolvent property and proposed an approac to identify a class of elastic materials for wic te linear and bilinear IFE functions are guaranteed to be uniquely constructed. Teir analysis indicated tat te bilinear IFE functions are guaranteed to be applicable for elasticity interface problems for a larger class of elasticity material configurations tan linear IFE functions. For time dependent problems, IFE metods ave been used to solve te parabolic interface problems wit static interfaces. In [8] te autors proposed a backward Euler sceme using IFEs to solve a semi-linear parabolic interface problem and studied corresponding error estimation. In [04], a novel approac was proposed using IFEs togeter wit te Laplace transform to solve a parabolic interface problem. Unlike classic time-marcing algoritms, for wic solution on new time step necessitates information from previous time step, te Laplace transformation in time led to a set of Helmoltz-like interface problems independent of eac oter, wic can be solved wit IFE functions in parallel. In [4], an immersed Eulerian-Lagrangian localized adjoint metod was developed for transient advection-diffusion interface problems. IFE metods ave been used also for time dependent problems wit moving interfaces. In [75], te autors developed several fully discrete Crank-Nicolson (CN) type IFE algoritms for solving parabolic equation wit moving interfaces. All of tese algoritms ave been observed to ave optimal convergence rates. Tese CN-IFE algoritms are consistent wit te standard CN-FE sceme for te parabolic equation in te sense tat tey become te standard CN sceme if te coefficient function is continuous or if te interface Γ does not cange wit time and a body-fitting mes is used. In [0], te autors demonstrated tat IFEs metods can be used togeter wit te metod of lines (MoL) to solve parabolic interface problems. Tey observed tat using IFEs and a suitably cosen ODE solver can efficiently and reliably solve parabolic moving interface problems on a fixed Cartesian mes. In particular, an adaptive IFE-MoL sceme can be a very effective and efficient approac for

24 Xu Zang Capter. Introduction a moving interface problem were te interface canges wit respect to time in a complicated way. Te IFE-MoL scemes were extended to andle te non-omogeneous flux jump in a moving interface problem [0]. For interface problems involving fourt order PDEs, IFE metods were used to solve te one dimensional beam interface problem and te two dimensional bi-armonic interface problem [00]. To avoid using iger degree polynomials (p ), tese metods used a mixed formulation wit linear IFE functions..4 Motivations to Study Nonconforming IFE Metods Tis dissertation focuses on nonconforming IFE metods for interface problems. First we would like to clarify te terminology nonconforming. Te nonconforming ere is different from te one used traditionally in te finite element literatures. Note tat most of te two and tree dimensional IFE spaces in te literatures, except for te only one in [96], are nonconforming in te traditional sense because teir IFE functions are discontinuous across interface edges. However, tese nonconforming IFE spaces are constructed using linear or bilinear polynomials wit teir degrees of freedom determined by te nodal values on vertices, wic is a key ingredient for conforming finite elements. In tis dissertation, we propose new IFE spaces wit teir DOF determined by te midpoint values or average integral values over edges, wic are nonconforming finite element ideas. One can see tis difference from anoter point of view. IFE spaces can be viewed as locally modified finite element spaces for interface problems in te sense tat finite element functions are adjusted to satisfy interface jump conditions. Most of te IFE spaces in te literatures modify conforming finite element spaces suc as te usual linear or bilinear finite element spaces. Functions in IFE spaces proposed in tis dissertation are revised nonconforming finite element functions according to specific interface problems; ence, tey are called nonconforming IFE spaces. To be specific, we adapt te simplest nonconforming finite elements on rectangular meses, i.e. te nonconforming rotated Q finite elements [36, 38, 85, 86, 3]. Nonconforming finite element metods ave been widely used especially in te solid mecanics and fluid mecanics due to teir better stability tan conforming finite element metods. Te simplest nonconforming finite element on a triangular mes, known as nonconforming P element or Crouzeix-Raviart element, was first introduced in 973 by Crouzeix and Raviart [47]. Te simplest nonconforming finite elements on a rectangular mes are called nonconforming rotated Q elements wic were first introduced by Rannacer and Turek [3] for solving te Stokes problem. Degrees of freedom for tese nonconforming finite elements are determined by te midpoint values on edges, or

25 Xu Zang Capter. Introduction 3 te average integral values over edges. In te following discussion, we will use S P (Ω) to denote te nonconforming FE/IFE space of te first type, wit P in S P (Ω) to empasize tat te degrees of freedom of tis finite element is determined by point values. We will use S I (Ω) to denote te nonconforming FE/IFE space of te second type, wit I in S I (Ω) to empasize tat te degrees of freedom of tis finite element is determined by integral values over element edges. Note tat te nonconforming P element [47] coincides for bot cases, i.e., S P (Ω) = SI (Ω). For nonconforming rotated Q finite elements, tese two types are different. Nonconforming finite element metods are used for solving elliptic problems [50, 85, 8] and elasticity problems [9, 53, 88]. We refer readers to [5, 8, 37, 44] and te reference terein for more details about nonconforming finite elements metods. In [87], Kwak, Wee and Cang developed IFE metods based on nonconforming P finite element for te second order elliptic interface problem. According to te autors, te difficulty of dealing wit nonconformity in te error analysis seems to be significantly reduced. Our motivations to study nonconforming IFE metods are from tree aspects. First, te convergence and error estimation for conforming type IFE metods are usually callenging. Tere ave been a few attempts [4, 69, 74, 43] in te study of error estimates of linear [95] and bilinear IFE [70] solutions to te two dimensional elliptic interface problem. Te difficulties in te analysis stem from te error estimation of te integrations over interface edges. Te estimation is callenging because conforming type IFE functions, suc as linear and bilinear IFE functions, are discontinuous across interface edges. Nonconforming finite elements, suc as Crouzeix-Raviart element and rotated Q element, impose te degrees of freedom, in oter words te continuity of IFE functions, troug edges instead of vertices; ence, we ope te continuity conditions of te IFE functions can be improved on interface edges. Tese nonconforming IFE functions wit improved continuity condition may furter elp us in te error analysis of IFE metods. Te second motivation to study nonconforming IFE metods is wit regard to improving te numerical performance. It as been observed tat conforming type IFE solutions are not as accurate around te interface as places away from te interface. For instance, we plot te error of a bilinear IFE solution to an elliptic problem wit a circular interface in Figure.. Te numerical solution error is observed to ave te crown effect, i.e., te bilinear IFE solution as muc larger error near te interface tan te rest of te domain. We guess tis is caused by te poor continuity condition of conforming type IFE functions on interface edges. Tis migt be te reason tat te convergence of conforming type IFE solutions cannot acieve optimal rate in L norm. Since nonconforming finite elements strengten continuity condition on edges, terefore we ope tat using nonconforming IFE metods can eliminate tis crown effect and improve te overall numerical performance. Te tird motivation to study nonconforming IFE metods is tat nonconforming finite elements are usually preferable to deal wit more complicated PDEs arising from solid and

26 Xu Zang Capter. Introduction 4 Figure.: Te left plot is a rectangular domain wit a circular interface. Te rigt plot is te point-wise error of a bilinear IFE solution to an elliptic interface problem define on te geometry on te left. Galerkin IFE x x Pointwise Error y x fluid mecanics due to teir reliability and simplicity. In linear elasticity system, low order conforming FE metods usually encounter te so-called locking penomenon as te elastic material becomes nearly incompressible [6, 45]. Nonconforming finite elements, on te oter and, are usually locking free and are tested to be more robust for tese applications. In Stokes system, te fluid velocity field is assumed to be divergence free wic caracterizes an incompressible fluid flow. Tere are no conforming finite element spaces wit piecewise polynomials of degree less tan five tat can satisfy te divergence free condition [37]; ence it is usually very expensive to use conforming finite element metods to solve Stokes equations. On te oter and, divergence free nonconforming finite element spaces can be constructed wit lowest order polynomials, suc as nonconforming P element [7] on triangular meses and nonconforming rotated Q element [5, 37] on rectangular meses. Wen we deal wit interface problems involving more complicated PDE models suc as te linear elasticity system and Stokes equations, it is desirable to develop IFE metods based on nonconforming rater tan conforming finite elements..5 Outline of te Dissertation In tis dissertation, we focus on te nonconforming rotated Q IFE metods and teir error analysis for elliptic and elasticity interface problems. Te rest of te dissertation is organized

27 Xu Zang Capter. Introduction 5 as follows. In Capter, we construct IFE spaces based on nonconforming rotated Q finite elements on Cartesian meses for second order elliptic interface problems. Te discussion covers bot midpoint value and integral value degrees of freedom of tese IFE spaces. In Capter 3, we investigate properties of tese nonconforming IFE spaces. Fundamental properties suc as partition of unity, trace inequalities, and inverse inequalities are proved. Te approximation capabilities of tese IFE spaces will be analyzed via te error estimation of interpolations. Some numerical examples will be provided at te end of tis capter. In Capter 4, we develop numerical scemes based on nonconforming IFE spaces for solving te elliptic interface problems. Error estimation for tese scemes will be carried out. Numerical experiments are provided to demonstrate features of tese new IFE metods. Related numerical metods are compared at te end of tis capter. In Capter 5, we extend tese nonconforming IFE metods to planar elasticity interface problems. We discuss ow to construct vector-valued IFE spaces for te elasticity system and investigate fundamental features of tese IFE spaces. Numerical examples are provided to demonstrate te locking free property of te new IFE metods. In Capter 6, we extend IFE metods to parabolic type moving interface problems. Bot semi-discrete and fully discrete scemes are developed. Numerical results are provided to demonstrate performance of tese scemes. In Capter 7, we briefly discuss our future researc plan.

28 Capter Nonconforming IFE Spaces In tis capter, we develop nonconforming IFE spaces S P (Ω) and SI (Ω) wit midpoint-value and integral-value degrees of freedom, respectively. Tese new IFE functions are derived from te well-known nonconforming rotated Q finite elements [3]. In Section., we introduce some preliminary results and notations tat will be frequently used trougout tis dissertation. In Section., we recall te basic framework of nonconforming rotated Q finite elements wic are used on non-interface elements of a Cartesian mes. In Section.3, we construct local IFE spaces S P (T ) on interface elements using te midpoint-value degrees of freedom, and ten form te corresponding global IFE space S P (Ω). In Section.4, we construct local IFE spaces S I (T ) on interface elements using te integral-value degrees of freedom, and ten form te corresponding global IFE space S I(Ω).. Preliminaries and Notations Trougout te dissertation, Ω R denotes a bounded domain formed by a union of rectangles. Let D(Ω) denote te space of C functions wit compact support in Ω. Te dual space D (Ω) is te space of distributions. For any multi-index α = (α, α ) and α = α +α, te weak derivative D α is defined by [8]: We use te Sobolev space D α v(φ) = ( ) α Ω α φ v x α y α dxdy, W k,p (Ω) = {v : v k,p,ω < }, φ D(Ω). wit non-negative integer index k and p. Te associated norm k,p,ω is defined 6

29 Xu Zang Capter. Nonconforming IFE Spaces 7 by v k,p,ω = D α v p 0,p,Ω α k were ( /p v 0,p,Ω = v dx) p, Ω in te case p <. If p =, we define v k,,ω = max α k Dα v 0,,Ω, /p, were v 0,,Ω = ess sup{ v(x) : X Ω}. Te Sobolev semi-norm k,p,ω associated wit W k,p (Ω) is defined by v k,p,ω = D α v 0,p,Ω, were α = k. In te case p =, te Sobolev space W k,p (Ω) becomes a Hilbert space and we denote it by H k (Ω) = W k, (Ω). We omit te index in associated norms for simplicity, i.e. v k,,ω = v k,ω, and v k,,ω = v k,ω. For interface problem, we assume tat te pysical domain Ω R is formed by multiple materials. Witout loss of generality, Ω is assumed to be separated by an interface curve Γ into two sub-domains Ω +, Ω, suc tat Ω = Ω + Ω Γ, see Figure.. Eac sub-domain contains only one material. Te solution spaces of interface problems usually ave low global regularity due to canges of solution properties across interfaces. If v Ω s W k,p (Ω s ), s = +,, and v / W k,p (Ω), ten te notations of norm v k,p,ω and semi-norm v k,p,ω sould be understood as follows v k,p,ω = ( v k,p,ω + + v k,p,ω ) /, v k,p,ω = ( v k,p,ω + + v k,p,ω ) /, (.) for p <, and v k,,ω = max ( v k,,ω +, v k,,ω ), v k,,ω = max ( v k,,ω +, v k,,ω ), (.) for p =. Let T be a Cartesian mes of te solution domain Ω wit te maximum lengt of edge. For eac element T T, we call it an interface element if te interior of T intersects wit te interface Γ; oterwise, we name it a non-interface element. Witout loss of generality, we assume tat interface elements in T satisfy te following ypoteses wen te mes size is small enoug [70]: (H) Te interface Γ cannot intersect an edge of any rectangular element at more tan two points unless te edge is part of Γ;

30 Xu Zang Capter. Nonconforming IFE Spaces 8 (H) If Γ intersects te boundary of a rectangular element at two points, tese intersection points must be on different edges of tis element. We let T i n be te collection of interface elements, and T = T /T i be te collection of noninterface elements. Moreover, we denote te collection of all edges in te mes T by E and denote te collections of interface edges and non-interface edges by E i and E n, respectively.. Nonconforming Rotated Q Functions In tis section, we recall te nonconforming rotated Q finite elements [3]. In our IFE metods, standard nonconforming rotated Q finite element functions are used on all noninterface elements. On interface elements, tese functions are locally modified to satisfy interface jump conditions. We first consider a typical non-interface element T T n We label four edges of T as follows: wit te following vertices: A = (0, 0) t, A = (, 0) t, A 3 = (0, ) t, A 4 = (, ) t. (.3) b = A A, b = A A 4, b 3 = A 4 A 3, b 4 = A 3 A, (.4) and let M i be te midpoint of b i, i =,, 3, 4, respectively. See Figure. for an illustration of a non-interface element. Figure.: A non-interface rectangular element. b 3 A 3 A 4 b 4 T b A A b

31 Xu Zang Capter. Nonconforming IFE Spaces 9 As proposed in [3], tere are two types of degrees of freedom for nonconforming rotated Q finite elements. Here we follow Ciarlet s triplet definition [44] of a finite element to differentiate tese two finite elements. Te first type of nonconforming rotated Q finite elements is defined by (T, Π T, Σ P T ), were and Π T = Span{, x, y, x y }, (.5) Σ P T = {ψ P T (M i ) : i =,, 3, 4, ψ P T Π T }. (.6) We use te superscript P to empasize tat te degrees of freedom are determined by te point values at te midpoints. It is easy to ceck tat tis Lagrange type finite element (T, Π T, Σ P T ) is unisolvent, i.e., for given values v i R, i =,, 3, 4, tere exists a unique function ψ P T Π T suc tat ψ P T (M i ) = v i, i =,, 3, 4. (.7) Te local finite element basis functions ψj,t P, j =,, 3, 4 are cosen suc tat In particular, if te vertices of T are given by (.3), ten ψ P j,t (M i ) = δ ij, i, j =,, 3, 4. (.8) ψ P,T = 4 (3 + 4x 8y 4(x y )), ψ P,T = 4 ( + 4y + 4(x y )), ψ P 3,T = 4 ( + 4x 4(x y )), ψ P 4,T = 4 (3 8x + 4y + 4(x y )). (.9a) (.9b) (.9c) (.9d) Te second type of nonconforming rotated Q finite elements is defined by te triplet (T, Π T, Σ I T ), were { } Σ I T = ψt I (X)ds : i =,, 3, 4, ψt I Π T, (.0) b i b i and b i denotes te lengt of te edge b i. Here we use superscript I to empasize tat te degrees of freedom are given by average integral values over edges. Again, it is not ard to verify tat (T, Π T, Σ I T ) is unisolvent. We coose te local basis functions ψi j,t, j =,, 3, 4, suc tat ψj,t I (X)ds = δ ij. (.) b i b i

32 Xu Zang Capter. Nonconforming IFE Spaces 0 In particular, if te vertices of T are given by (.3), ten ψ I,T = 4 (3 + 6x 0y 6(x y )), ψ I,T = 4 ( x + 6y + 6(x y )), ψ I 3,T = 4 ( + 6x y 6(x y )), ψ I 4,T = 4 (3 0x + 6y + 6(x y )). (.a) (.b) (.c) (.d) We define nonconforming rotated Q local finite element spaces wit midpoint-value and integral-value degrees of freedom by S P,n (T ) = Span{ψP i,t : i =,, 3, 4}, (.3) S I,n (T ) = Span{ψI i,t : i =,, 3, 4}. (.4) Direct comparison sows tat tese local basis function ψj,t P, and ψi j,t, j =,, 3, 4 are different. Neverteless, on eac non-interface element T T n, it can be easily verified tat S P,n (T ) = SI,n (T ) = Π T. From now on, we use te uniform notation S n (T ) to represent te local nonconforming rotated Q finite element space on an element T..3 Nonconforming IFE Space S P (Ω) In an IFE metod, we modify te local basis functions and local IFE spaces on interface elements to accomodate te interface jump conditions. In tis section, we construct IFE basis functions wose degrees of freedom are determined by midpoint values on eac interface element T T i, and ten use tem to form local and global IFE spaces. We exemplify te procedure of constructing an IFE function wit a typical interface element T T i wose geometrical configuration is specified in (.3) - (.4). Based on te Hypoteses (H) and (H), we assume tat an interface curve Γ intersects T at two different points D, E, and te segment DE separates T into two sub-elements T + and T. We call an element T a Type I interface element if Γ intersects wit T at two adjacent edges, or a Type II interface element if Γ intersects wit T at two opposite edges. For te degrees of freedom determined by midpoint values, tere are tree geometric configurations of te midpoints of a Type I interface element: Case : all te midpoints M i, i =,, 3, 4 are on one side of DE. Case : tree midpoints are on one side of DE, and one midpoint is on te oter side. Case 3: two midpoints are on one side of DE, and two midpoints are on te oter side.

33 Xu Zang Capter. Nonconforming IFE Spaces Similarly, tere are two geometric configurations of te midpoints of a Type II interface element: Case : tree midpoints are on one side of DE, and one midpoint is on te oter side. Case : two midpoints are on one side of DE, and two midpoints are on te oter side. See Figures. and Figure.3 for illustrations of interface elements of different types and cases. Figure.: Type I interface rectangles: Case,,3 (midpoint-value degrees of freedom). A 3 A 4 M 3 A 3 A 4 M 3 E A 3 A 4 M 3 M 4 E T + Γ T M M 4 E T + T Γ M M 4 T + T Γ M A D A A D A A D A M M M Figure.3: Type II interface rectangles: Case, (midpoint-value degrees of freedom). A 3 E A 4 M 3 A 3 E M 3 A 4 T T M 4 T + Γ M M 4 T + Γ M A D M A A M D A IFE basis functions are constructed via a similar approac for interface elements wit different types and cases. In te following discussion we exemplify te approac for a Type I Case interface element. We assume tat te interface points D and E in tis case are specified as follows D = (d, 0) t, E = (0, e) t,

34 Xu Zang Capter. Nonconforming IFE Spaces were / d, and 0 < e /. We define an IFE function φ P T rotated Q polynomial: to be a piece-wise φ P T (x, y) = { φ P,+ T (x, y) = v ψ,t P + c ψ,t P + c 3ψ3,T P + c 4ψ4,T P, if (x, y)t T +, φ P, T (x, y) = c ψ,t P + v ψ,t P + v 3ψ3,T P + v 4ψ4,T P, if (x, y)t T. (.5) Here v i, i =,, 3, 4 are point values at midpoints M i, i =,, 3, 4. Te function ψi,t P, i =,, 3, 4 are nonconforming rotated Q finite element local basis functions defined in (.8). Te coefficients c i, i =,, 3, 4 are determined by imposing te following interface jump conditions to interpret (.4) and (.5). We enforce te continuity of φ P T troug two intersection points D and E: te second order terms in φ P T : te flux in te following sense: [φ P T (x D, y D )] = 0, [φ P T (x E, y E )] = 0, (.6) DE [ ] φ P T = 0, (.7) x [β φ P T (x, y) n DE ]ds = 0. (.8) We will sow tat tese conditions are linearly independent so tat tey can uniquely determine a nonconforming rotated Q local IFE function φ P T on an interface element T. Note tat, to maintain te continuity across DE, instead of (.7), it seems to be more natural to impose te following condition: [ ( φ P xd + x E T, y )] D + y E = 0, (.9) because [ φ P T ] DE is a quadratic polynomial wic can usually be determined by its values at tree points. However, wen te slope of DE is in a Type I Case interface element, conditions (.6) and (.9) lose teir linear independence. On te oter and, conditions (.6) and (.7) are always linearly independent. In addition, wen te slope of DE is not, conditions (.6) and (.7) are equivalent to (.6) and (.9). Tese observations suggest us to use (.7) instead of (.9). Equations (.6), (.7), and (.8) lead to te following algebraic system M c C = M v V to solve coefficients c i, i =,, 3, 4, i.e., (3/4) + d d /4 d /4 d + d (3/4) + d d c (3/4) e + e /4 e + e /4 e (3/4) e + e c / / (/ ) / c 3 = β d(d e) β + d β + de β + d(d e) c 4

35 Xu Zang Capter. Nonconforming IFE Spaces 3 (3/4) + d d /4 d /4 d + d (3/4) + d d (3/4) e + e /4 e + e /4 e (3/4) e + e / / (/ ) / β + d(d e) β d β de β d(d e) v v v 3 v 4. (.0) Note tat for all numbers d, e, satisfying 0 < e / d <, te determinant of M c is strictly negative, i.e., det(m c ) = d ( β (d )(d e)e + β + (d ( 4e) + e + de( + e)) ) < 0. (.) Tus, te IFE functions φ P T is uniquely determined by its midpoint values v i, i =,, 3, 4. To form IFE basis functions, we let V = V i = (v,, v 4 ) t R 4 be te i-t canonical unit vector suc tat v i = and v j = 0 if j i. Ten we solve for C i = (c,, c 4 ) t from (.0) and apply its values in (.5) to form te i-t nonconforming rotated Q local IFE basis function φ P i,t. A nonconforming rotated Q local FE basis function and te corresponding IFE basis functions on Type I and Type II interface elements are illustrated in Figure.4. Note tat te IFE basis functions are made by piece-wise polynomials and satisfying interface jump conditions (.6) - (.8). Figure.4: Nonconforming rotated Q FE (left), Type I (middle) and Type II (rigt) IFE local basis functions wit midpoint-value degrees of freedom. We define te nonconforming rotated Q local IFE space on an interface element T T i to be S P,i (T ) = Span{φP i,t : i =,, 3, 4}. (.) Te corresponding global IFE space is defined by enforcing te continuity troug interior midpoints { S P (Ω) = v L (Ω) : v T S(T n ) if T T n, v T S P,i i (T ) if T T ; } if T T = b wose midpoint is M b, ten v T (M b ) = v T (M b ). (.3)

36 Xu Zang Capter. Nonconforming IFE Spaces 4 Eac global IFE function φ P i is associated wit an edge b i E. Te support of a global IFE basis function is one or two elements depending on weter te associated edge is a boundary edge or an interior edge. Note tat a global nonconforming rotated Q IFE basis function is usually supported on a smaller region compared wit te global bilinear IFE basis function wose support is four elements tat saring a common vertex [70]. A comparison of a nonconforming rotated Q global FE and IFE basis functions associated wit an interior edge is provided in Figure.5. We note tat a global IFE basis function is continuous at te midpoint on te associated edge but not continuous trougout te entire edge as te standard FE basis function. Figure.5: Nonconforming rotated Q FE (left) and IFE (rigt) global basis functions wit midpoint-value degrees of freedom..4 Nonconforming IFE Space S I (Ω) In tis section, we construct te nonconforming rotated Q IFE function φ I T on eac interface element T T i using degrees of freedom determined by integral values over edges, and ten form te corresponding IFE space S I(Ω). Likewise, tere are two types of interface elements depending on te location of interface points D and E. However, it is unnecessary to subdivide eac type of interface elements into different cases as we did for IFE functions wit midpoint-value degrees of freedom in Section.3, because it does not matter ow many midpoints are in eac piece of an interface element if te degrees of freedom are determined by edge integral values. As before, we denote an element as Type I interface element if Γ intersects wit it at two adjacent edges, and Type II interface element if Γ intersects wit it at two opposite edges. See Figure.6 for an illustration of te different types of interface elements.

37 Xu Zang Capter. Nonconforming IFE Spaces 5 Figure.6: Type I and Type II interface rectangles (integral-value degrees of freedom). A 3 A 4 b 3 b 3 A 3 E A 4 E b 4 Γ T b b 4 T + Γ T b T + A D A b A D b A We use Type II interface element as an example to demonstrate te construction of its local IFE space. Assume tat te interface points D and E are D = (d, 0) t, E = (e, ) t, wit 0 < d < and 0 < e <. Te IFE function φ I T is defined as follows: { φ I,+ φ I T T (x, y) = (x, y) = c ψ,t I + c ψ,t I + c 3ψ3,T I + v 4ψ4,T I, if (x, y)t T +, φ I, T (x, y) = c 4ψ,T I + v ψ,t I + c 5ψ3,T I + c 6ψ4,T I, if (x, y)t T. (.4) Here v i, i =, 4 are average values over edges b i, i =, 4. Coefficients c i, i =,, 6 are determined by te following conditions: average values v i, i =, 3: te continuity at two intersection points: φ I T (x, y)ds = v i, i =, 3, (.5) b i b i [φ I T (x D, y D )] = 0, [φ I T (x E, y E )] = 0, (.6) te second derivative continuity: [ φ I T x ] = 0, (.7) te weak flux jump continuity: DE [β φ I T (x, y) n DE ]ds = 0. (.8)

38 Xu Zang Capter. Nonconforming IFE Spaces 6 Combining (.5) - (.8) leads to te following algebraic system M c C = M v V, (.9) were C = (c, c,, c 6 ) t, V = (v,, v 4 ) t, M c = (Mc, Mc ), and d(3 + 3d d ) d( d + d ) d( 3d + d ) e( 3e + e ) e( e + e ) e(3 + 3e e ) Mc = 3( d + d ) + d 6d 6d + 6d 6e + 6e + e 6e 3( e + e ), / (/ ) / β + (3 5d e) β + ( + 3d + 3e) β + (3 d 5e) and Mc = 4 3d 3d + d 3 d( 3d + d ) d(3 5d + d ) e( 3e + e ) 4 3e 3e + e 3 e(3 5e + e ) 3 + 6d 6d + 6d 6d 3 0d + 6d + 6e 6e 3 + 6e 6e 3 0e + 6e (/ ) (/ ) / β ( 3 + 5d + e) β ( 3 + d + 5e) β (5 3d 3e) M v = 4 d( d + d ) 0 d(3 5d + d ) 0 e( e + e ) 4 e(3 5e + e ) 0 + d 6d 0 3 0d + 6d 0 + e 6e 0 3 0e + 6e 0 (/ ) 0 / 0 β ( + 3d + 3e) 0 β + ( 5 + 3d + 3e) Note tat for 0 < d < and 0 < e <, te determinant of M c satisfies det(m c ) = 3 ( β + (4 5d + 0d 5d 3 5e de d e + 0e de 5e 3 ) ) +β (5d 6d + 5d 3 + 5e 6de + d e 6e + de + 5e 3 ) > 0. (.30)., Tus, te IFE function φ I T is uniquely determined by its average values v i over edges b i, i =,, 3, 4. To form te local IFE basis function φ I i,t, we let V = V i = (v,, v 4 ) t R 4 be te i-t canonical unit vector suc tat v i = and v j = 0 for j i. Ten we solve for C i = (c,, c 6 ) t in (.9) and use tese values in (.4) to form φ I i,t. See Figure.7 for an illustration of nonconforming rotated Q local FE and IFE basis functions. Denote te nonconforming rotated Q local IFE space to be S I,i (T ) = Span{φI j,t : j =,, 3, 4}. (.3)

39 Xu Zang Capter. Nonconforming IFE Spaces 7 Figure.7: Nonconforming rotated Q FE (left), Type I (middle) and Type II (rigt) IFE local basis functions wit integral-value degrees of freedom. Te global IFE space wit integral-value degrees of freedom is defined as follows { S(Ω) I = v L (Ω) : v T S(T n ) if T T n, v T S I,i i (T ) if T T ; } if T T = b, ten v T ds = v T ds. (.3) b Eac global IFE function φ I i is associated wit an interior or boundary edge b i E. A comparison of te nonconforming rotated Q global FE and IFE basis functions associated wit an interior edge is illustrated in Figure.8. We can observe tat a global IFE basis function is discontinuous across te common edge of te two elements in its support. However, te average values of a global IFE basis on te common edge calculated from eiter of tese two elements in its support are te same. Remark.. It can be easily sown tat for every interface element, te nonconforming rotated Q local IFE spaces wit midpoint-value and integral-value degrees of freedom coincide, i.e., S P,i (T ) = SI,i(T ), T T i. (.33) From now on, we use te uniform notation S i (T ) to denote te local IFE space on an interface element T. Remark.. Te nonconforming rotated Q global IFE spaces wit midpoint-value and integral-value degrees of freedom differ, i.e., b S P (Ω) S I (Ω). (.34) To see (.34), Assume φ I S I (Ω) is a global IFE basis function associated wit an interior edge, as illustrated on te rigt plot in Figure.8. It can be easily observed tat φ I is not zero at te midpoint on rigt side edge of te element on te rigt. On te oter and, tis IFE function is zero outside tese two elements. Tis means φ I is not continuous at te midpoint mentioned above; ence, it is not in S P (Ω). Terefore, (.34) is true.

40 Xu Zang Capter. Nonconforming IFE Spaces 8 Figure.8: Nonconforming rotated Q FE (left) and IFE (rigt) global basis functions wit integral-value degrees of freedom.

41 Capter 3 Properties of IFE Spaces In tis capter, we investigate properties of nonconforming rotated Q IFE spaces S P (Ω) and S I (Ω). In Section 3., we sow tat tese IFE functions inerit a few desirable features from te standard nonconforming rotated Q finite element spaces suc as unisolvency, continuity witin an element, partition of unity and so fort. We derive trace inequalities and inverse inequalities for IFE functions wic play important roles in error estimation in Capter 4. In Section 3., we discuss te approximation capabilities of tese IFE spaces by analyzing te corresponding IFE interpolation errors. In Section 3.3, we use numerical examples to confirm our error analysis of IFE interpolations. 3. Fundamental Properties In tis section, we present several fundamental and useful properties for local IFE spaces S i (T ) and global IFE spaces SP (Ω) and SI (Ω). First, as we mentioned in Section.3 and.4, a nonconforming rotated Q IFE function possesses te unisolvent property. Lemma 3.. (Unisolvency) On eac interface element T T i, an IFE function φp T S i (T ) (resp. φi T Si (T )) can be uniquely determined by its midpoint values on edges (resp. average values over edges) and interface jump conditions. Proof. It can be verified straigtforwardly tat te coefficient matrix M c in (.) for te midpoint-value degrees of freedom or (.30) for te integral-value degrees of freedom is non-singular for arbitrary configuration of te diffusion coefficient β and interface location, reflected by d and e. Tus, an IFE function φ P T Si (T ) (resp. φi T Si (T )) can be uniquely determined by its midpoint values (resp. average values) and interface jump conditions. 9

42 Xu Zang Capter 3. Properties of IFE Spaces 30 Te following lemma sows tat an IFE function is continuous witin eac interface element. Lemma 3.. (Continuity) On eac interface element T T i, te local IFE space Si (T ) is a subspace of C(T ). Proof. Since eac function φ T S i (T ) is a piece-wise polynomial on T + and T, ten it suffices to sow tat φ T is continuous across te straigt line DE. For eac function φ T S i (T ), using continuity of te second order derivative condition (.7) or (.7) we can write it as follows: { φ + T φ T (x, y) = (x, y) = a+ + b + x + c + y + d(x y ), if (x, y) t T +, φ T (x, y) = a + b x + c y + d(x y ), if (x, y) t T, were a +, a, b +, b, c +, c and d are coefficients. Terefore, te jump of te function φ T is a linear function as follows [φ T (x, y)] = φ + T (x, y) φ T (x, y) = (a+ a ) + (b + b )x + (c + c )y. By definition, te function [φ T ] vanises at two different points D and E; ence, it vanises on te wole line segment DE. Tus φ T is continuous across DE. Te next lemma sows tat te nonconforming IFE functions inerit te partition of unity property from standard finite element functions. Lemma 3.3. (Partition of Unity) On eac interface element T T i, IFE basis functions φ P i,t and φi i,t satisfy te partition of unity property, i.e., and 4 φ P i,t (x, y) =, (x, y) t T, (3.a) i= 4 φ I i,t (x, y) =, (x, y) t T. (3.b) i= Proof. We prove (3.a) for te Type I Case interface elements, and te oter types and cases can be verified similarly. We define an IFE function φ P T to be te sum of four local IFE basis functions, i.e., 4 φ P T (x, y) = φ P i,t (x, y). i= Ten function φ P T can be determined by letting V = (,,, )t in (.0), solving for C and plugging tese values to (.5). Direct calculation leads to C = (,,, ) t. Hence, we obtain te two pieces φ P,+ T and φ P, T of φ P T satisfying φ P,+ T (x, y) = φ P, T (x, y) = 4 ψi,t P (x, y). i=

43 Xu Zang Capter 3. Properties of IFE Spaces 3 Note tat te standard nonconforming finite element basis functions ψi,t P possess te partition of unity wic can be verified by summing up te four basis functions given in (.9a) - (.9d); ence, φ P T (x, y) =. Te result (3.b) can be verified similarly. One of te important features of nonconforming rotated Q IFE functions is tat tey can weakly preserve te flux continuity across te actual interface curve Γ altoug te flux continuity is enforced on te line segment DE. Lemma 3.4. (Flux continuity on Γ) Let T be an interface element. function φ T S i (T ), we ave were n is te normal vector of Γ. Γ T For every IFE [β φ T n]ds = 0, (3.) Proof. Let T be te region enclosed by Γ T wit DE. Ten we ave [ div (β φ T )] dxdy T )) = dxdy = = 0. ( ( div β + φ + T T ( ( β + φ + T x T Te last equality is due to )) ( ( div β φ T + φ + )) ( ( T β φ T y x + φ T y )) dxdy φ s T x + φ s T y = 0, s = +,, for every function φ T S i (T ). By divergence teorem, we obtain [β φ T n]ds = [β φ T n]ds [β φ T n]ds = [ div (β φ T )] dxdy = 0. Γ T T DE T Te following teorem states tat te nonconforming rotated Q IFE functions are consistent wit standard FE functions.

44 Xu Zang Capter 3. Properties of IFE Spaces 3 Teorem 3.. (Consistency) IFE basis functions on eac interface element T T i consistent wit te finite element basis functions in te sense tat are. if te coefficient as no discontinuity, i.e., β + = β, te IFE basis functions φ P i,t and φ I i,t become te standard FE basis functions ψp i,t and ψi i,t, respectively.. if min{ T +, T } srinks to zero, ten IFE basis functions φ P i,t and φi i,t become te standard FE basis functions ψi,t P and ψi i,t, respectively. Here T s denotes te area of T s, s = +,. Proof. We prove te consistency of te IFE basis functions φ P i,t only, and te consistency of IFE basis functions φ I i,t can be sown similarly. For te first property, if we let β + = β in (.0) for te Type I Case interface elements, ten by straigtforward calculation we obtain M c = M v. Solve te linear system (.0) we obtain C = V in tese equations, and tis leads to φ P,+ i,t = φp, i,t = ψp i,t, i =,, 3, 4. Te first property follows from tis result and te definition of φ P i,t. Te consistency of interface elements of oter types and cases can be verified similarly. For te second property, witout loss of generality, we assume tat te sub-element T + srinks to zero. We prove te consistency for Type I Case interface elements, and te oter types and cases can be sown in a similar argument. Note tat T + 0 implies e 0. Letting e = 0 in (.0), we obtain c = v ; ence, φ P T = φ P, T = 4 v i ψi,t P. By coosing suitable values for v i, we ave φ P i,t = ψp i,t. i= Remark 3.. Te first consistency property in Teorem 3. indicates tat if β + = β, ten IFE spaces S P (Ω) and SI (Ω) become standard FE spaces. Corresponding IFE metods, wic will be introduced in Section 4., become standard finite element metods. Remark 3.. Te second consistency property in Teorem 3. is critical wen applying IFE functions to solve moving interface problems, wic will be discussed in Capter 6, because te consistency implies tat once an interface curve moves out of an element, te IFE functions on tat element will continuously become te standard finite element functions. Te following teorem is about bounds on te rotated Q IFE basis functions. Teorem 3.. (Bound of IFE basis functions) Tere exists a constant C, independent of interface location but depending on te diffusion coefficient, suc tat for i =,, 3, 4, and k = 0,,, te IFE basis functions on every interface element T ave te following bounds: φ P i,t k,,t C k, (3.3a)

45 Xu Zang Capter 3. Properties of IFE Spaces 33 and φ I i,t k,,t C k. (3.3b) Proof. We prove (3.3a) only, and (3.3b) can be verified using similar arguments. For te Type I Case interface elements, note tat te coefficients c i of te IFE function φ P T can be calculated explicitly from (.0) as follows were c = c = c 3 = c 4 = ( (β β + )(v + v 4 )d e + (β β + )(v 3 v 4 )de W I, +β + v d + (β + β )(v + v 4 )de + ((β + β )(v 3 v 4 ) + β + v )e ), ( (β β + )(v + v v 4 )d e + (β β + )(v 3 v 4 )de W I, +β + v d + (β + β )(v + v 4 )de + ((β + β )(v 3 v 4 ) + β + v )e ), ( (β β + )(v v + v 3 v 4 )d e + (β β + )(v 3 v 4 )de W I, +β + v d + (β + β )(v + v 4 )de + ((β + β )(v 3 v 4 ) + β + v )e ), ( (β β + )(v v + v 4 )d e + (β β + )(v 3 v 4 )de W I, +β + v d + (β + β )(v + v 4 )de + ((β + β )(v 3 v 4 ) + β + v )e ), W I, = (β ( + d)(d e)e + β + (d ( 4e) + e + de( + e))) > 0. Note tat { β + W I, β + (d ( 4e) + e (d ( 4e)) β + d 4 + de( + e)) β+, if 0 < e, 4 β + (e + de( + e)) 3 8 β+, if e. 4 Tus tere exists a constant C depends only on β s, s = +,, and midpoint values v i, i =,, 3, 4, suc tat c i C. Note tat for standard nonconforming FE basis ψi,t P, it can be verified directly from (.9a) - (.9d) tat ψi,t P k,,t C k, k = 0,,. Te bound of IFE basis functions (3.3a) follows from tat te IFE basis function φ P i,t are linear combination of ψi,t P wit coefficient v i, and c i. Interface elements of oter types and cases can be verified by similar arguments.

46 Xu Zang Capter 3. Properties of IFE Spaces 34 Trace Inequalities Trace inequalities are important for te finite element analysis, especially for te estimation on te boundary terms. It is well-known [5] tat te following trace inequalities old on eac non-interface element T T n : Tere exists a constant C suc tat v 0,b C b / T / ( v 0,T + T v 0,T ), v H (T ), (3.4) v n 0,b C b / T / ( v 0,T + T v 0,T ), v H (T ). (3.5) Here b T is an edge of T and T is te diameter of T. b and T denotes te lengt of b and area of T, respectively. We note from Lemma 3. tat on eac T T i, te IFE space Si (T ) H (T ); ence te trace inequality (3.4) is true for IFE functions in S i (T ). However, Si (T ) H (T ) since an IFE function φ T S i (T ) is not C ; ence, te result (3.5) cannot be applied to IFE function v S i (T ) in general. In te following discussion, we derive a trace inequality similar to (3.5) for IFE functions. Let φ T S i (T ) be a rotated Q IFE function defined on T = A A A 3 A 4 wose vertices are given by A = (x 0, y 0 ) t, A = (x 0 +, y 0 ) t, A 3 = (x 0, y 0 + ) t, A 4 = (x 0 +, y 0 + ) t. Ten, we can write φ T in te following form: φ T (x, y) = { c + c (x x 0 ) + c 3 (y y 0 ) + c 4 ((x x 0 ) (y y 0 ) ), in T, c + + c + (x x 0 ) + c + 3 (y y 0 ) + c + 4 ((x x 0 ) (y y 0 ) ), in T +. (3.6) To simplify te notations, we only consider te case x 0 = y 0 = 0 and a general interface element can be mapped to tis element via translation. Tus (3.6) becomes φ T (x, y) = { φ T (x, y) = c + c x + c 3 y + c 4 (x y ), if (x, y) T, φ + T (x, y) = c+ + c + x + c + 3 y + c + 4 (x y ), if (x, y) T +. (3.7) Witout loss of generality, we assume tat te interface points D, E satisfy D = (d, 0) t A A, E = (0, e) t A A 3, for Type I interface element wit 0 < d, 0 < e, and D = (d, 0) t A A, E = (e, ) t A 3 A 4, for Type II interface element wit 0 < d <, 0 < e <. First we derive a bound for te coefficients c s j, j =,, 3, 4, s = +, of an IFE function φ T defined in (3.7).

47 Xu Zang Capter 3. Properties of IFE Spaces 35 Lemma 3.5. Tere exists a constant C >, depending on te diffusion coefficient β but independent of interface location D, E and te element size, suc tat for every IFE function φ T defined by (3.7) on any interface element T, we ave and ( c + C + c c + 4 ) c + c 3 + c 4 C ( c + + c c + 4 ), (3.8) c + C ( c + c + c 3 + c 4 ), c C ( c + + c + + c c + 4 ). (3.9) Proof. Since φ T S i (T ), ten te two pieces φ T and φ+ T satisfy te interface jump conditions (.6) - (.8), or (.6) - (.8). By direct calculation, we ave c (β β + )de (β β + )d e 0 β (d +e ) β (d +e ) c + c β c 3 = 0 d +β + e (β + β )de 0 β (d +e ) β (d +e ) c + (β 0 + β )de β + d +β e 0 c c β (d +e ) β (d +e ) + 3, (3.0) 4 c for Type I interface element, and c (β β + )d (β β + )d(d e) β (+(d e) ) β (+(d e) ) c β c 3 = 0 (d e) +β + (β + β )(d e) β (+(d e) ) β (+(d e) ) (β 0 + β )(d e) β +β + (d e) c β (+(d e) ) β (+(d e) ) (β β + )de β (+(d e) ) (β + β )e β (+(d e) ) (β + β )(d e)e β (+(d e) ) c + c + c + 3 c + 4, (3.) for Type II interface element. It is easy to ceck tat every entry in te coefficient matrices in (3.0) and (3.), regardless of, can be bounded above by a constant C independent of d, e, wit te assumption d, e [0, ]. Ten using (3.0) or (3.) we can obtain te second parts of te estimates in (3.8) and (3.9). Te first parts of te estimates (3.8) and (3.9) can be proved similarly. To extend te trace inequality (3.5) to an interface element T, we first work on a sub-element T T. We partition T into four congruent squares T i, i =,, 3, 4, by connecting two pairs of midpoints on opposite edges of T. It is easy to verify tat, for eiter Type I or Type II interface element, tere exists a sub-element T i wic is completely inside of eiter T or T +, and we denote tis sub-element by T. Te following lemma provides an estimate of φ T on T. Lemma 3.6. Tere exists a constant C independent of interface location and te element size T suc tat for every IFE function φ T defined by (3.7) on any interface element T, we ave T T φ T 0, T C ( T (c s ) + (c s 3) + T (c s 4) ), s = +,, (3.)

48 Xu Zang Capter 3. Properties of IFE Spaces 36 were T is a sub-element defined above. Proof. Witout loss of generality, we assume T = T 4 wic is te upper rigt sub-element and T T. By Young s inequality, tere exists a positive σ suc tat x φ T 0, T = ( ) T 3(c ) + 9c c 4 T + 7(c 4 ) T T Similarly, tere exists a positive σ suc tat (( 3 9 ) ( (c ) + 7 9σ ) ) (c 4 ) T. σ y φ T 0, T = ( ) T 3(c 3 ) 9c 3 c 4 T + 7(c 4 ) T T (( 3 9 ) ( (c 3 ) + 7 9σ ) ) (c 4 ) T. σ Let σ i ( 3, 4 ), i =,, ten by direct calculation we obtain 9 φ T 0, T ( C T (c ) + (c 3 ) + T (c 4 ) ). (3.3) Applying te coefficients equivalence result (3.8) in Lemma 3.5 to te above inequality, we obtain φ T 0, T ( C T (c + ) + (c + 3 ) + T (c + 4 ) ). (3.4) Note tat T = T 4, ten (3.) follows from (3.3) and (3.4). Now we are ready to establis te trace inequality for IFE functions on an interface element T. Teorem 3.3. (Trace Inequality) Tere exists a constant C depending only on te diffusion coefficient β suc tat for every nonconforming rotated Q IFE function φ T on any interface element T, we ave β φ T n 0,b C / T T / β φ T 0,T, (3.5) were b T is an interface edge of T, and n is te unit outer normal of T. Proof. Witout loss of generality, we consider an interface element T wose vertices are given by (.3); ence, φ T can be written as (3.7). Let b = A A 3, wit b + = A E, and b = EA 3.

49 Xu Zang Capter 3. Properties of IFE Spaces 37 By direct calculations, we obtain φ + T,x 0,b + = (c+ ) e T (c + ) T, φ + T,y 0,b = + (c+ 3 ) e T c + 3 c + 4 e T (c+ 4 ) e 3 3 T 7 3 φ T,x 0,b = (c ) ( e) T (c ) T, ( (c + 3 ) + (c + 4 T ) ) T, φ T,y 0,b = (c 3 ) ( e) T c 3 c 4 ( e ) T (c 4 ) ( e 3 ) 3 T 7 3 ( (c 3 ) + (c 4 T ) ) T. Applying te estimate (3.) to te above inequalities, we obtain Hence, φ T,p 0,b C T s T φ T, p = x, y, s = +,. 0, T β φ T n 0,b β max ( φ T,x 0,b + φ T,y 0,b) C β max β min 4 T T β φ T 0, T C T T β φ T 0,T, (3.6) were β max = max{β, β + } and β min = min{β, β + }. Te estimate in (3.5) follows from taking square root of bot sides in (3.6). Inverse Inequalities Anoter important property of finite element functions is te inverse inequalities [8]: Tere exists a constant C suc tat for all v S (T ) v l,p,t C m l+ n p n q v m,q,t, (3.7) were S (T ) is a finite element subspace of W l,p (T ) W m,q (T ) wit p, q, and 0 m l. In (3.7), n is te dimension of T, i.e., T R n and is te diameter of T. Te above inverse estimates are true for finite element functions wic are uniform polynomials in eac element. However, te IFE functions are piece-wise polynomials in interface elements; ence, te inverse inequality (3.7) cannot be applied for IFE functions directly. Te following teorem establises inverse inequalities for IFE functions.

50 Xu Zang Capter 3. Properties of IFE Spaces 38 Teorem 3.4. (Inverse Inequalities) Tere exists a constant C depending only on te diffusion coefficient β suc tat every nonconforming rotated Q IFE function φ T on any interface element T satisfies φ T k,,t C T φ T k,t, (3.8) and φ T k,t C l k T φ T l,t (3.9) for all 0 l k. Proof. For (3.8), we prove te case k =, te oter cases can be verified similarly. Consider te typical interface element T = A A A 3 A 4 wose vertices are given in (.3). By te coefficient equivalent property (3.8) in Lemma 3.5 and (3.) in Lemma 3.6, we obtain, φ T,,T = max{ φ T,,T, φ T,,T +} max{ c + c c 4, c + + c c + 4 } C( c + c 3 + c 4 ) C T φ T 0, T C T φ T,T. For (3.9), we prove te case k =, l = 0, and te oter cases can be verified similarly. Using te coefficient equivalent property (3.8) again, we obtain φ T,T = φ T,T + φ + T,T = (c + + c + 4 x) + (c + 3 c + 4 y) dxdy + (c + c 4 x) + (c 3 c 4 y) dxdy T + T = (c + + c + 4 x) + (c + 3 c + 4 y) + (c + c 4 x) + (c 3 c 4 y) dxdy T ( = (c + ) + (c + 3 ) + c + c + 4 c + 3 c ) 3 (c + 4 ) + ( (c ) + (c 3 ) + c c 4 c 3 c (c 4 ) ) C ( (c ) + (c 3 ) + (c 4 ) ). C φ T, T. (3.0) On te oter and, we note tat φ T on T is a uniform polynomial; ence te standard inverse inequality (3.7) can be applied on T to ave Combining te estimates (3.0) and (3.), we obtain φ T, T C φ T 0, T C φ T 0,T. (3.) φ T,T C φ T 0,T.

51 Xu Zang Capter 3. Properties of IFE Spaces 39 Te trace inequality (3.5) and inverse inequalities (3.8), (3.9) for IFE functions play important roles in te error estimation of related IFE metods, wic we will discuss in Capter 4. Remark 3.3. Similar trace inequalities and inverse inequalities for linear and bilinear IFE functions can be proved in a similar approac. We refer a recent article [03] for more details. 3. Approximation Capabilities In tis section, we discuss te approximation capabilities of nonconforming rotated Q IFE spaces S P (Ω) and SI (Ω). Recall tat on eac non-interface element T T n, te local interpolation can be defined in two ways based on two different types of finite element basis {ψi,t P }4 i= and {ψi,t I }4 i=. Te first type is associated wit midpoint-value degrees of freedom and te local interpolation operator I,T P : H (T ) S n (T ) is defined as follows: I P,T u = 4 u(m i )ψi,t P, (3.) i= were M i, i =,, 3, 4 are midpoints of edges of T. Using te local basis of integral-value degrees of freedom, te local interpolation operator I,T I : H (T ) S n (T ) is defined as follows: 4 ( ) I,T I u = uds ψi,t I. (3.3) b i b i i= Error estimation for bot of tese interpolations are analyzed in [3] using te standard scaling argument: I P,T u u 0,T + I P,T u u,t C u,t, T T n, (3.4a) and I I,T u u 0,T + I I,T u u,t C u,t, T T n. (3.4b) Te interpolations operators I,T P and II,T can be defined similarly on an interface element T T i by replacing te FE local basis functions wit te IFE local basis functions. In te following subsections, we focus on te error analysis for I,T P u u k,t and I,T I u u k,t on an interface element T T i. Te main ideas of our analysis are Using multi-point Taylor expansion tecniques to derive a uniform bound for I P,T u u on eac interface element T I P,T u u k,t C k u,t, k = 0,,

52 Xu Zang Capter 3. Properties of IFE Spaces 40 were C is independent of te interface location. Deriving a uniform error bound for te difference of te two interpolations I P,T u I I,T u k,t C k u,t, k = 0,, ten we apply te triangular inequality and te two estimates above to obtain a uniform error bound for I I,T u u: I I,T u u k,t C k u,t, k = 0,. 3.. Error Analysis of Interpolation on S P (Ω) We use te multi-point Taylor expansion idea [8, 35] to analyze te interpolation error I,T P u u. We note tat tis idea as been used for error analysis of te linear [95] and bilinear IFE interpolations [70]. On eac interface element T T i, we assume tat te interface Γ intersects T at two points, denoted by D and E. We use te line segment DE to approximate te actual interface Γ T. Te segment DE separates T into two sub-elements, denoted by T + and T were T = T + T DE. Note tat tere is a small region enclosed by Γ T and DE, denoted by T were T = (Ω + T ) (Ω T + ). Ten, an interface element T is subdivided into up to four pieces, i.e., T = ( Ω + T +) ( Ω T ) ( Ω + T ) ( Ω T +). (3.5) For every interface element T T i, we define te following space P H m (T ) = {u : u Ω s T H m (Ω s T ), s = +, }. Te equipped norm for every function u P H m (T ) is defined as follows and te semi-norm is defined by u m,t = u m,ω T + u m,ω + T, u m,t = u m,ω T + u m,ω + T. Ten we define te following spaces on an interface element T T i te interface jump conditions (.4) and (.5). wose functions satisfy P H m int(t ) = {u C(T ), u Ω s T H m (Ω s T ), s = +,, [β u n Γ ] = 0 on Γ T }, P C m int(t ) = {u C(T ), u Ω s T C m (Ω s T ), s = +,, [β u n Γ ] = 0 on Γ T }.

53 Xu Zang Capter 3. Properties of IFE Spaces 4 Similarly, we define te following spaces on te wole domain Ω: P H m int(ω) = {u C(Ω), u Ω s H m (Ω s ), s = +,, [β u n Γ ] = 0 on Γ}, P C m int(ω) = {u C(Ω), u Ω s C m (Ω s ), s = +,, [β u n Γ ] = 0 on Γ}. Here m. Clearly, we ave te inclusion P C int(t ) P H int(t ), for every T T i. On an interface element T, we consider te interpolation I,T P : P H int(t ) S i (T ) defined by 4 I,T P u(x) = u(m i )φ P i,t (X), (3.6) i= were φ P i,t, i =,, 3, 4, are nonconforming rotated Q local IFE basis functions wit midpoint-value degrees of freedom. We define te global IFE interpolation I P : P H int(ω) S P (Ω) as follows (I P u) T = I,T P u, T T, (3.7) were te local interpolation I,T P is defined in (3.) or (3.6) depending on weter T is a non-interface or an interface element. We assume tat te interface Γ and te partition T satisfy te following assumptions [70, 95]: (H3) Te interface Γ is defined by a piecewise C function, and te partition T is formed suc tat te subset of Γ in every interface element T T i is C. (H4) Te interface Γ is smoot enoug so tat P C 3 int(t ) is dense in P H int(t ) for every interface element T T i. Note tat te ypotesis (H4) olds if te interface is smoot enoug [4, 46]. Let ρ = β β +, and ρ = β+ β. For any point M Γ T, let M be te ortogonal projection of M onto DE. We first recall several results in [70]. Lemma 3.7. Let M be an arbitrary point on Γ T and X DE be an arbitrary point on DE. Assume n( M) = (n x ( M), n y ( M)) t is te unit normal vector of Γ at M, and n(de) = ( n x, n y ) t is te unit normal vector of DE. Ten, for any u P Cint(T ), we ave u + ( M) = N ( M) u ( M), u ( M) = N + ( M) u + ( M), (3.8) were ( N ( M) = ( N + ( M) = n y ( M) + ρn x ( M) (ρ )n x ( M)n y ( M) (ρ )n x ( M)n y ( M) n x ( M) + ρn y ( M) n y ( M) + ρn x ( M) ( ρ )n x ( M)n y ( M) ( ρ )n x ( M)n y ( M) n x ( M) + ρn y ( M) ), ).

54 Xu Zang Capter 3. Properties of IFE Spaces 4 Moreover, for any v S i (T ), we ave v + (X DE ) = N DE v (X DE ), v (X DE ) = N + DE v+ (X DE ), (3.9) were N DE = ( n y + ρ n x (ρ ) n x n y (ρ ) n x n y n x + ρ n y N + DE = ( n y + ρ n x ( ρ ) n x n y ( ρ ) n x n y n x + ρ n y ), ). Lemma 3.8. For any point M Γ T, tere exists a constant C > 0 suc tat M M C, (3.30) and N s DE N s ( M) C, s = +,. (3.3) Figure 3.: A sketc of interface rectangle: Type I, Case. M 3 A 3 A 4 X T M 4 M E X DE Γ T + M M M A M D A We use te Type I Case interface element, illustrated in Figure 3., to present our analysis and interface elements of oter types and cases can be discussed similarly. First we consider te interpolation error estimation on Ω T. Let X = (x, y) t Ω T. Assume te line segments XM i, i =, 3, 4 do not intersect wit te interface Γ and DE, wereas XM

55 Xu Zang Capter 3. Properties of IFE Spaces 43 meets Γ and DE at M, and M, respectively. One can find an illustration of te related geometry in Figure 3.. We also assume tat M = tm + ( t)x = ( x, ỹ ) t, M = tm + ( t)x = ( x, ȳ ) t, for some positives 0 t, t. Terefore, t(m M ) = ( t)( M X), t(m M ) = ( t)(m X). We define te operator by for any two points X = (x, y ) t, X = (x, y ) t. X X (x x, y y ) t, (3.3) Lemma 3.9. For any number p, r R, any vector q R, and any point X Ω T, X DE DE, tere exists an IFE function v S i (T ) suc tat and were q v(x) = p, v(x) = q, v(x) x = r, (3.33) 4 (M i X)φ P i,t (X) (3.34) i= = (N DE I)q (M M )φ P,T (X) (N DE I)q ( M X DE )φ P,T (X) rf 0 (X), F 0 (X) = 4 ((x i x) (y i y) )φ P i,t (X) i= +N (X DE DE X) (M X DE )φ P,T (X) (3.35) +((x DE x) (y DE y) )φ P,T (X) + ((x x DE ) (y y DE ) )φ P,T (X). Proof. Note tat a local IFE function v S i (T ) can be written as { v v(y ) = (Y ) = a + b x + c y + d (x y ), if Y T, v + (Y ) = a + + b + x + c + y + d + (x y ), if Y T +, for every point Y = (x, y) t T. Since te point X is in T, ten we can uniquely determine coefficients a, b, c, d by solving a small linear system defined by (3.33). Te coefficients a +, b +, c +, d + in te plus piece v + of te IFE function v, can be determined by te following interface jump conditions v + (D) = v (D), v + (E) = v (E), d + = d,

56 Xu Zang Capter 3. Properties of IFE Spaces 44 and DE β + v + (Y ) n DE ds = DE β v (Y ) n DE ds. We ten apply te Taylor expansion to express v(m i ), i =, 3, 4 at X as follows v(m i ) = v (M i ) = v (X)+ v (X) (M i X)+d ((x i x) (y i y) ), i =, 3, 4. (3.36) Next we expand v(m ) at X using te interface jump conditions (.4) - (.5) and te derivative relation (3.9) given in Lemma 3.7 as follows v(m ) = v + (M ) = v + (X DE ) + v + (X DE ) (M X DE ) + d + ((x x DE ) (y y DE ) ) = v (X DE ) + N DE v (X DE ) (M X DE ) + d ((x x DE ) (y y DE ) ) = v (X) + v (X) (X DE X) + d ((x DE x) (y DE y) ) +N DE v (X DE ) (M X DE ) + d ((x x DE ) (y y DE ) ) = v (X) + v (X) (M X) v (X) (M X DE ) +d ((x DE x) (y DE y) ) + N DE v (X DE ) (M X DE ) +d ((x x DE ) (y y DE ) ) = v (X) + v (X) (M X) + (N DE I) v (X) (M X DE ) +N DE ( v (X DE ) v (X)) (M X DE ) +d ((x DE x) (y DE y) ) + d ((x x DE ) (y y DE ) ) = v (X) + v (X) (M X) + (N DE I) v (X) (M M ) +(N DE I) v (X) ( M X DE ) + d N DE (X DE X) (M X DE ) +d ((x DE x) (y DE y) ) + d ((x x DE ) (y y DE ) ). (3.37) Here te last equality is due to v (X DE ) v (X) = ( b + d x DE c d y DE ) ( b + d x c d y ) = d (X DE X).

57 Xu Zang Capter 3. Properties of IFE Spaces 45 Applying (3.36) and (3.37) to te IFE local interpolation (3.6) yields I P,T v(x) = = 4 v(m i )φ P i,t (X) i= 4 v(x)φ P i,t (X) + i= i= 4 v(x) (M i X)φ P i,t (X) 4 +d ((x i x) (y i y) )φ P i,t (X) i= +(N DE I) v (X) (M M )φ P,T (X) +(N DE I) v (X) ( M X DE )φ P,T (X) +d N (X DE DE X) (M X DE )φ P,T (X) +d ((x DE x) (y DE y) )φ P,T (X) +d ((x x DE ) (y y DE ) )φ P,T (X). (3.38) Using te partition of unity property (3.a) in Lemma 3.3, we obtain 4 v(x)φ P i,t (X) = v(x) = I,T P v(x). (3.39) i= Applying (3.39) to te first term on te rigt and side of (3.38) yields 4 v(x) (M i X)φ P i,t (X) i= = (N DE I) v (X) (M M )φ P,T (X) (N DE I) v (X) ( M X DE )φ P,T (X) rf 0 (X). (3.40) Te identity (3.34) follows from substituting q for v(x) in (3.40). Te next lemma provides upper bounds of F 0 (X) and F 0 (X). Lemma 3.0. Tere exists a constant C > 0 suc tat F 0 (X) given in (3.35) satisfies F 0 (X) C, (3.4) and F 0 (X) C, (3.4) for every point X Ω T, and X DE DE.

58 Xu Zang Capter 3. Properties of IFE Spaces 46 Proof. We recall te upper bounds of te IFE basis functions (3.3a) in Teorem 3.. Ten by direct calculations we obtain F 0 (X) 4 ((x i x) (y i y) ) φ P i,t (X) i= + N DE X DE X M X DE φ P,T (X) + (x DE x) (y DE y) φ P,T (X) + (x x DE ) (y y DE ) φ P,T (X) 4 C ((x i x) (y i y) ) + C N X DE DE X M X DE i= +C (x DE x) (y DE y) + C (x x DE ) (y y DE ) C, wic proves (3.4). For estimate (3.4), first we note tat (3.43) = x F 0 (X) 4 4 (x i x)φ P i,t (X) + ((x i x) (y i y) )φ P i,t,x(x) i= i= +N (, DE 0)t (M X DE )φ P,T (X) + N (X DE DE X) (M X DE )φ P,T,x(X) + (x DE x)φ P,T (X) + ((x DE x) (y DE y) )φ P,T,x(X) +((x x DE ) (y y DE ) )φ P,T,x(X). Ten we use Teorem 3. again and follow similar arguments in (3.43) to obtain x F 0 (X) C. (3.44) Similarly, we can obtain te bound y F 0 (X) C. Te estimate (3.4) follows immediately. Now we are ready to derive an expansion of te IFE interpolation error. Teorem 3.5. Let T T i be an interface element, and u P C int(t ). For every point X Ω T, and X DE DE, we ave I P,T u(x) u(x) = (F + F )φ P,T (X) F 0u xx (X) + 4 I i φ P i,t (X), (3.45) i=

59 Xu Zang Capter 3. Properties of IFE Spaces 47 were F (X) = (N ( M ) N DE ) u (X) (M M ), (3.46a) F (X) = (N DE I) u (X) ( M X DE ), (3.46b) I, (X) = (N ( M ) I) I, (X) = t 0 0 d dt ( u (t M + ( t)x))(m M ) dt, ( t) d dt u(tm + ( t)x)dt, (3.46c) (3.46d) I,3 (X) = ( t) d t dt u(tm + ( t)x)dt, (3.46e) I (X) = I, (X) + I, (X) + I,3 (X), (3.46f) I i (X) = 0 ( t) d dt u(tm i + ( t)x)dt, i =, 3, 4. (3.46g) Proof. First we note tat u(m i ), i =, 3, 4 can be expanded at X as follows u(m i ) = u(x) + u(x) (M i X) + For u(m ), te expansion is derived as follows 0 ( t) d dt u(tm i + ( t)x)dt, i =, 3, 4. (3.47) u(m ) t d = u(x) + 0 dt u(tm d + ( t)x)dt + t dt u(tm + ( t)x)dt = u(x) + u (X) (M X) ( t) u ( M ) (M X) + + t 0 t ( t) d dt u (tm + ( t)x)dt + ( t) u + ( M ) (M X) ( t) d dt u+ (tm + ( t)x)dt = u(x) + u (X) (M X) + (N ( M ) I) u ( M ) (M X)( t) + t 0 ( t) d dt u (tm + ( t)x)dt + t ( t) d dt u+ (tm + ( t)x)dt = u(x) + u (X) (M X) + (N ( M ) I) u (X) (M X)( t) (N d ( M ) I) dt ( u (t M + ( t)x)) (M X)( t)dt (3.48) + t 0 0 ( t) d dt u (tm + ( t)x)dt + t ( t) d dt u+ (tm + ( t)x)dt.

60 Xu Zang Capter 3. Properties of IFE Spaces 48 Using te expansions (3.47) and (3.48) to substitute u(m i ) in (3.6), and letting p = u(x), q = u (X), r = u xx(x) in (3.34), ten we can see (3.45) follows from applying (3.34) to (3.6). Now we are ready to derive te interpolation error in L norm. Teorem 3.6. Tere exists a constant C > 0 independent of interface location suc tat for every u P H int(t ) on any interface element T T i. I P,T u u 0,Ω T C u,t, (3.49) Proof. First we assume u P Cint(T ). Ten by Teorem 3.5 we can expand I,T P u(x) u(x) as (3.45). Taking te absolute value of bot sides on (3.45) and applying triangle inequality we ave I P,T u(x) u(x) ( F + F ) φ P,T (X) + F0 u xx (X) +( I, + I, + I,3 ) φ P,T (X) 4 + I i φ P i,t (X) Note tat in Teorem 3. we ave sown tat IFE basis functions are bounded (3.3a); ence I P,T u u ( C F 0,Ω T 0,Ω T + F 0,Ω T + F 0 0,,Ω T u xx 0,Ω T i= I, 0,Ω T + I, 0,Ω T + I,3 0,Ω T + 4 I i 0,Ω T ). i= (3.50) To estimate te term involving F in (3.50), we apply (3.3) from Lemma 3.8 to obtain F 0,Ω T N ( M ) N M DE M u 0,Ω T C u,ω T. (3.5) To bound te term involving F in (3.50), we let X DE = M (3.30) from Lemma 3.8 to obtain in (3.46b), and use te estimate F 0,Ω T N I M DE M u 0,Ω T C u,ω T. (3.5) Applying te estimate of F 0 in (3.4) from Lemma 3.0, we ave F 0 0,,Ω T u xx 0,Ω T C u,ω T. (3.53)

61 Xu Zang Capter 3. Properties of IFE Spaces 49 To bound te term involving I,, we let ξ = t x + ( t)x, η = tỹ + ( t)y, and note tat d dt ( u (t M + ( t)x)) (M M ) = d ( ) u ξ (t M + ( t)x), u η (t M + ( t)x) (M dt M ) = u ξξ (ξ, η)( x x)(x x ) + u ξη (ξ, η)(ỹ y)(x x ) +u ηξ (ξ, η)( x x)(y ỹ ) + u ηη (ξ, η)(ỹ y)(y ỹ ) u ξξ (ξ, η) + u ξη (ξ, η) + u ηη (ξ, η). Terefore, by Caucy-Scwarz inequality, we ave ( ) I, C N ( M ) I u ξξ (ξ, η) + u ξη (ξ, η) + u ηη (ξ, η) dt 0 C 4 u ξξ (ξ, η) + u ξη (ξ, η) + u ηη (ξ, η) dt. (3.54) 0 Note tat te point (ξ, η) is in Ω T since te entire line segment X M is inside Ω T. Tus, we integrate (3.54) over Ω T to ave te following bound I, 0,Ω T = I Ω, dξdη T ( ) C 4 u ξξ (ξ, η) + u ξη (ξ, η) + u ηη (ξ, η) dξdη dt 0 Ω T C 4 u,ω T, (3.55) Similarly, we bound I, as follows I, ( ) t C ( t) d 0 dt u (tm + ( t)x)dt ( t ( t) ( u ξξ (ξ, η)(x x) + u ξη (ξ, η)(x x)(y y) + u ηη(ξ, η)(y y) ) dt 0 t C 4 ( t) ( u ξξ (ξ, η) + u ξη (ξ, η) + u ηη(ξ, η) ) dt. (3.56) 0 Hence, by integrating (3.56) over Ω T t I, 0,Ω T C4 ( t) 0 C 4 u,ω T ( u ξξ (ξ, η) + u ξη (ξ, η) + u ηη(ξ, η) ) dξdηdt Ω T. (3.57) )

62 Xu Zang Capter 3. Properties of IFE Spaces 50 We bound I,3 as follows I,3 ( ) ( t) d t dt u+ (tm + ( t)x)dt ( ( t) ( u + ξξ (ξ, η)(x x) + u + ξη (ξ, η)(x x)(y y) + u + ηη(ξ, η)(y y) ) dt t C 4 ( t) ( u + ξξ (ξ, η) + u + ξη (ξ, η) + u + ηη(ξ, η) ) dt. (3.58) t Hence, integrating (3.58) over Ω T yields I,3 0,Ω T C 4 ( t) ( u + ξξ (ξ, η) + u + ξη (ξ, η) + u + ηη(ξ, η) ) dtdξdη T t ) C 4 ( t) (T/Ω u + ξξ (ξ, η) + u + ξη (ξ, η) + u + ηη(ξ, η) dξdη dt C 4 u,t/ω, t ) i.e., Similarly, we can sow I,3 0,Ω T C u,t/ω C u,t. (3.59) I i 0,Ω T C u,ω T, i =, 3, 4. (3.60) Applying te above estimates (3.5), (3.5), (3.53), (3.55), (3.57), (3.59), and (3.60) to (3.50), we obtain (3.49) for u in P C int(t ). Finally, we apply te density ypotesis (H4) to obtain te estimate (3.49) for u in P H int(t ). Now we estimate te interpolation error using semi-h norm. Te following teorem gives te Taylor expansion of te first order derivatives of te interpolation error. Teorem 3.7. Let T T i be an interface element, and u P C3 int(t ). For every point X Ω T, and X DE DE, we ave and x ( I P,T u(x) u(x) ) = (F + F )φ P,T,x(X) F 0,xu xx (X) + y ( I P,T u(x) u(x) ) = (F + F )φ P,T,y(X) F 0,yu xx (X) + 4 I i φ P i,t,x(x), (3.6) i= 4 I i φ P i,t,y(x), (3.6) were F 0, F, F, and I i, i =,, 4 are defined in (3.35), (3.46a) - (3.46g), and F 0,x = x F 0, and F 0,y = y F 0. i=

63 Xu Zang Capter 3. Properties of IFE Spaces 5 Proof. We only prove (3.6), and similar arguments can be used to establis (3.6). Taking te x-derivative on bot sides of te equation (3.45), we obtain x ( I P,T u(x) u(x) ) = (F,x + F,x )φ P,T (X) + (F + F )φ P,T,x(X) F 0,xu xx (X) F 0u xxx (X) + 4 ( Ii,x φ P i,t (X) + I i φ P i,t,x(x) ). (3.63) i= Taking te x-derivative of (3.47) and (3.48) yields I,x = u x (X) (M X) ( (N ( M ) I)( u (X)) (M x M ) ), I i,x = u x (X) (M i X), i =, 3, 4. Terefore, 4 I i,x φ P i,t (X) = i= 4 u x (X) (M i X)φ P i,t (X) i= ( (N ( M ) I)( u (X)) (M x M ) ) φ P,T (X). (3.64) We let q = u x (X) = (u xx (X), u xy (X)) t, d = u xxx (X) and use tem in (3.34) from Lemma 3.9, ten we obtain 4 u x (X) (M i X)φ P i,t (X) i= = (N DE I) u x(x) (M M )φ P,T (X) (N DE I) u x(x) ( M X DE )φ P,T (X) u xxx(x)f 0 (X). (3.65) Replacing te first term on te rigt and side of (3.64) by (3.65) yields 4 I i,x φ P i,t (X) i= = (N DE I) u x(x) (M M )φ P,T (X) +(N I) u DE x(x) ( M X DE )φ P,T (X) + u xxx(x)f 0 (X) x ((N ( M ) I)( u (X)) (M M ) ) φ P,T (X). (3.66) Plugging (3.66) in (3.63) we ave ( x I P,T u(x) u(x) ) = (F,x + F,x + F 4 + F 5 + F 6,x )φ P,T (X) + (F + F )φ P,T,x(X) 4 F 0,xu xx (X) + I i φ P i,t,x(x), (3.67) i=

64 Xu Zang Capter 3. Properties of IFE Spaces 5 were F 4 = (N I) u DE x(x) (M M ), F 5 = (N I) u DE x(x) ( M X DE ), F 6,x = x ((N ( M ) I)( u (X)) (M M ) ). Note tat F,x + F,x + F 6,x ) = x ((I N DE ) u (X) (M M ) (N DE I) u (X) ( M X DE ) = (N DE I) u x (X) (X DE M ). Ten, it can be verified by direct calculations tat F,x + F,x + F 4 + F 5 + F 6,x = 0. (3.68) Plugging (3.68) in (3.67), we obtain te result (3.6). Now we can derive te semi-h norm error for interpolation function. Teorem 3.8. Tere exists a constant C > 0 independent of interface location suc tat for every u P H int(t ) on any interface element T T i. I P,T u u,ω T C u,t, (3.69) Proof. First we assume u P Cint(T 3 ). By Teorem 3.7 we can expand te first order derivatives of I,T P u u as (3.6) and (3.6). Following a similar approac in te proof of Teorem 3.6, we can derive upper bounds for F, F, and I i, i =,, 3, 4 from (3.5) - (3.60). A bound of F 0,x is given by (3.4) in te Lemma 3.0. Applying tese estimates to (3.6) and (3.6) establises (3.69) for u P Cint(T 3 ). Ten (3.69) follows for u P Hint(T ) according to te density ypotesis (H4). For estimates on Ω + T +, Ω + T and Ω T +, similar results can be obtained. For interface elements of oter types and cases, te analysis is also similar. Putting all of tese error estimates togeter leads to te following teorem. Teorem 3.9. Tere exists a constant C > 0 independent of interface location suc tat I P,T u u 0,T + I P,T u u,t C u,t, (3.70) for every u P H int(t ) on any interface element T T i.

65 Xu Zang Capter 3. Properties of IFE Spaces 53 Te global interpolation error bound, stated in te next teorem, follows from summing over te estimate (3.4a) on non-interface elements and estimate (3.70) on interface elements. Teorem 3.0. For u P H int(ω), we ave te following estimate for te interpolation error I P u u 0,Ω + I P u u,ω C u,ω. (3.7) We will provide numerical verification of te interpolation error estimates (3.7) in Section Error Analysis of Interpolation on S I (Ω) In tis subsection, we analyze interpolation error I I,T u u k,t, k = 0, on an interface element T, and ten derive a global interpolation error bound for I I u u. For an interface element T T i, we define te local interpolation II,T : P H int(t ) S i (T ) using te set of IFE basis function {φ I i,t }4 i= as follows I I,T u(x) = 4 i= ( ) u(x)ds φ I i,t (X). (3.7) b i b i Ten we define te global IFE interpolation I I : P H int(ω) S (Ω) piece-wisely by First we derive an error bound for IFE basis function φ I i,t (I I u) T = I I,T u. (3.73) in te following lemma. Lemma 3.. Tere exists a constant C > 0, independent of te interface location, suc tat φ I i,t 0,T + φ I i,t,t C, i =,, 3, 4, (3.74) for every interface element T T i. Proof. Using te point-wise bound (3.3b) of te IFE basis function stated in Teorem 3., we ave te following estimate ( φ I i,t 0,T = φ I i,t (x, y) ) dxdy φ I i,t 0,,T dxdy C. Similarly φ I i,t,t = T T φ I i,t (x, y) φ I i,t (x, y)dxdy φ I i,t,,t T T dxdy C. Here we use te fact tat te size of an element is T = O( ). Finally, te estimate (3.74) follows from combining te above estimates.

66 Xu Zang Capter 3. Properties of IFE Spaces 54 Now we are ready to derive an error bound for te interpolation error I I,T u u. Teorem 3.. Tere exists a constant C > 0 independent of interface location suc tat I I,T u u 0,T + I I,T u u,t C u,t, (3.75) for every u P H int(t ) on any interface element T T i. Proof. By triangular inequality, we ave I I,T u u k,t I I,T u I P,T u k,t + I P,T u u k,t, k = 0,, (3.76) were te notation 0,T means te regular L norm on T, i.e., 0,T. A bound of te second term on te rigt and side of (3.76) is given in (3.70); ence, it suffices to derive an error bound for I,T I u IP,T u k,t. Using te Caucy-Scwarz inequality on (3.7) yields I I,T u(x) = 4 i= 4 i= 4 i= b i b i u(x)ds φ I i,t (X) b i / u(x) ds ds / φ I i,t (X) b i b i b i / u 0,b i φ I i,t (X). (3.77) For k = 0,, we integrate (3.77) on T, using te IFE basis function bounds (3.74) stated in Lemma 3. and standard trace inequality (3.4) to obtain I I,T u k,t 4 i= C k 4 b i / u 0,b i φ I i,t k,t i= C k 4 i= b i / u 0,b i b i C ( ) / u / 0,T + / u,t C ( k u 0,T + k u,t ). (3.78) Here te last inequality is due to b i = O(). We note tat I I,T u I P,T u = I I,T u I I,T (I P,T u) = I I,T (u I P,T u). Also, function u I,T P u is in H (T ). Substituting u by u I,T P u in (3.78) yields I I,T u I P,T u k,t C ( k u I P,T u 0,T + k u I P,T u,t ). (3.79)

67 Xu Zang Capter 3. Properties of IFE Spaces 55 Applying te error bounds (3.70) in Teorem 3.9, we obtain I I,T u I P,T u k,t C ( k u,t + k u,t ) C k u,t. (3.80) Ten (3.75) follows immediately from combining (3.80) and (3.70). Te global interpolation error bound, stated in te next teorem, follows from summing over te estimate (3.4b) on non-interface elements and estimate (3.75) on interface elements. Teorem 3.. For u P H int(ω), we ave te following estimate for te interpolation error I I u u 0,Ω + I I u u,ω C u,ω. (3.8) 3.3 Numerical Experiments In tis section, we provide numerical verifications for error estimates of te nonconforming rotated Q IFE interpolations. We let te simulation domain be a unit square, i.e., Ω = (, ) (, ), and assume tat te interface curve Γ is a circle centered at te origin wit radius r 0 = π/6.8. Te interface curve Γ separates te domain into te following two sub-domains Ω = {(x, y) t : x + y < r 0}, Ω + = {(x, y) t : x + y > r 0}. We compute two types IFE interpolants I P u and II u of te following function u(x, y) = r α β, if r < r 0, ( r α β + + β ) r α β + 0, if r > r 0, (3.8) were α = 5, r = x + y. Te interface jump conditions (.4) and (.5) can be easily verified for tis function u. We note tat tis numerical example ave been used in [69, 70]. In our computation, we use a family of Cartesian meses {T }. Eac mes T is formed by partitioning Ω into N N congruent squares suc tat te edge lengt of square is = /N. Errors of interpolations are given in L, L, and semi-h norms. Errors in te L norm are defined by ( ) I k u u 0,,Ω = max T T max Iu(x, k y) u(x, y) (x,y) T T, k = P, I, (3.83) were T consists of te 49 uniformly distributed points in T as illustrated in Figure 3.. Te L and semi-h norms are computed by suitable Gaussian quadratures. In te following

68 Xu Zang Capter 3. Properties of IFE Spaces 56 error tables, rates of convergence are computed by applying te formulas: ( ) ln() ln I ku u I/ k u u, k = P, I, (3.84) for a specific norm. Figure 3.: Points selected to calculate te L norm on a rectangular element T. We compute errors of tese interpolants wit te coefficient (β, β + ) = (, 0) wic represents a moderate discontinuity in te diffusion coefficient. Table 3. and Table 3. contain errors of nonconforming rotated Q IFE interpolations I Iu and IP u, respectively. Data in tese tables confirm tat bot interpolants I Iu and IP u ave optimal approximation capabilities in L and semi-h norms wic are consistent wit our error estimates given in Teorem 3.0 and Teorem 3.. Moreover, te optimal point-wise convergence of tese interpolations can be observed in L norm. Table 3.: Errors of IFE interpolations I I u wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate E E 3.960E E E E E E E E E E E E E E E E E E E E E E We also experiment wit a larger coefficient discontinuity, i.e., (β, β + ) = (, 0000). Table 3.3 and Table 3.4 contain errors of nonconforming rotated Q IFE interpolations I Iu and I P u, respectively. Data in tese tables again agree wit our error estimates given in Teorem 3.0 and Teorem 3..

69 Xu Zang Capter 3. Properties of IFE Spaces 57 Table 3.: Errors of IFE interpolations I P u wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate 0 3.4E 6.797E E E E E E E E E E E E E E E E E E E E E E E Table 3.3: Errors of IFE interpolations I I u wit β =, β + = N 0,,Ω rate 0,Ω rate,ω rate E E E 0.54E E E E E E E E E E E E E E E E E E E E E Table 3.4: Errors of IFE interpolations I P u wit β =, β + = N 0,,Ω rate 0,Ω rate,ω rate E E E E E E E E E E E E E E E E E E E E E E E E

70 Capter 4 IFE Metods and Error Estimation In tis capter, we propose several computational scemes using nonconforming rotated Q IFE functions for solving te elliptic interface problem and carry out error estimation for tese metods. In Section 4., we consider tree classes of computational scemes including te Galerkin scemes, partially penalized scemes and interior penalty discontinuous Galerkin scemes for solving te elliptic interface problem. In Section 4., we carry out error estimation for te partially penalized Galerkin scemes and interior penalty discontinuous Galerkin scemes. Also, we provide numerical verifications for our error analysis. In Section 4.3, we provide numerical experiments of related IFE metods to compare teir performances. 4. IFE Metods In tis section, we consider tree classes of computational scemes using nonconforming rotated Q IFE functions for solving te elliptic interface problem. Recall tat we use E to denote te collection of all te edges in a Cartesian mes T. For every edge b E, we let M b be te midpoint of b. Moreover, we let E, and E b be te collections of interior edges and boundary edges, respectively. Te sets of interior interface edges and interior non-interface edges are denoted by E i and E n, respectively. To simplify te notations in te following discussion, we assume tat te interface curve Γ does not intersect wit te boundary, i.e., Γ Ω =. As a result, te set of interface edges is a subset of te set of interior edges and E i = E i ; te set of boundary edges is a subset of te set of non-interface edges and E n,b = E b. 58

71 Xu Zang Capter 4. Error Analysis Galerkin IFE Metods Te Galerkin IFE sceme as been used for solving elliptic interface problems [69, 74, 87, 95, 96]. Here we briefly discuss te Galerkin IFE sceme wit nonconforming rotated Q IFE functions. Multiplying te elliptic differential equation (.) by a test function v H0(Ω) and integrating over eac sub-domain Ω s, s = +,, we ave (β s u)v dx dy = Ω s fv dx dy, Ω s v H0(Ω). (4.) Applying te Green s formula to (4.) leads to β s u v dx dy Ω s (β s u n)v ds = Ω s fv dx dy, Ω s v H0(Ω). (4.) Summing (4.) over sub-domains and applying te jump condition (.5) yield te following weak form β u v dx dy = fvdxdy, v H0(Ω). (4.3) Te equation (4.3) is equivalent to β u v dx dy = T T Ω T Ω Ω fv dx dy, v H 0(Ω). (4.4) On eac element T, we use nonconforming rotated Q IFE functions to approximate functions in H (Ω) in te weak form (4.4), ten we obtain two Galerkin IFE scemes: Find u k Sk (Ω), k = P or I, tat satisfies β u k v k dx dy = fv k dx dy, v k S (Ω), k (4.5) T T and te following boundary conditions: or T b Ω u P (M b ) = g(m b ), if b E b, (4.6) u I (X)ds = g(x)ds, if b E. b (4.7) Here te nonconforming rotated Q IFE test function spaces are defined by b S P (Ω) = {v S P (Ω) : v(m b ) = 0, if b E, b and M b is te midpoint of b}. (4.8) S (Ω) I = {v S(Ω) I : vds = 0, if b E}. b (4.9) b

72 Xu Zang Capter 4. Error Analysis Partially Penalized Galerkin IFE Metods Te second group of IFE scemes to be considered are called partially penalized Galerkin (PPG) scemes. To introduce tese scemes, we need to introduce a few more notations. For every interior edge b E, we let T b, and T b, be te two elements saring te common edge b. For a function u defined on T b, T b,, we define its jump and average values on b as follows: [u] b = u Tb, u Tb,, {u} b = ( u Tb, + u ) T b,. (4.0) For every boundary edge b E b, we let T b be te element suc tat b is part of its boundary. We define its jump and average on b as follows [u] b = {u} b = u Tb. (4.) We usually omit te subscript in { } and [ ] if tere is no confusion. To derive te PPG scemes, we multiply te elliptic differential equation (.) by a test function v S k(ω), k = I or P, and integrate over eac element T T, (β u)v dx dy = fv dx dy, v S (Ω). k (4.) T Applying Green s formula to (4.) yields β u v dx dy (β u n T )v ds = T T T T fv dx dy, v S k (Ω). (4.3) Here n T is te unit outward normal of T. Summing (4.3) over all elements leads to β u v dx dy (β u n T )v ds = fv dx dy, v S (Ω). k (4.4) T T T T T T For every interior edge b E wit b = T b, T b,, we let n ij be te unit normal vector pointing from T b,i to T b,j ; ence n = n. We assign n to be te normal of b, i.e. n b = n, ten we ave te following identity β u n v ds + β u n v ds = [β u n b v] ds. (4.5) T b, b Using te following algebraic identity T b, b Ω ab cd = (a + b)(c d) + (a b)(c + d), (4.6) in te rigt and side of (4.5), we obtain [β u n b v] ds = {β u n b }[v] ds + [β u n b ]{v} ds. (4.7) b b b b

73 Xu Zang Capter 4. Error Analysis 6 Replacing te rigt and side of (4.5) by (4.7) and summing (4.5) over all te elements, we obtain (β u n T )v ds = {β u n b }[v] ds + [β u n b ]{v} ds T T T b b b E b E + β u n b v ds. (4.8) b E b b Substituting te second term in (4.4) by (4.8), we ave β u v dx dy {β u n b }[v]ds [β u n b ]{v}ds T T T b b b E b E β u n b v ds = fv dx dy, v S (Ω). k (4.9) b E b b T T T Assume tat u is smoot enoug so tat β u n b is continuous at almost all points on every interior edge b; ence, [β u n b ] = 0 almost every were. Terefore te tird term in (4.9) is zero and (4.9) becomes β u v dx dy {β u n b }[v]ds = fv dx dy, v S (Ω). k (4.0) b E T T T T T b Note te second term in (4.0) includes bot interior and boundary edges and it equals te sum of te second and fourt terms in (4.9). Here we recall tat te jump and average on a boundary edge b is specified in (4.). We assume tat u is continuous almost everywere in te interior of Ω, ten [u] = 0 almost everywere on eac interior edge b. We add two stabilization and penalty terms defined only on interface edges to (4.0), ten we obtain β u v dx dy n b }[v]ds + ɛ T T T b E b{β u {β v n b }[u]ds (4.) b E i b + b E i b σb 0 b [u][v]ds = fv dx dy, α T T T T v S k (Ω). We assume tat on non-interface edges, te following quantity {β u n b }[v]ds (4.) b E n b

74 Xu Zang Capter 4. Error Analysis 6 is not large wic suggests to ignore te term described in (4.) in our algoritms. Ten te weak form (4.) becomes β u v dx dy n b }[v]ds + ɛ T T T b{β u {β v n b }[u]ds (4.3) b E i b E i b + b E i b σ 0 b b α [u][v]ds T T T fv dx dy, v S k (Ω). According to (4.3), we can define te partially penalized Galerkin IFE scemes: Find u k Sk (Ω), k = P, I tat satisfies subject to te following boundary conditions: or a ɛ (u k, v k ) = L(v k ), v k S k (Ω), (4.4) b u P (M b ) = g(m b ), if b E b, (4.5) u I (X)ds = g(x)ds, if b E. b (4.6) b Te bilinear form and linear form in (4.4) are defined by a ɛ (u, v) = β u v dx dy {β u n b }[v]ds T T T b E i b +ɛ n b }[u]ds + b{β v σb 0 [u][v]ds, (4.7) b E i b E i b b α L(v) = fv dx dy. (4.8) T T T In te bilinear form (4.7), α and σb 0 are penalty parameters and we will discuss te possible coices for tese parameters in related error estimation later. Te parameter ɛ in (4.7) as te following tree popular coices: ɛ = : in tis case te bilinear form a ɛ (, ) is symmetric, and we call te corresponding sceme a symmetric partially penalized Galerkin (SPPG) IFE metod. ɛ = : in tis case te bilinear form a ɛ (, ) is nonsymmetric, and we call te corresponding sceme a nonsymmetric partially penalized Galerkin (NPPG) IFE metod. ɛ = 0: in tis case te bilinear form a ɛ (, ) is also nonsymmetric, and we call te corresponding sceme an incomplete partially penalized Galerkin (IPPG) IFE metod.

75 Xu Zang Capter 4. Error Analysis 63 In te above derivation, we ave followed te idea of interior penalty discontinuous Galerkin (IPDG) scemes [5] to design te weak form (4.4) of te PPG IFE scemes. Te idea of partially penalization is for alleviating discontinuity of IFE functions across interface edges. We note tat tis partially penalization idea was used in designing quadratic IFE metods [9] for te elliptic interface problem IPDG IFE Metods Nonconforming rotated Q IFE functions can be used in IPDG scemes [3, 48, 5, 7, 44] to generate metods for solving interface problems wit mass conservation feature and prefinement capability. To introduce te IPDG IFE scemes, we first define te broken IFE spaces as follows: S DG (Ω) = {v L (Ω) : v T S(T n ), if T T n ; v T S(T i ), if T T i }, (4.9) S I,DG (Ω) = {v S DG (Ω) : vds = 0, if b E}, b (4.30) b S P,DG (Ω) = {v S DG (Ω) : v(m b ) = 0, if b E b }. (4.3) Ten we define te IPDG IFE sceme: Find u DG a DG ɛ (u DG, v DG ) = L(v DG ), v DG S DG (Ω), tat satisfies S k,dg (Ω), k = I, P, (4.3) and subject to one of te following boundary conditions: u DG (X)ds = g(x)ds, b E, b (4.33a) or b b u DG (M b ) = g(m b ), b E b. (4.33b) Here te bilinear form and linear form in (4.3) are defined as a DG ɛ (u, v) = β u v dx dy {β u n b }[v]ds T T b E L(v) = T T T +ɛ {β v n b }[u]ds + b E b E T b b b σb 0 [u][v]ds, (4.34) b α fv dx dy. (4.35) Here, α and σb 0 are penalty parameters and we will discuss te possible coices for tese parameters in related error estimation later. Similar to PPG IFE scemes, te parameter ɛ in te bilinear form of IPDG IFE sceme (4.34) as te following tree popular coices:

76 Xu Zang Capter 4. Error Analysis 64 ɛ = : in tis case te bilinear form a DG ɛ (, ) is symmetric, and we call tis sceme a symmetric interior penalty discontinuous Galerkin (SIPDG) IFE metod. ɛ = : in tis case te bilinear form a DG ɛ (, ) is nonsymmetric, and we call tis sceme a nonsymmetric interior penalty discontinuous Galerkin (NIPDG) IFE metod. ɛ = 0: in tis case te bilinear form a DG ɛ (, ) is also nonsymmetric, and we call tis sceme an incomplete interior penalty discontinuous Galerkin (IIPDG) IFE metod. We note tat te differences between PPG IFE scemes and IPDG IFE scemes are teir coices of IFE spaces and collections of edges were penalization is applied. Te number of global degrees of freedom in PPG scemes is muc less tan te IPDG IFE scemes. In fact, PPG IFE scemes ave te same number of global degrees of freedom as te Galerkin IFE scemes. In addition, penalty terms are added only on interfaces edges for PPG scemes instead being added at all edges for IPDG scemes. 4. Error Estimation In tis section, we carry out te error estimation for PPG IFE and IPDG IFE scemes using nonconforming rotated Q IFE functions wit midpoint-value degrees of freedom and integral-value degrees of freedom. Trougout te error analysis, we assume tat te collection of interface elements T i in a Cartesian mes T satisfies te following ypotesis: (H5) Tere exists a constant C suc tat te number of interface element in a mes T, denoted by T i satisfies T i C. (4.36) 4.. Error Estimation for PPG IFE Solutions in S I (Ω) We first derive an error estimate for te partially penalized Galerkin IFE scemes. Define te energy norm,ω on S P (Ω) S I (Ω), wic is a semi-norm on SP (Ω) SI (Ω): v,ω = T T T β v v dx dy + b E i b σ 0 b b [v][v]ds α /. (4.37) Te coercivity of te bilinear forms a ɛ (, ) wit respect to te energy norm,ω is given in te following lemma.

77 Xu Zang Capter 4. Error Analysis 65 Lemma 4.. Assume α in te bilinear form (4.7) and te energy norm (4.37). Tere exists a positive constant κ suc tat κ v,ω a ɛ (v, v), v S P (Ω) S I (Ω), (4.38) for any positive σ 0 b if ɛ =, or for σ0 b large enoug if ɛ = 0 or. Proof. Note tat te coercivity result (4.38) is trivial for ɛ = and te corresponding coercivity constant κ = ; ence, our proof below focuses on te oter two cases ɛ = or 0. For eac interior interface edge b E i, applying te trace inequality of IFE functions stated in Teorem 3.3 and te Caucy-Scwarz inequality, we obtain te following estimate n b }[v]ds b{β v ( (β v nb ) Tb, 0,b + ) (β v nb ) 0,b Tb, [v] 0,b ( ) b α/ C / T b, β v 0,Tb, + C / T b, β v 0,Tb, b [v] α/ 0,b ( ) / C β v 0,T b, + β v 0,T b, b [v] α/ 0,b. (4.39) Summing up (4.39) over all te interior interface edges, and applying Young s inequality, we obtain ( ) / ( ) {β v n b }[v]ds C b E i b b α [v] 0,b ( β v ) / 0,Tb, + β v 0,Tb, b E i b E ( i ) ) ( + C δ δ T T ( β v 0,T b E i ) b α [v] 0,b, for every δ > 0. Tus, ( ) ( δ ɛ a ɛ (v, v) β v 0,T + σb 0 ) C ɛ δ [v] b α 0,b T T b E ( i δ ɛ min, σ0 b ) C ɛ δ v b,ω. (4.40) α Coosing δ and σ 0 b in (4.40) suc tat δ ɛ <, σb 0 > C ɛ, δ

78 Xu Zang Capter 4. Error Analysis 66 we obtain (4.38) wit κ = min ( δ ɛ, σ0 b ) C ɛ δ > 0. (4.4) b α For eac edge b E, and v L (b), we define v b to be te average integral value of v over te edge b as follows v b = vds. (4.4) b b Lemma 4.. Tere exists a constant C suc tat [v ] 0,b C / ( v,tb, + v,tb, ), v S P (Ω) S I (Ω), (4.43) for every interior edge b E, and for every boundary edge b E b. v 0,b C / v,tb, v S P (Ω) S I (Ω), (4.44) Proof. We first prove tat te estimate (4.43) olds for every v S P (Ω). Note tat v is continuous at te midpoint M b of an interior (interface or non-interface) edge b. Witout loss of generality, we assume tat b is a vertical line segment wose endpoints are (x 0, y 0 ) and (x 0, y 0 + ), ten by te triangle inequality we ave te following estimate [v ] 0,b = v Tb, v Tb, 0,b = (v Tb, v Tb, (M b )) + (v Tb, (M b ) v Tb, ) 0,b v Tb, v Tb, (M b ) 0,b + v Tb, v Tb, (M b ) 0,b ( y0 + ( v Tb,j (x 0, y) v Tb,j (x 0, y 0 + ) ) dy j= y 0 ( y0 + ( y ( z v Tb,j (x 0, z) ) dz) dy y 0 y 0 +/ ( y0 + ) / v,,t b,j dy j= j= y 0 ) / ) / C 3/ ( v,,tb, + v,,tb, ) (4.45) We apply te standard inverse inequality (3.7) or te IFE inverse inequality (3.8) to v on tese two elements T b,, T b,, depending on weter tey are non-interface elements or interface elements. Ten we ave v,,tb,j C v Tb,j, j =,. (4.46)

79 Xu Zang Capter 4. Error Analysis 67 Combining te above estimates (4.45) and (4.46), we obtain [v ] 0,b C / ( v,tb, + v,tb, ), v S P (Ω). (4.47) For every v S I (Ω), te average integral value on every interior edge b is continuous. Note tat v T is in H (T ) for every element T T, ten by standard approximation results stated in [37] we obtain [v ] 0,b = v Tb, v Tb, 0,b = (v Tb, v Tb, b ) + (v Tb, b v Tb, ) 0,b v Tb, v Tb, b 0,b + v Tb, v Tb, b 0,b C / ( v,tb, + v,tb, ). (4.48) Te estimate (4.43) follows from (4.47) and (4.48). We can prove (4.44) similarly if b E b is a boundary edge. In te following error analysis in tis subsection, we need te following ypotesis, (H6) Te interface Γ is smoot enoug so tat P C 3 int(ω) is dense in P H 3 int(ω). Te interpolation error on an interface edge is analyzed in te following lemma. Lemma 4.3. For every u P H 3 int(ω), tere exists a constant C suc tat for every interface element T T i β (u I P,T u) n b 0,b C( u 3,Ω + u,t ). (4.49) were b T is one of its interface edges. Proof. We consider te Type I Case interface element as illustrated in Figure 3., and oter cases can be discussed similarly. Witout loss of generality, we let b = A A 3. Recall from Teorem 3.7 tat for every function u P C 3 int(ω), its interpolation error at point X EA 3 can be expanded as follows (I,T P u(x) u(x)) x = (F + F )φ P,T,x(X) F 0,xu xx (X) + 4 I i φ P i,t,x(x). (4.50) i= Here F 0, F, F, and I i, i =,, 4 are defined in (3.35), (3.46a) - (3.46g). To estimate te first term in (4.50), we use estimates (3.3), (3.30) in Lemma 3.8 and te bound of IFE functions (3.3a) in Teorem 3. to obtain (F i φ P,T,x(X)) ds C EA 3 u ds C u,ω C u 3,Ω, EA 3 i =,. (4.5)

80 Xu Zang Capter 4. Error Analysis 68 Also, applying te bound of F 0 (3.4) to estimate te second term in (4.50) we obtain ( ) EA 3 F 0,xu xx(x) ds F 0,x u xx(x) ds C u xx(x) ds C u 3,Ω. EA 3 EA 3 (4.5) Using te bound of IFE functions (3.3a) in Teorem 3., we ave te following estimation involving te term I, : ( I, φ P,T,x(X) ) ds EA 3 ( N ( M ) I u ξξ (ξ, η)( x x)(x x ) + u ξη (ξ, η)(ỹ y)(x x ) EA 3 0 ) φ P +u ηξ (ξ, η)( x x)(y ỹ ) + u ηη (ξ, η)(ỹ y)(y ỹ ),T,x(X) dt ds ) C ( u ξξ (ξ, η) + u ξη (ξ, η) + u ηξ (ξ, η) + u ηη (ξ, η) dt ds EA 3 0 C u 3,Ω. Similarly, for te term involving I,, we use Teorem 3. to obtain ( I, φ P,T,x(X) ) ds EA 3 C C ( t EA 3 0 t EA 3 0 ( t) d dt u(tm + ( t)x)dt ( t) ( + u xy (tm + ( t)x) (x x) (y y) + u yx (tm + ( t)x) (y y) (x x) ) ds u xx (tm + ( t)x) (x x) (x x) (4.53) + u yy (tm + ( t)x) (y y) (y y) )dt ds C + u yx e t 0 ( ) t ( t) u xx, ( t)y + u xy ) + u yy ( t, ( t)y ( ) t, ( t)y ( ) t, ( t)y dt dy. (4.54) Let p = t, q = ( t), ten te Jacobian of tis substitute is (t, y) (p, q) = ( t) = q.

81 Xu Zang Capter 4. Error Analysis 69 Te inequality (4.54) becomes ( I, φ P,T,x(X) ) ds EA 3 ( C uxx (p, q) + u xy (p, q) + u yx (p, q) + u yy (p, q) ) dp dq A 3 E M ) C ( u xx 0, A3E + u M xy 0, A3E + u M yx 0, A3E + u M yy 0, A3E M C u,t. (4.55) Similarly for te term involving I,3, using te bound of IFE functions (3.3a) again, we obtain ( I,3 φ P,T,x(X) ) ds EA 3 ( ) C ( t) d dt u(tm + ( t)x)dt ds EA 3 t ( ) t ( ) C ( t) u xx e t, ( t)y + t u xy, ( t)y ( ) + t ( ) u yx, ( t)y + t u yy, ( t)y dt dy ( C uxx (p, q) + u xy (p, q) + u yx (p, q) + u yy (p, q) ) dp dq EM M ) C ( u xx 0, EM + u M xy 0, EM + u M yx 0, EM + u M yy 0, EM M C u,t +. For te term involving I, we ave ( I φ P,T,x(X) ) ds EA 3 C C e EA 3 0 ( ) ( t) d 0 dt u(tm + ( t)x)dt ds ( t) u xx (t, t + ( t)y) + u xy(t, t + ( t)y) + u yx(t, t + ( t)y) + u yy(t, t + ( t)y) dt dy (4.56) Let p = t, q = t + ( t)y, te Jacobian of tis substitution is te following (t, y) (p, q) = ( t) = p.

82 Xu Zang Capter 4. Error Analysis 70 Hence, ( I φ P,T,x(X) ) ds EA 3 ( ) p ( = C uxx(p, q) + u xy(p, q) A 3 EM + u yx (p, q) + u yy (p, q) ) dp dq p ( C uxx (p, q) + u xy (p, q) + u yx (p, q) + u yy (p, q) ) p dp dq A 3 EM ( C uxx (p, q) + u xy (p, q) + u yx (p, q) + u yy (p, q) ) dp dq A 3 EM C ( ) u xx 0, A 3 EM + u xy 0, A 3 EM + u yx 0, A 3 EM + u yy 0, A 3 EM C u,t Similarly, for I 3 we ave (4.57) ( I3 φ P 3,T,x(X) ) ds C u,t. (4.58) EA 3 Te term involving I 4 ave te following bounds: ( I4 φ P 4,T,x(X) ) ds EA 3 ( ) C ( t) d dt u(tm 4 + ( t)x)dt ds C EA 3 0 e 0 ( t) u yy (0, t + ( t)y) ( y ) 4 dt dy Let z = t + ( t)y, ten dz = ( y) dt, and t = ( z)/( y). Hence, ( I4 φ P 4,T,x(X) ) ds EA 3 / ( ) ( ) C u e yy (0, z) y y z dz dy ( / ) C u yy (0, z) dz dy + u yy (0, z) dz dy C e ( / e e e u yy (0, z) dy dz + ( ) C u yy (0, z) dz e e / / e u yy (0, z) dy dz C u 3,Ω (4.59) )

83 Xu Zang Capter 4. Error Analysis 7 Combining te estimates (4.5) - (4.59) and using te density ypotesis (H4), we obtain for every u P H 3 int(ω), x (I P,T u u) 0,EA 3 C ( u,t + u 3,Ω ). Use similar arguments by letting X A E, we can also prove tat x (I P,T u u) 0,A E C ( u,t + u 3,Ω + ). Note tat b = EA 3 A E; ence, we obtain te following estimate x (I P,T u u) 0,b C ( u,t + u 3,Ω). (4.60) Similar arguments leads to te error estimate for y derivative: y (I P,T u u) 0,b C ( u,t + u 3,Ω). (4.6) Combining (4.60) and (4.6) yields te error estimate (4.49) for u P C 3 int(ω). Applying te density assumption (H6), we obtain (4.49) for u P H 3 int(ω). Now we derive an error estimate for te interpolation I I,T on interface edges. Lemma 4.4. For every u P H 3 int(ω), tere exists a constant C suc tat were T T i β (u I I,T u) n b 0,b C( u 3,Ω + u,t ), (4.6) is an interface element and b T is one of its interface edges. Proof. By te triangle inequality, we ave β (u I I,T u) n b 0,b β (u I P,T u) n b 0,b + β (I P,T u I I,T u) n b 0,b. (4.63) We ave derived an error bound (4.49) in Lemma 4.3 for te first term on te rigt and side of (4.63). It suffices to estimate te second term on te rigt and sides of (4.63). Note tat I,T P u II,T u Si (T ), ten by te IFE trace inequality (3.5) and te estimate (3.80) we obtain β (I P,T u I I,T u) n b 0,b C / T / β (I P,T u I I,T u) 0,T Combining (4.64) and (4.49) leads to te estimate (4.6). C / T / u,t C / u,t. (4.64)

84 Xu Zang Capter 4. Error Analysis 7 Te following lemma provides an error bound for te IFE interpolation in te energy norm,ω. Lemma 4.5. Let u P H int(ω), ten tere exists a constant C independent of and interface location, suc tat I k u u,ω C( + (3 α)/ ) u,ω, k = I or P. (4.65) Proof. By te interpolations error estimates stated in Teorem 3.9 and Teorem 3., and te standard trace inequality (3.4), we obtain for k = P or I tat I k u u,ω = T T T β (I k u u) (I k u u)dx + b E i T T I k u u,t + C α b E i I k u u 0,b b σ 0 b b α [(Ik u u)] ds C T T u,t + C α T T ( I k u u 0,T + I k u u,t C( + 3 α ) u,ω. (4.66) Taking square root on bot sides of (4.66) leads to (4.65). ) Remark 4.. Te error estimate (4.65) in Lemma 4.5 indicates tat nonconforming IFE functions ave optimal approximation property in energy norm if α. On te oter and, te coercivity property (4.38) in Lemma 4. requires α for te bilinear form (4.7) to be coercive. Te only value of α tat satisfies bot of tese conditions is α =. Terefore, from now on, we let α = in bot error analysis and numerical experiment. Now we are ready to derive a bound for te error in te PPG IFE solution u I. Teorem 4.. Assume u P Hint(Ω) 3 is te solution to te interface problem (.) - (.4), and u I SI (Ω) is te partially penalized Galerkin IFE solution to (4.4) and (4.6), ten tere exists a constant C suc tat u I u,ω C u 3,Ω. (4.67) Proof. Note tat te solution u P H 3 int(ω) is continuous in Ω, and te flux β u n b is

85 Xu Zang Capter 4. Error Analysis 73 continuous across every interior edges b E. By Green s formula and (4.8), we ave {β u n b }[v ]ds b E i = b E b = b E b b b b b E n = T T β u n b v ds + n b }[v ]ds b{β u {β u n b }[v ]ds b E b E n b β u n b v ds + {β u n b }[v ]ds + {β v n b }[u]ds b b b E b E {β u n b }[v ]ds T b β u n T v ds b E n {β u n b }[v ]ds, (4.68) b for every v S I (Ω). Te equation (4.68) implies a ɛ (u, v ) = β u v dx {β u n b }[v ]ds T T T b E i b = β u v dx β u n T v ds + {β u n b }[v ]ds T T T T T T b E n b = (β u)v dx + {β u n b }[v ]ds T T T b E n b = (f, v ) + {β u n b }[v ]ds. (4.69) b E n b Hence, subtracting (4.69) from (4.4), we ave a ɛ (u I, v ) = a ɛ (u, v ) {β u n b }[v ]ds, b E n b v S (Ω). I (4.70) For every function w in S I(Ω) = {w S I(Ω) : w b ds = gds, b E b b }, we subtract a ɛ (w, v ) from bot sides of (4.70) a ɛ (u I w, v ) = a ɛ (u w, v ) {β u n b }[v ]ds, v S (Ω), I w S (Ω). I b E n b We let v = u I w S I (Ω) in te above equation, ten a ɛ (u I w, u I w ) = a ɛ (u w, u I w ) {β u n b }[u I w ]ds, b E n b w S I (Ω). (4.7)

86 Xu Zang Capter 4. Error Analysis 74 Applying te coercivity result (4.38) to (4.7), we ave κ u I w,ω a ɛ (u w, u I w ) + {β u n b }[u I w ]ds b E n b β (u w ) (u I w )dx T T T + {β (u w ) n b }[u I w ]ds b E i b + ɛ {β (u I w ) n b }[u w ]ds b E i b + σb 0 b E i b b [u w ][u I α w ]ds + {β u n b }[u I w ]ds b b E n Q + Q + Q 3 + Q 4 + Q 5. (4.7) To bound Q we use Caucy-Scwarz inequality and Young s inequality to obtain te following estimate Q κ 6 ( T T β / (u w ) ) / ( 0,T β / (u I w ) ) / 0,T T T β / (u I w ) 0,T + C (u w ) 0,Ω T T κ 6 ui w,ω + C (u w ) 0,Ω. (4.73) To bound Q, using Young s inequality we ave Q κ 6 ( ) / ( {β (u w ) n b } [ ) / ] 0,b u I w 0,b b E i b E i b E i σb 0 [ ] u I b α w + C b α {β (u w 0,b σ 0 ) n b } 0,b b E i b κ 6 ui w,ω + C b E i b α σ 0 b {β (u w ) n b } 0,b. (4.74) To bound Q 3, applying te trace inequality (3.4) for H function u w and trace inequality

87 Xu Zang Capter 4. Error Analysis 75 (3.5) for IFE function u I w, we ave Q 3 ɛ C ( b E i ( ( b E i b E i C ( b E i ( b E i { ) / ( } β (u I w ) n b 0,b (β (u I w ) n b ) Tb, b E i 0,b + b E i (u w ) Tb, + 0,b (u w ) Tb, 0,b b E i [u w ] 0,b ) / (β (u I w ) n b ) Tb, ) / (β / (u I w ) + 0,T (β / (u I b, w ) ) / 0,T b, b E i ( (u w ) 0,T b, + (u w ) 0,T b, + (u w ) 0,T b, + (u w ) 0,T b, ) ) / 0,b ) / ) / C ( T T β / (u I w ) 0,T ( T T ( ) ) / (u w ) L (T ) + (u w ) 0,T κ 6 ui w,ω + C ( T T ( (u w ) 0,T + (u w ) 0,T ) ). (4.75) To bound Q 4, we ave Q 4 ( b E i b ) / ( σb 0 b [u w ] ds α κ 6 ui w,ω + 3 κ b E i b E i σ 0 b b α [u w ] 0,b b σ 0 b b α [ui w ] ds ) / κ 6 ui w,ω + C T T (+α) ( T T ( (u w ) 0,T + (u w ) 0,T ) ). (4.76)

88 Xu Zang Capter 4. Error Analysis 76 For Q 5, note tat te flux β u n b is continuous across every interior edges b E ; ence {β u n b } = β u n b. Using te estimates (4.43) and (4.44) in Lemma 4. and te standard approximation result [37], we ave [ ] Q 5 = β u n b u I w ds b = b E n b E n C b E n b ( β u n b β u n b b ) [ u I w ] ds β u n b β u n b b 0,b [ (u I w ) ] 0,b C T T / β u n,t / u I w,t C T T u,t u I w,t C ( T T u,t ) / ( T T u I w,t ) / κ 6 ui w,ω + C u,ω. (4.77) Applying te estimates (4.73) - (4.77) to (4.7), we ave κ 6 ui w,ω C u,ω + C (u w ) 0,Ω + C b E i +C( + (+α) ) ( T T b α σ 0 b {β (u w ) n b } 0,b ( (u w ) 0,T + (u w ) 0,T ) ). (4.78) Ten, letting w = I I u in (4.78), using te estimate (3.8) in Teorem 3. and (4.6) in Lemma 4.4 we ave te following estimate: u I I I u,ω C u,ω + C b E i C u,ω + C T T i b σ 0 b { β (u I I u) n b } 0,b ( ) u 3,Ω + u,t C u 3,Ω. (4.79) Te last inequality in (4.79) is because of te ypotesis (H5). Note tat te interpolation error is bounded as follows using te energy norm (4.66) u I I u,ω C u,ω. (4.80)

89 Xu Zang Capter 4. Error Analysis 77 Finally, (4.67) follows from applying (4.79) (4.80) to te triangle inequality u I u,ω u I I u,ω + u I I I u,ω. Remark 4.. Te error estimate (4.67) is optimal from te point view of te degree of polynomials tat we use to solve te elliptic interface problem. To confirm our error analysis, we present several numerical examples in te following discussion. We use te function u given in (3.8) as te exact solution. Te Diriclet boundary function g and te source function f are cosen correspondingly. Errors of IFE solutions are given in L, L, and semi-h norms. Errors in te L norm are defined by ( ) u u 0,,Ω = max T T max u (x, y) u(x, y) (x,y) T T, (4.8) were T consists of te 49 uniformly distributed points in T as illustrated in Figure 3.. Te L and semi-h norms are computed by suitable Gaussian quadratures. In te following data tables, rates of convergence in a numerical solution u are computed by applying te formulas: ln() ln ( u u u / u ), (4.8) for a specific norm, were u denotes IFE solution based on te mes T. Example 4.. (Integral-Value Degrees of Freedom): In tis experiment, we test te accuracy of IFE solutions u I generated from PPG IFE scemes. We test all te tree PPG IFE scemes based on IFE space S I (Ω) for te elliptic interface problem (.) - (.5). First, we consider te case (β, β + ) = (, 0) wic represents a moderate discontinuity in te diffusion coefficient. Table 4., Table 4., and Table 4.3 contain errors of NPPG solutions, SPPG solutions and IPPG solutions, respectively. In te NPPG sceme, we coose σb 0 =,. In SPPG and IPPG scemes, we coose σ0 b = 0 max(β, β + ) for all te interface edges b E i. Te different coices of σ0 b are due to te coercivity requirement described in Lemma 4.. Convergence rates in semi-h norm in Tables 4. troug 4.3 confirm our error analysis in te energy norm (4.67). In fact, errors in te semi-h norm can be bounded by te energy norm wit a constant C depending only on te diffusion coefficient β, i.e., u I u,ω C u I u,ω. Te data also suggest tat te convergence rates in L and L norms are approximately O( ), wic are also optimal from te point view of te degree of polynomials used to construct IFE spaces. However, te error analysis of L and L norms is still open.

90 Xu Zang Capter 4. Error Analysis 78 Table 4.: Errors of NPPG IFE solutions u u I wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate 0.684E.379E.953E E E E E E E E E E E E E E E E E E E E E E Table 4.: Errors of SPPG IFE solutions u u I wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate 0.665E.39E.9570E E E E E E E E E E E E E E E E E E E E E E Table 4.3: Errors of IPPG IFE solutions u u I wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate 0.665E.30E.9570E E E E E E E E E E E E E E E E E E E E E E We also test PPG IFE scemes wit an interface problem wose coefficient as a larger

91 Xu Zang Capter 4. Error Analysis 79 jump, i.e., (β, β + ) = (, 0000). Errors of nonsymmetric, symmetric and incomplete PPG IFE solutions are listed in Table 4.4, Table 4.5 and Table 4.6, respectively. Tese data demonstrate te optimal convergence in L, and semi-h for all tese PPG IFE scemes. Errors in L norm are sligtly less tan optimal rate O( ). We also note tat NPPG solutions are more accurate tan SPPG and IPPG scemes in tis example. In addition, te convergence rate of NPPG solutions seems to be closer to optimal in L norm compared to SPPG and IPPG scemes. Table 4.4: Errors of NPPG IFE solutions u u I wit β =, β + = N 0,,Ω rate 0,Ω rate,ω rate E E E E E E E E E E E E E E E E E E E E E E E E Table 4.5: Errors of SPPG IFE solutions u u I wit β =, β + = N 0,,Ω rate 0,Ω rate,ω rate E 3.690E E 0.574E E E E E E E E E e E E E E E E E E E E E Error Estimation for IFE Solutions in S P (Ω) In tis subsection, we derive an error bound for te PPG IFE solutions u P. Te discussion in tis section is based on te assumption tat te solution u to te elliptic interface problem is piece-wise W,, i.e., u Ω s W, (Ω s ), s = +,. First, we define te following space P W, int (Ω) = { u C(Ω), u Ω s W, (Ω s ), s = +,, [β u n Γ ] = 0 on Γ }. (4.83)

92 Xu Zang Capter 4. Error Analysis 80 Table 4.6: Errors of IPPG IFE solutions u u I wit β =, β + = N 0,,Ω rate 0,Ω rate,ω rate E 3.690E E 0.574E E E E E E E E E e E E E E E E E E E E E In te following error analysis in tis subsection, we need te following ypotesis, (H7) Te interface Γ is smoot enoug so tat P Cint(Ω) 3 is dense in P W, int (Ω). Ten, we derive te interpolation error bound on an interface edge using te norm,,ω. Lemma 4.6. For every u P W, (Ω), tere exists a constant C suc tat were T T i int β (u I P,T u) n b 0,b C 3 u,,ω, (4.84) is an interface element and b T is one of its interface edges. Proof. We consider te Type I Case interface element as illustrated in Figure 3., and te discussion for oter cases are similar. Witout loss of te generality, we let b = A A 3. Recall from (3.6) tat for every function u P C 3 int(ω), its interpolation error at a point X EA 3 can be expanded as follows (I,T P u(x) u(x)) x = (F + F )φ P,T,x(X) F 0,xu xx (X) + 4 I i φ P i,t,x(x), (4.85) i= were F 0, F, F, and I i, i =,, 4 are defined in (3.35), (3.46a) - (3.46g). To bound te first term on te rigt and side of (4.85), we use (3.30), (3.3) and te bound of IFE functions (3.3a) in Teorem 3. to obtain (F i φ P,T,x(X)) ds C EA 3 u ds C 3 u,,ω C3 u,,ω, EA 3 i =,. (4.86)

93 Xu Zang Capter 4. Error Analysis 8 Also, applying te bound of F 0 from (3.4) to te second term on te rigt and side of (4.85) yields ( ) F 0,xu xx(x) ds F 0,x u xx(x) ds EA 3 EA 3 C EA 3 u xx(x) ds C 3 u,,ω. (4.87) For te term involving I, we ave te following estimate from (3.54) and te bound of IFE functions (3.3a) given in Teorem 3. ( I, φ P,T,x(X) ) ds EA 3 ) C ( u ξξ (ξ, η) + u ξη (ξ, η) + u ηξ (ξ, η) + u ηη (ξ, η) dt ds EA 3 0 C 3 u,,ω. (4.88) Similarly to (4.88), we obtain ( I,i φ P,T,x(X) ) ds C 3 u,,ω, EA 3 i =, 3. (4.89) For te term involving I we ave te following estimate using (3.3a) ( I φ P,T,x(X) ) ds EA 3 C ( t) u xx (t, t e 0 + ( t)y) + u xy(t, t + ( t)y) + u yx(t, t + ( t)y) + u yy(t, t + ( t)y) dt dy C 3 u,,ω. (4.90) Similarly to (4.90), we obtain ( Ii φ P 3,T,x(X) ) ds C 3 u,,ω, EA 3 i = 3, 4. (4.9) Combining all estimates (4.86) - (4.9) yields x (I P,T u u) 0,EA 3 C 3 u,,ω. Similarly, we can also sow tat x (I P,T u u) 0,A E C3 u,,ω +.

94 Xu Zang Capter 4. Error Analysis 8 Note tat b = EA 3 A E, ence x (I P,T u u) 0,b C3 u,,ω. (4.9) Using a similar argument we can derive an error estimate for y-derivative: y (I P,T u u) 0,b C3 u,,ω. (4.93) Combining (4.9) and (4.93) leads to te error estimate (4.84) for u P Cint(Ω). 3 Applying te density assumption (H7), we obtain (4.49) for u P W, int (Ω). Te following lemma is useful for te error analysis. Te proof of te tis lemma follows a similar approac in te proof of Teorem 3.3 in [85]. Lemma 4.7. Let u H (Ω) and u Ωs H (Ω s ), s = +,. Tere exists a constant C suc tat (β u n b ) (v v (M b )) ds C u,t v,t, v S P (Ω), (4.94) b T b for every non-interface element T T n, were b is an edge of T and M b is te midpoint of b. Proof. We consider an element T T n wit vertices A = (x 0, y 0 ) t, A = (x 0 +, y 0 ) t, A 3 = (x 0, y 0 + ) t, A 4 = (x 0 +, y 0 + ) t. Ten we denote te edges of T as follows: b = A A, b = A A 4, b 3 = A 4 A 3, b 4 = A 3 A, Witout loss of generality, we consider te two vertical edges b and b 4. Note tat te normal vectors associated to tese edges satisfy n b = n b4. Ten we ave te following estimate

95 Xu Zang Capter 4. Error Analysis 83 by using te inverse inequality (3.7): ( u n b ) (v v (M b )) ds + ( u n b4 ) (v v (M b4 )) ds b b 4 = u x (x, y)(v (x, y) v (M b ))ds u x (x, y)(v (x, y) v (M b4 ))ds b b 4 = = = = y0 + y 0 y0 + y 0 y0 + u x (x 0 +, y)(v (x 0 +, y) v (x 0 +, y 0 + ))dy u x (x 0, y)(v (x 0, y) v (x 0, y 0 + ))dy ( y u x (x 0 +, y) y 0 y0 + ( y u x (x 0, y) y 0 y0 + y 0 y0 + ( x0 + y 0 y 0 +/ y 0 +/ (u x (x 0 +, y) u x (x 0, y)) x 0 C v,t x0 + x 0 C v,t u,t. ) ( y u xx (x, y)dx y0 + y 0 ) v (x 0 +, y ) dy dy y ) v (x 0, y ) dy dy y ( y ) v (x 0 +, y ) dy dy y u xx (x, y) dx dy y 0 +/ ) v,,t dy dy y 0 +/ Similarly, we can derive an estimate on two orizontal edges b and b 3. Finally, te result (4.94) follows immediately from combining tese estimates togeter. Now we are ready to prove te convergence of u P. Teorem 4.. Assume tat u P W, int (Ω) is te solution to te interface problem (.) - (.4), and u P SP (Ω) is te partially penalized Galerkin IFE solution to (4.4) and (4.5), ten tere exists a constant C suc tat u P u,ω C / u,,ω (4.95) Proof. Note tat te solution u W, int (Ω) is continuous inside Ω, and te flux β u n b are continuous across every interior edge b E. Similar to te derivation (4.68) in te proof of Teorem 4., for every v S P (Ω), we ave {β u n b }[v ]ds = β u n T v ds {β u n b }[v ]ds. (4.96) b T T T b E n b b E i

96 Xu Zang Capter 4. Error Analysis 84 Similar to te derivation (4.69), te equation (4.96) yields a ɛ (u, v ) = (f, v ) + {β u n b }[v ]ds. (4.97) b E n b Hence, subtracting (4.97) from (4.4), we ave a ɛ (u P, v ) = a ɛ (u, v ) {β u n b }[v ]ds, b E n b v S P (Ω). (4.98) For every function w in S P (Ω) = {w S P (Ω) : w (M b ) = g(m b ), b E b }, we subtract a ɛ (w, v ) from bot sides of (4.98), a ɛ (u P w, v ) = a ɛ (u w, v ) {β u n b }[v ]ds, v S P (Ω), w S P (Ω). b E n b We let v = u P w S P (Ω) in te above equation, ten a ɛ (u P w, u P w ) = a ɛ (u w, u P w ) {β u n b }[u P w ]ds, w S P (Ω). b E n b (4.99) We can write te last term in (4.99) as te summation wit respect to te element T : {β u n b }[u P w ]ds b E n = T T n b + T T i (β u n b )(u P w (u P (M b ) w (M b )))ds b T b b T,b/ E i (β u n b )(u P w (u P (M b ) w (M b )))ds. (4.00) b

97 Xu Zang Capter 4. Error Analysis 85 Applying te identity (4.00) and te coercivity result (4.38) to (4.99) yields κ u P w,ω a ɛ (u w, u P w ) + {β u n b }[u I w ]ds b E n b = β (u w ) (u P w )dx T T T + {β (u w ) n b }[u P w ]ds b E i b + ɛ {β (u P w ) n b }[u w ]ds b E i b + σ 0 b b E i b b [u w ][u P α w ]ds + (β u n b )(u P w (u P (M b ) w (M b )))ds T T n b T b + (β u n b )(u P w (u P (M b ) w (M b )))ds b T T i b T,b/ E i Q P + Q P + Q P 3 + Q P 4 + Q P 5 + Q P 6. (4.0) Following similar arguments (4.73) - (4.76) as for analyzing Q troug Q 4, we obtain te following estimates for Q P i, i =,, 3, 4: Q P κ 6 up w,ω + C (u w ) 0,Ω, (4.0) Q P κ 6 up w,ω + C b E i b α σ 0 b {β (u w ) n b } 0,b, (4.03) Q P 3 κ 6 up w,ω + C ( T T ( (u w ) 0,T + (u w ) 0,T ) ), (4.04) Q P 4 κ 6 up w,ω + C ( ) ( (u ) (+α) w ) 0,T + (u w ) 0,T. T T T T (4.05) By Lemma 4.7, we ave te following bounds for Q P 5 : Q P 5 (β u n b ) ( u P w (u P (M b ) w (M b )) ) ds T T n b T ( C T T n b u,t ) / ( T T u P w,t ) / κ 6 up w,ω + C u,ω. (4.06)

98 Xu Zang Capter 4. Error Analysis 86 To bound Q P 6, we let T be te collection of non-interface elements wic are adjacent to interface elements and let Ω = T T T. By standard trace teorem (3.5) we ave Q P 6 (β u n)(u P w (u P (M b ) w (M b )))ds b T T i b T,b/ E i C T T i C b T,b/ E i β u n 0,b u P w (u P (M b ) w (M b )) 0,b / u,t / u P w,t C T T ( T T u,t ) / ( κ up w,ω + C u,ω T T u P w,t ) / Using te ypotesis (H5), we may conclude tat meas(ω ) C. As a result, we ave u,ω = D α u dx u,,ω dx C u,,ω. Ω Ω α = Hence, Q P 6 κ up w,ω + C u,,ω (4.07) Applying estimates (4.0) - (4.07) to (4.0), we ave κ up w,ω C u,,ω + C u,ω + C (u w ) 0,Ω + C b E i +C( + (+α) ) ( T T b α σ 0 b {β (u w ) n b } 0,b ( (u w ) 0,T + (u w ) 0,T ) ). (4.08) Ten we let w = I P u in (4.08) and use te estimate (3.7) in Teorem 3.0 and (4.49) in Lemma 4.3 to ave te following estimate: u P I P u,ω C u,ω + C u,,ω + C b E i C u,ω + C u,,ω + C T T b σ 0 b { β (u I P u) n b } 0,b ( ) 3 u,,ω C u,,ω. (4.09)

99 Xu Zang Capter 4. Error Analysis 87 Again, we use te ypotesis (H5) to derive te last inequality in (4.09). Ten we apply te interpolation error estimate wit energy norm (4.66) to ave Since u P W, (Ω), ten it can be easily verified tat int u I P u,ω C u,ω. (4.0) u,ω C u,,ω. (4.) Finally, te result (4.95) follows from applying te error bounds (4.09), (4.0) and (4.) to te following triangle inequality u P u,ω u I P u,ω + u P I P u,ω. Note tat te error bound we ave derived for te PPG IFE solution u P is at least suboptimal in te energy norm,ω. Next we provide a few numerical experiments to test te performance of tis PPG IFE sceme. We consider te same example as we use in Example 4.. Example 4.. (Midpoint-Value Degrees of Freedom): In tis experiment, we test te accuracy IFE solutions u P generated from PPG IFE scemes. We carry out numerical experiments for te moderate coefficient discontinuity (β, β + ) = (, 0) using te same Cartesian meses T as we used in Example 4.. Errors of nonsymmetric, symmetric, and incomplete PPG IFE solutions u P are listed in Table 4.7, Table 4.8, and Table 4.9, respectively. Data in tese tables demonstrate convergence patterns of PPG IFE solutions wit midpointvalue degrees of freedom. Errors in semi-h norm seem to maintain an optimal rate O() for all tree PPG IFE scemes, altoug suboptimal convergence rate (4.95) as been teoretically establised for te PPG IFE scemes wit midpoint-value degrees of freedom. Errors in L norm seem to obey an optimal rate O( ) for symmetric and incomplete PPG IFE scemes but only suboptimal rate for nonsymmetric sceme. Errors in L norm can only acieve suboptimal rates for all of tese tree PPG IFE scemes. Comparisons of tese two PPG IFE scemes wit different types of degrees of freedom will be provided in Section Error Estimation for IPDG IFE Solutions In tis section, we discuss te error estimation of te interior penalty discontinuous Galerkin IFE solution u DG. We note tat te analysis for udg is quite similar to te PPG IFE solution ; ence, we omit some details in te following discussion. u I

100 Xu Zang Capter 4. Error Analysis 88 Table 4.7: Errors of NPPG IFE solutions u u P wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate E 8.467E E E E E E E E E E E E E E E E E E E E E E E Table 4.8: Errors of SPPG IFE solutions u u P wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate E 7.667E 3.976E E E E E E E E E E E E E E E E E E E E E E Table 4.9: Errors of IPPG IFE solutions u u P wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate E E 3.974E E E E E E E E E E E E E E E E E E E E E E We define te energy norm, still denoted by,ω, on te broken IFE space S DG (Ω) as

101 Xu Zang Capter 4. Error Analysis 89 follows ( v,ω = T T T β v v dx dy + b E b σ 0 b b α [v][v]ds ) /. (4.) Using similar arguments to tose for proving Lemma 4., we can sow tat te bilinear form a DG ɛ (, ) defined in (4.34) is coercive wit te above energy norm (4.). Lemma 4.8. Assume α in te bilinear form (4.34) and te energy norm (4.). Tere exists a positive constant κ suc tat κ v,ω a DG ɛ (v, v), v S DG (Ω), (4.3) for any positive σ 0 b if ɛ =, or for σ0 b large enoug if ɛ = or 0. Te following lemma provides an estimate of interpolation errors on non-interface edges. Lemma 4.9. Tere exists a constant C suc tat β (u I k,t u) n b 0,b C u,t, u H (T ), k = P, I. (4.4) for every non-interface element T T n Proof. By trace inequality (3.5) we obtain were b T is one of its edges. β (u I k,t u) n b 0,b C( u I k,t u,t + u I k,t u,t ). (4.5) Using standard scaling argument [44] to analyze te interpolation error u I,T k u, k = P, I, we ave u I,T k u m,t C m u,t, m =,. (4.6) Te estimate (4.4) follows from (4.5) and (4.6). An error estimate of te IPDG IFE scemes is given in te following teorem. Teorem 4.3. Assume u P Hint(Ω) 3 is te solution to te interface problem (.) - (.4), and u DG S DG (Ω) is te interior penalty discontinuous Galerkin IFE solution to (4.3) wit boundary condition (4.33a) or (4.33b), ten tere exists a constant C suc tat u DG u,ω C u 3,Ω. (4.7)

102 Xu Zang Capter 4. Error Analysis 90 Proof. Note tat te solution u P Hint(Ω) 3 is continuous inside Ω, and te flux β u n b is continuous across every interior edges b E. By Green s formula and (4.8) we ave {β u n b }[v ]ds b E b = β u n b v ds + {β u n b }[v ]ds b E b b b b E = β u n b v ds + {β u n b }[v ]ds + {β v n b }[u]ds b E b b b b b E b E = β u n T v ds, (4.8) T T T k,dg for every v S (Ω), k = P, I. Te equation (4.8) implies a DG ɛ (u, v ) = β u v dx β u n T v ds = (f, v ). (4.9) T T T T T T Hence, subtracting (4.9) from (4.3) we obtain a DG ɛ (u DG For every function w in S P,DG, v ) = a DG ɛ (u, v ), v S k,dg (Ω), k = P, I. (4.0) (Ω) = {w S DG (Ω) : w (M b ) = g(m b ), b E}, b or S I,DG (Ω) = {w S DG (Ω) : w ds = gds, b E}, b b b we subtract a DG ɛ (w, v ) from (4.0) ten a DG ɛ (u DG w, v ) = a DG ɛ (u w, v ), v We let v = u DG w a DG ɛ (u DG w, u DG k,dg S (Ω), w k,dg S (Ω) in te above equation, ten w ) = a DG ɛ (u w, u DG w ), w Applying te coercivity result (4.3) to (4.), we ave S k,dg (Ω), k = P, I. S k,dg (Ω), k = P, I. (4.) κ u DG w,ω β (u w ) (u DG w )dx T T T + {β (u w ) n b }[u DG w ]ds b E b + ɛ {β (u DG w ) n b }[u w ]ds b E b + σ 0 b b E b b [u w ][u DG α w ]ds Q DG + Q DG + Q DG 3 + Q DG 4. (4.)

103 Xu Zang Capter 4. Error Analysis 9 Following similar arguments (4.73) - (4.76) in analyzing Q troug Q 4, we obtain te following estimates for Q DG i, i =,, 3, 4: Q DG κ 6 udg w,ω + C (u w ) 0,Ω, (4.3) Q DG κ 6 udg w,ω + C b α {β (u w σ 0 ) n b } 0,b, (4.4) b E b Q DG 3 κ 6 udg w,ω + C ( T T ( (u w ) 0,T + (u w ) 0,T ) ), (4.5) Q DG 4 κ 6 udg w,ω +C T T (+α) ( T T ( (u w ) 0,T + (u w ) 0,T ) ). (4.6) Applying te estimates (4.3) - (4.6) to (4.), we ave κ 3 udg w,ω C (u w ) 0,Ω + C b E b α +C( + (+α) ) ( T T σ 0 b {β (u w ) n b } 0,b (4.7) ( (u w ) 0,T + (u w ) 0,T ) ). Ten, we let w be te IFE interpolant I Iu or IP u in (4.7) and apply interpolation error estimates (3.7) if k = P or (3.8) if k = I to obtain u DG Iu k,ω C u,ω + C b { } β (u I σ u) k n 0 b 0,b b E i b b +C b E n σ 0 b { β (u I k u) n b } 0,b. (4.8) We use te estimate (4.4) or (4.6) to bound te second term on te rigt and side of (4.8) and apply te estimate (4.7) to bound te tird term on te rigt and side of (4.8). Ten we obtain u DG I k u,ω C u,ω + C T T i ( ) u 3,Ω + u,t C u 3,Ω. (4.9) Te last inequality in (4.9) is due to te ypotesis (H5). Finally, te estimate (4.7) follows from applying te interpolation error estimate (4.80) if k = I or (4.0) if k = P and (4.9) to te following triangle inequality u DG u,ω u I k u,ω + u DG I k u,ω.

104 Xu Zang Capter 4. Error Analysis 9 Again, we use some numerical experiments to confirm our error analysis. Example 4.3. (IPDG IFE Scemes): In tis experiment, we test te accuracy of interior penalty DG IFE solutions u DG using nonconforming rotated Q IFE functions. In tis experiment, we andle te boundary conditions by matcing te midpoint values of IPDG IFE solutions wit te boundary function g following te approac (4.33b). We ave also conduct te experiment by matcing te integral values of te boundary condition (4.33a), and te numerical results are almost exactly te same; ence, tese results are skipped in te discussion below. Table 4.0, Table 4., and Table 4. contain te errors of nonsymmetric, symmetric, and incomplete IPDG IFE solutions, respectively. Convergence rates in semi-h norm seem to be optimal for all tese tree IPDG IFE scemes, wic confirms our error estimate (4.7). Moreover, convergence rates of nonsymmetric IPDG IFE solutions seem to be optimal in L norm. Convergence rates of symmetric and incomplete IPDG IFE scemes are observed suboptimal using coarse meses but tey tend to become optimal wen mes size is sufficient small. One may find similar observation for convergence rates in L norm. Table 4.0: Errors of NIPDG IFE solutions u u DG wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate E 6.467E 3.995E E E E E E E E E E E E E E E E E E E E E E Discussions on Related Scemes In tis section, we consider oter related IFE scemes and compare teir numerical performance wit PPG IFE scemes in Section 4.. Related IFE metods considered ere include tose using different computational scemes, suc as Galerkin IFE scemes, or tose using different IFE functions, suc as bilinear IFE functions [69]. Example 4.4. (Comparison wit Galerkin IFE Solutions): In tis example, we solve te interface problem using Galerkin IFE metod (4.5) wit nonconforming rotated Q IFE functions, and compare teir numerical performance to te PPG IFE metods.

105 Xu Zang Capter 4. Error Analysis 93 Table 4.: Errors of SIPDG IFE solutions u u DG wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate E E.405E 0.30E E E E E E E E E E E E E E E E E E E E E Table 4.: Errors of IIPDG IFE solutions u u DG wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate E E.395E 0.933E E E E E E E E E E E E E E E E E E E E E We start by using nonconforming rotated Q IFE functions wit midpoint-value degrees of freedom. We test te Galerkin IFE scemes for interface problem wose diffusion coefficient as a moderate jump, i.e., (β, β + ) = (, 0). Te exact solution is cosen te same as we used in Example 4.. Errors in L, L, and semi-h norms are listed in Table 4.3. Data in Table 4.3 suggest tat Galerkin IFE solutions cannot acieve optimal rates of convergence in L norm. For L, and semi-h norms, optimal convergence rates are observed up to mes size = /30 and tese rates deteriorate as we keep refining te mes. Tese observation indicates tat te Galerkin IFE sceme may ave difficulty in retaining te optimal convergence rates in L, and semi-h norms wen te mes size is sufficient small. Comparing data in Table 4.3 wit tose in Table 4.7 troug Table 4.9, we can observe tat adding penalty terms over interface edges leads to prominent improvement of te numerical accuracy in te sense tat convergence rates are closer to optimal and numerical errors are significantly reduced. A more illustrative comparison between Galerkin IFE solution errors and NPPG IFE solution errors can be found in Figure 4.. We observe tat errors in NPPG

106 Xu Zang Capter 4. Error Analysis 94 Table 4.3: Errors of Galerkin IFE solutions u u P wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate E 7.830E 3.966E E E E E E E E E E E E E E E E E E E E E E IFE sceme illustrated by green das line, are smaller tan errors in Galerkin IFE sceme illustrated by red solid line wen mes size is small enoug. Consequently, te convergence rates are elevated for NPPG IFE sceme due to te decrease of errors. Figure 4.: Comparison of errors in different nonconforming rotated Q IFE metods wit β =, β + = L Norm Errors L Norm Errors H Norm Errors Galerkin IFE (integral) NPPG IFE (integral) Galerkin IFE (midpoint) NPPG IFE (midpoint) Galerkin IFE (integral) 0 6 NPPG IFE (integral) Galerkin IFE (midpoint) NPPG IFE (midpoint) Galerkin IFE (integral) NPPG IFE (integral) Galerkin IFE (midpoint) NPPG IFE (midpoint) We also compare te point-wise error e P (X) = up (X) u(x) of te Galerkin IFE sceme and NPPG IFE sceme on te same mes containing elements. Error functions e P of tese two scemes are plotted in Figure 4.. Tese figures are generated by plotting te maximum error on eac element. It can be observed from te left plot tat te point-wise accuracy of te Galerkin IFE metod is quite poor around te interface and we suspect tis is because te discontinuity of IFE functions across interface could be large wit different configurations of te interface location and diffusion coefficient. By penalizing te discontinuity in IFE functions, te NPPG IFE sceme can produce muc better approximations around te interface as illustrated on te rigt plot in Figure 4.. Tese observations suggest tat adding penalty around interface can effectively reduce te IFE solution errors and improve te overall solution accuracy.

107 Xu Zang Capter 4. Error Analysis 95 Figure 4.: Point-wise error comparison of Galerkin solution and NPPG solution u P. Next we use nonconforming rotated Q IFE functions wit integral-value degrees of freedom in te Galerkin IFE sceme. Bot moderate discontinuity (β, β + ) = (, 0) and larger discontinuity (β, β + ) = (, 0000) of diffusion coefficients are considered in our experiments. For bot coefficient configurations, errors in L, L, and semi-h norms are listed in Table 4.4, and Table 4.5, respectively. Table 4.4: Errors of Galerkin IFE solutions u u I wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate 0.683E.395E.9585E E E E E E E E E E E E E E E E E E E E E E According to data in Table 4.4 and 4.5, optimal convergence rates are observed in L, L, and semi-h norms for Galerkin IFE solutions wit integral-value degrees of freedom. Comparing te data in Table 4.4 wit data in Table 4. troug Table 4.3 for te example wit moderate jump, we do not see major differences between Galerkin IFE solutions and PPG IFE solutions. Similar penomenon can be observed for example wit larger coefficient jump by comparing data in Table 4.5 wit data in Tables 4.4 troug 4.6. A more illustrative

108 Xu Zang Capter 4. Error Analysis 96 Table 4.5: Errors of Galerkin IFE solutions u u I wit β =, β + = N 0,,Ω rate 0,Ω rate,ω rate E E E E E E E E E E E E E E E E E E E E E E E E comparison of tese two scemes can be found in Figure 4. in wic te plots of errors in Galerkin IFE sceme illustrated by blue solid line and errors in NPPG IFE sceme illustrated by black das line almost coincide. We also compare te point-wise error e I (X) = ui (X) u(x) in Galerkin IFE sceme and NPPG IFE sceme in Figure 4.3. It can be observed tat tese two plots are very similar. In bot of tese plots, point-wise errors around interface are comparable to errors far away from te interface. All of tese comparisons indicate an interesting and also important feature, tat is, for nonconforming rotated Q IFE functions wit integral-value degrees of freedom, te penalty terms in te PPG IFE scemes enable us to derive optimal error bounds, but tey seem to be unnecessary for actual computation. How to teoretically prove tat te Galerkin IFE sceme wit nonconforming rotated Q IFE functions using integral-value degrees of freedom does converge optimally is an interesting future researc topic. Example 4.5. (Comparison wit Bilinear IFE Metods): In tis example, we compare te numerical performances of nonconforming rotated Q IFE metods wit bilinear IFE metods [69]. Bilinear IFE spaces are defined on Cartesian meses [70, 98]; ence it is natural to compare bilinear IFE metods wit nonconforming rotated Q IFE metods using te same meses. First we solve te elliptic interface problem wose exact solution is given in (3.8) using Galerkin sceme wit bilinear IFE functions. Related numerical errors are listed in Table 4.6. Data in tis table indicate tat errors in bilinear Galerkin IFE metods ave a suboptimal convergence rate in L norm. Also, te convergence rates in L and semi-h norms seem to be optimal for a moderately small mes size. As we continue refining meses, te rates of convergence in L and semi-h norms tend to degenerate. Note tat we observe similar penomenons in Galerkin IFE solutions using nonconforming rotated Q IFE functions wit midpoint-value degrees of freedom as listed in Table 4.3. We solve te elliptic interface problem again using PPG scemes wit bilinear IFE functions.

109 Xu Zang Capter 4. Error Analysis 97 Figure 4.3: Point-wise error comparison of Galerkin solution and NPPG solution u I. Errors in nonsymmetric, symmetric, and incomplete PPG scemes are listed in Table 4.7, Table 4.8, Table 4.9, respectively. We observe tat all tese PPG IFE solutions can acieve optimal convergence rates in L, L, and semi-h norms. Tese observations reaffirm te importance of penalizing discontinuities at interface edges for bilinear IFE solutions. Table 4.6: Errors of bilinear Galerkin IFE solutions u u wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate E E.706E E E E E E E E E E E E E E E E E E E E E E In Figure 4.4, we compare point-wise error e (X) = u (X) u(x) in Galerkin IFE solution and NPPG IFE solution using bilinear IFE functions. Tese plots are generated by plotting te maximum error on eac element in te mes containing elements. We observe tat te error in IFE solution produced by te Galerkin sceme is muc larger around te interface tan oter places. Tis migt be caused by te discontinuities of IFE functions on interface edges, wic can be quite large, if, for instance, te jump ratio in te coefficient is large. On te oter and, we see from te plot on te rigt in Figure 4.4 tat te IFE

110 Xu Zang Capter 4. Error Analysis 98 Table 4.7: Errors of bilinear NPPG IFE solutions u u wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate E E.67E 0.008E E E E E E E E E E E E E E E E E E E E E Table 4.8: Errors of bilinear SPPG IFE solutions u u wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate E 3.670E.703E E E E E E E E E E E E E E E E E E E E E E Table 4.9: Errors of bilinear IPPG IFE solutions u u wit β =, β + = 0. N 0,,Ω rate 0,Ω rate,ω rate E.6660E.709E E E E E E E E E E E E E E E E E E E E E E solution generated by NPPG sceme is significantly more accurate around te interface wic demonstrates again te effectiveness of te penalty terms introduced on interface edges.

111 Xu Zang Capter 4. Error Analysis 99 Figure 4.4: Point-wise error comparison of bilinear Galerkin IFE solution and NPPG IFE solution u. Galerkin IFE x NPPG IFE x x x Pointwise Error Pointwise Error y x y x In Figure 4.5, we provide a comparison of Galerkin IFE scemes using bilinear IFE functions and nonconforming rotated Q functions wit bot integral-value and midpoint-value degrees of freedom. Gauged in L norm, te nonconforming rotated Q Galerkin IFE metod wit integral-value degrees of freedom seem to be te most reliable one among tese tree metods, because its convergence rate seems to retain optimal O( ) and te error produced by tis metod is muc smaller tan te oter two IFE metods wen is sufficiently small. In L norm, te numerical solutions produced by bilinear IFE and nonconforming rotated Q IFE metods wit midpoint-value degrees of freedom seem to converge optimally over meses wit moderately small mes size. However, teir convergence rates in L norm seem to deteriorate on finer meses. On te oter and, te Galerkin IFE metod wit nonconforming rotated Q IFE functions wit integral-value degrees of freedom beave more stable and its optimal convergence rate is observed. In semi-h norm, we ave te similar observation. A comparison of NPPG IFE scemes using tese tree types of IFE functions is given in Figure 4.6. In L norm, te convergence rates of NPPG scemes using nonconforming rotated Q functions wit integral-value degrees of freedom and te bilinear IFE functions seem to retain optimal rate O( ). By comparing te magnitudes of errors, we observe tat bilinear IFE metods generate more accurate numerical solutions tan nonconforming rotated Q IFE metods in tis example. In L and semi-h norms, te performance of tese metods are quite similar. Optimal convergence is observed in NPPG IFE scemes using eac of tese tree IFE functions. Now let us take a closer look at tese IFE functions. We compare te discontinuity in global basis functions for tese IFE spaces in Figure 4.7. In tis figure, eac column contains plots

112 Xu Zang Capter 4. Error Analysis 00 Figure 4.5: Comparison of errors in different Galerkin IFE metods L Norm Errors L Norm Errors H Norm Errors Galerkin IFE (integral) Galerkin IFE (midpoint) Galerkin IFE (bilinear) Galerkin IFE (integral) Galerkin IFE (midpoint) Galerkin IFE (bilinear) Galerkin IFE (integral) Galerkin IFE (midpoint) Galerkin IFE (bilinear) Figure 4.6: Comparison of errors in different NPPG IFE metods L Norm Errors L Norm Errors H Norm Errors NPPG IFE (integral) NPPG IFE (midpoint) NPPG IFE (bilinear) NPPG IFE (integral) NPPG IFE (midpoint) NPPG IFE (bilinear) NPPG IFE (integral) NPPG IFE (midpoint) NPPG IFE (bilinear) of global basis functions in tese tree IFE spaces corresponding to one configuration of diffusion coefficient. In eac configuration, β is, but β + is, 5, 000, respectively. Plots in te first row represent alf of bilinear IFE global basis. In te second and tird rows, plots represent nonconforming rotated Q global basis wit midpoint-value and integralvalue degrees of freedom, respectively. Since tese IFE basis functions ave discontinuities on interface edges, we plot te difference of global IFE functions on te common interface edge in te fourt row. Plots in te first column represent te IFE global basis functions wen coefficient as no discontinuity i.e., β = β + =. As a matter of fact, tese global IFE bases become standard finite element global basis functions because of te consistency of IFE and FE functions stated in Teorem 3.. Consequently, in te plot at te bottom of tis column, te differences of tese global FE basis functions on te common interface edge are completely zero. Plots in te second column demonstrate te discontinuity of IFE global basis functions wit a moderate coefficient discontinuity (β =, β + = 5). Note tat all te tree global IFE basis functions ave discontinuity across te interface edge. In te plot at te bottom of tis

113 Xu Zang Capter 4. Error Analysis 0 Figure 4.7: Comparison of discontinuity for different IFE global basis functions wit fixed value β = and different values of β + =, 5, bilinear RQ(integral) RQ(midpoint).5 bilinear RQ(integral) RQ(midpoint).5 bilinear RQ(integral) RQ(midpoint) column, te blue line represents te jump of te nonconforming rotated Q global IFE basis wit midpoint-value degrees of freedom, wic as te value of zero (no discontinuity) at te midpoint. Te green line associates to te bilinear global IFE basis, wic vanises at two endpoints on te interface edge due to te continuity imposed at vertices. Te red line sows te jump of te nonconforming rotated Q global IFE basis wit integral-value degrees of

114 Xu Zang Capter 4. Error Analysis 0 freedom in wic te average integral value of te jump is zero. By direct comparison, we observe tat te nonconforming rotated Q global IFE basis wit te integral-value degrees of freedom as smaller discontinuity over te interface edge tan te oter two types of IFE bases. Plots in te tird column demonstrate te discontinuity of IFE global basis functions wit a large coefficient discontinuity (β =, β + = 000). Note tat te discontinuity of eac IFE global basis on te interface edge is larger tan tose wit a moderate coefficient discontinuity in te second column. Observing from te plot at te bottom of tis column, te discontinuity of te nonconforming rotated Q global IFE basis wit integral-value degrees of freedom is muc smaller compared to te oter two global IFE bases. We also note tat even toug te discontinuity in te coefficient increases greatly from te ratio of 5 to 000, te discontinuity of te nonconforming rotated Q global IFE basis wit integral-value degrees of freedom does not increase dramatically as te oter two IFE bases. Te comparison of te global IFE basis functions indicates tat te nonconforming rotated Q IFE functions wit integral-value degrees of freedom usually ave smaller discontinuity tan bilinear IFE functions and te nonconforming rotated Q IFE functions wit midpointvalue degrees of freedom. We believe te reason for tis penomenon is te tat te integralvalue degrees of freedom impose te continuity over te wole interface edge in a global sense compared wit te point-wise continuity locally imposed on te interface edge for te oter two types of IFE functions. In oter words, te discontinuity in a nonconforming rotated Q IFE function wit integral-value degrees of freedom is less prominent across an interface edge because te discontinuity is scattered trougout te interface edge wic leads to a less impact. Less discontinuity on interface edges is possibly te reason wy te partial penalization is unnecessary for nonconforming rotated Q IFE functions wit integral-value degrees of freedom.

115 Capter 5 Nonconforming IFE Metods for Elasticity Interface Problems So far we ave considered te application of nonconforming IFE metods to interface problems of scalar second order elliptic equations. In tis capter, we discuss te extension of nonconforming IFE metods for solving interface problems involving a system of PDEs. In particular, we plan to develop vector-valued nonconforming rotated Q IFE functions for solving planar elasticity interface problems in solid mecanics. Finite element metods ave been widely employed to calculate deformation and stress of elastic bodies subject to loads [5, 37, 45, 57]. If we model an object tat consists of multiple elasticity materials separated by a definite interface, te material parameters in elasticity equilibrium equations are usually discontinuous. Tis leads to te elasticity interface problems. Tis capter is organized as follows. In Section 5., we describe te elasticity interface model problems and review existing numerical metods for tese elasticity interface problems. In Section 5., we introduce vector-valued nonconforming rotated Q IFE spaces based on te integral-value degrees of freedom for te planar elasticity interface problems. Ten we investigate fundamental properties of tese new IFE spaces. In Section 5.3, we provide numerical experiments to demonstrate tat nonconforming rotated Q Galerkin IFE metods can solve te elasticity interface problem effectively. In particular, te metods can circumvent locking effect wen te elastic material is nearly incompressible. Some of te materials in tis capter ave been reported in articles [05, 08]. 03

116 Xu Zang Capter 5. Elasticity Interface Problems Introduction We consider te following planar elasticity pure displacement boundary value problem: σ(u) = f in Ω, (5.) u = g on Ω. (5.) Witout loss of generality, te solution domain Ω R is assumed to be a rectangle (or a union of several rectangles) formed wit two types of elastic materials separated by an interface Γ wic is assumed to be a smoot curve. Tat means te domain Ω is te union of two disjoint sub-domains Ω and Ω +, eac formed by one of te materials, suc tat Ω = Ω Ω + Γ, as illustrated in Figure 5.. Across te material interface Γ, te displacement and traction are assumed to be continuous, i.e., [u)] Γ = 0, (5.3) [ ] σ(u) n = 0. (5.4) Γ Figure 5.: Te domain of planar elasticity interface problems. Ω + Γ Ω Ω Here, we use letters in bold font to denote vector-valued functions and teir associated function spaces. Te function u(x) = (u (x, y), u (x, y)) t denotes te displacement vector at a point x = (x, y) in te elastic body Ω. Te function f = (f, f ) t represents te given body force and g = (g, g ) t is te given displacement on te boundary Ω. Te vector n denotes te outward normal of Γ. Also, we use te matrix function ɛ(u)) = (ɛ ij (u))) i,j to denote te linearized strain tensor ɛ ij (u) = ( ui + u ) j. (5.5) x j x i Let λ and µ denote te Lamé parameters given by λ = Eν ( + ν)( ν), µ = E ( + ν),

117 Xu Zang Capter 5. Elasticity Interface Problems 05 were E and ν are Young s modulus and Poisson s ratio, respectively. Ten, te stress tensor σ(u) = (σ ij (u)) i,j of a linear isotropic elastic material is assumed to fulfill te following linear constitutive relation: σ ij (u) = λ( u)δ ij + µɛ ij (u), were δ ij denotes te Kronecker delta suc tat { if i = j, δ ij = 0 if i j. (5.6) Te Lamé parameters λ, µ are assumed to be discontinuous across te interface Γ. For te sake of simplicity, we assume tat tey are piece-wise constants suc tat (λ(x), µ(x)) = { (λ, µ ), if x Ω, (λ +, µ + ), if x Ω +. (5.7) Elasticity interface problems appear in many applications suc as te topology optimization of solid structures wic, as one of te important applications of te elasticity interface problems, as been studied bot teoretically and numerically in te past decades, see [, 3] and te reference terein. Topology optimization is to determine certain features suc as size, location and sape of oles of te target domain for finding te optimal lay-out of certain structures in te prescribed domain. In te minimum compliance design problem [], for example, one desires to minimize te compliance of a structure to increase its stiffness. Consider a mecanic body occupying a domain wic is a subset of Ω R. Te reference domain Ω is cosen to define te load and boundary conditions. Te aim for tis design is to find te optimal coice of stiffness tensor E ijkl (x, y), wic is a variable over te domain. Define te following energy bilinear form, a(u, v) E ijkl (x, y)ɛ ij (u)ɛ kl (v) dx dy, (5.8) Ω wit te linearized strain tensor ɛ(u) defined in (5.5). Also define te load linear form, F (v) f v dx dy. Ten te minimum compliance problem is described as: subject to: Ω min F (u) u V a(u, v) = F (v), v V, (5.9) were V denotes te space of kinematically admissible displacement fields.

118 Xu Zang Capter 5. Elasticity Interface Problems 06 If te elastic body is isotropic [33], i.e., te stiffness tensor as no preferred direction (an applied force will give te same displacements no matter te direction in wic te force is applied), ten te stiffness tensor can be written as E ijkl (x, y) = λ(x, y)δ ij δ kl + µ(x, y)(δ ik δ jl + δ il δ jk ). (5.0) were δ ij is te Kronecker delta. Terefore te bilinear energy form (5.8) becomes ( ) a(u, v) = µ(x, y) ɛ(u) : ɛ(v) + λ(x, y) div(u) div(v) dxdy. (5.) Ω Hence te energy form (5.9) corresponds to te equilibrium equation (5.). Wen multiple material pases design problems are considered, see [57, 3], te Laḿe parameters λ(x, y), and µ(x, y) distribution are discontinuous, and tis set-up leads to te elasticity interface problems (5.) - (5.4). Oter applications for te elasticity interface problems include problems in te crystalline materials [36], te simulation in te microstructural evolution [79, 89], and te atomic interactions [55], etc. Tere are many numerical metods developed to solve elasticity interface problems. Conventional finite element metods [8, 45, 57], as one of te most popular approaces, can work satisfactorily provided tat meses are tailored to fit interfaces, known as body-fitting meses, as illustrated in te plot on te left in Figure.. Te body-fitting restriction makes conventional metods excessively expensive if interfaces evolve in a simulation. It is terefore attractive to develop numerical metods based on non-body-fitting meses, suc as Cartesian meses, as illustrated in te middle and on te rigt of Figure.. Tere ave been quite a few numerical metods developed to solve elasticity interface problems based on Cartesian meses. In finite difference formulation, Yang, Li, and Li ave developed an immersed interface metod for te planar linear elasticity interface problem [50, 5]. However, te linear systems arising from tis metod are nonsymmetric and become ill-conditioned as elastic materials become nearly incompressible, i.e., ν /. In finite element formulation, Hansbo and Hansbo ave proposed a bilinear finite element metod wic employees a Nitsce s idea and a modified weak formulation using weigted average traction across interfaces [66]. Becker, Burman, and Hansbo ave extended tis Nitsce finite element metod for incompressible elastic materials using a mixed formulation []. Hou, Li, Wang, and Wang [76] ave modified te traditional finite element metod by designing trial functions to be a piecewise polynomial to fit jump condition across te interface wile keeping te test functions independent of interface. Due to te inconsistency of trial and test function spaces, te resulting linear system in tis metod is nonsymmetric, altoug positive-definiteness can be guaranteed under certain conditions. IFE metods also ave been applied to solve te elasticity interface problems. In [6, 97], Gong, Li, and Yang ave proposed a linear IFE metod to solve elasticity interface problems on triangular meses. Point-wise convergence as been investigated in teir articles. Teir numerical results indicated tat linear IFE solutions can acieve at least an O() convergence in L norm. Recently, Lin and Zang ave developed a bilinear IFE metod [08]

119 Xu Zang Capter 5. Elasticity Interface Problems 07 based on te rectangular Cartesian meses. Tis article as studied accuracy of linear and bilinear IFE metods numerically, and te autors ave reported tat bot linear and bilinear IFE metods could converge optimally in bot L and H norms. Neverteless, bot of tese IFE metods ave limitations. First, te existence of tese IFE functions cannot be guaranteed for arbitrary configuration of elastic materials in an interface problem. Moreover, bot of tese conforming type IFE metods can only solve te elasticity interface problem satisfactorily for compressible elastic materials; once elastic materials become nearly incompressible, i.e., te Poisson s ratio ν of elastic material approaces 0.5, tese IFE metods encounter te volume locking effect [6]. Tis locking effect arises wen displacements of te elastic body are approximated by using te lowest-order conforming type finite elements even for solving non-interface problems. As we know, in eiter te linear or te bilinear IFE metod, te majority of elements do not intersect wit te material interface, were standard conforming type finite element functions are utilized; ence, te locking can be considered inevitable for tese IFE metods. Tere are many approaces developed to circumvent te locking effect, suc as te mixed finite element metods [4, 5, 6, 30,, 34], te nonconforming finite element metods [9, 53, 67, 88, 3, 53], and te discontinuous Galerkin metods [46, 68, 6, 45]. In te following sections in tis capter, we follow te route of nonconforming finite element metods to eliminate te locking effect for elasticity interface problems. We first construct te vector-valued nonconforming rotated Q IFE functions wit integral-value degrees of freedom for elasticity interface problems, and ten use tese IFE functions in displacement Galerkin formulation to solve te elasticity interface problems. 5. Vector-Valued Nonconforming IFE Spaces In tis section, we introduce te vector-valued nonconforming rotated Q IFE functions wit integral-value degrees of freedom for elasticity interface problems. Note tat for planar elasticity problem, te displacement u of an elastic body as two components u = (u, u ) t ; ence, eac finite element basis is a vector-valued polynomial function wic contains two components. Similar to te elliptic interface problem, standard vector-valued nonconforming rotated Q finite element functions are utilized on non-interface elements, and IFE functions are constructed only on interface elements wit appropriately interpreted interface jump conditions. In our following discussion, we consider te integral-value degrees of freedom only, since nonconforming rotated Q finite element functions wit integral-value degrees of freedom perform better and ave more desirable features tan te midpoint-value degrees of freedom in te elliptic interface problem as sown in Section 4.3. Consequently, we omit superscript I on FE/IFE functions and corresponding spaces.

120 Xu Zang Capter 5. Elasticity Interface Problems Nonconforming IFE functions On eac non-interface element T T n, we define te vector-valued nonconforming rotated Q finite element space S n (T ) as follows: S n (T ) = { Ψ T = (Ψ,T, Ψ,T ) t : Ψ j,t S n (T ), j =, }, (5.) were S n(t ) = Span{, x, y, x y } is te standard local rotated Q finite element space as introduced in Section.. Tere are eigt local basis functions Ψ j,t S n (T ), j =,, 8, on eac non-interface element T T n, wic are cosen to satisfy te following average integral-value restrictions: ( ) δij Ψ j,t (x, y) ds =, j =,, 3, 4, (5.3) b i b i 0 and Ψ j,t (x, y) ds = b i b i ( 0 δ i,j 4 ), j = 5, 6, 7, 8, (5.4) were b i, i =,, 3, 4 are four edges of T, as illustrated in Figure.. Ten te vector-valued nonconforming rotated Q local FE space S n (T ) on a non-interface element T, as defined in (5.), can be also written as follows S n (T ) = Span {Ψ j,t : j =,, 8}. (5.5) On an interface element T T i, we assume tat te interface curve Γ intersects te boundary of T at points D and E. Similar to te discussion in Section.4, we classify interface elements in two types: if D and E locate at two adjacent edges, we classify tis element as Type I interface element; if D and E locate at two opposite edges, we classify tis element as Type II interface element. Figure.6 provides illustrations for tese two types of interface elements. Te line segment DE separates T into two sub-elements T and T +. Witout loss of generality, we consider a typical interface element T = A A A 3 A 4 wit vertices ( ) ( ) ( ) ( ) 0 0 A =, A 0 =, A 0 3 =, A 4 =. (5.6) We label te four edges b i, i =,, 3, 4, of T as follows: We also assume tat b = A A, b = A A 4, b 3 = A 4 A 3, b 4 = A 3 A. (5.7) D = ( d 0 ) ( 0, E = e for a Type I interface element, were 0 < d, 0 < e, and ( ) ( ) d e D =, E = 0 )

121 Xu Zang Capter 5. Elasticity Interface Problems 09 for a Type II interface element, were 0 < d < and 0 < e <. Note tat every interface element can be mapped into one of te above interface elements via an ortogonal affine mapping. On eac interface element T T i, we use piecewise vector-valued polynomials to construct nonconforming local IFE functions. Specifically, on an interface element T, a vector-valued IFE function Φ T is piece-wisely defined as follows: ( ) ( φ Φ T (x) =,T (x, y) a φ = + b x + c y + d (x y ) ),T (x, y) a + b x + c y + d in T, (x y ) Φ T (x) = ( ) ( φ Φ + + T (x) =,T (x, y) a + φ + = + b + x + c + y + d + (x y ) ),T (x, y) a + + b + x + c + y + d + in T +. (x y ) (5.8) Note tat for every local IFE function Φ T defined in (5.8), tere are 6 coefficients, i.e., a s j, b s j, c s j, d s j, were j =,, and s = +,. Tese coefficients are determined by average integral-value of Φ T on edges togeter wit te interface jump conditions described as follows: Average integral values v i, i =,, 8, over te edges: ( ) vi Φ T (x) ds =, i =,, 3, 4. (5.9) b i b i v i+4 Tese integral values provide eigt restrictions. Displacement continuity at te intersection points D and E: Tese equations provide four restrictions. Traction continuity: Φ + T (D) = Φ T (D), Φ+ T (E) = Φ T (E). (5.0) DE σ(φ + T ) n DE ds = Tis equation provides two restrictions. DE σ(φ T ) n DE ds. (5.) Second derivative continuity: Φ + T x Tis equation provides two restrictions. = Φ T x. (5.) Overall we ave sixteen restrictions matcing te number of undetermined coefficients in (5.8). We will sow tat tese conditions (5.9) - (5.) are linearly independent so tat

122 Xu Zang Capter 5. Elasticity Interface Problems 0 tey can uniquely determine a vector-valued nonconforming rotated Q IFE function Φ T on an interface element T. Combining te conditions in (5.9)-(5.) leads to te following algebraic system to determine a s j, b s j, c s j, d s j, j =,, s = +, : M C C = V, (5.3) were C = ( ) a, a +, a, a +, b, b +, b, b +, c, c +, c, c +, d, d +, d, d + t, V = (v, v, v 3, v 4, v 5, v 6, v 7, v 8, 0, 0, 0, 0, 0, 0, 0, 0) t. (5.4) For a Type I interface element, te coefficient matrix M C = M I C = (mi i,j) 6 6 wit te normalization =, as te following form: M C = M I C = (5.5) d d d 0 0 d d d e e e e 0 0 e3 e d 0 0 d d 0 0 d d d e 0 0 e e e 0 0 e3 e d d d d d d d d e e 0 0 e e e e 0 0 e e were te components denoted by are specified as follows, m I 5,5 = de(λ + µ ), m I 5,6 = de(λ + + µ + ), m I 5,7 = d µ, m I 5,8 = d µ +, m I 5,9 = d µ, m I 5,0 = d µ +, m I 5, = deλ, m I 5, = deλ +, m I 5,3 = d e(λ + µ ), m I 5,4 = d e(λ + + µ + ), m I 5,5 = d( e λ + d µ ), m I 5,6 = d(e λ + d µ + ), m I 6,5 = d λ, m I 6,6 = d λ +, m I 6,7 = deµ, m I 6,8 = deµ +, m I 6,9 = deµ, m I 6,0 = deµ +, m I 6, = d (λ + µ ), m I 6, = d (λ + + µ + ), m I 6,3 = d(d λ e µ ), m I 6,4 = d(d λ + e µ + ), m I 6,5 = d e(λ + µ ), m I 6,6 = d e(λ + + µ + ). = (mii i,j) 6 6 for a Type II interface element wit normal- Te coefficient matrix M C = MC II ization = is given by M C = M II C = (5.6)

123 Xu Zang Capter 5. Elasticity Interface Problems d d d 0 0 d d d e e e 0 0 e e 0 0 e e e 3e e d 0 0 d d 0 0 d d d e 0 0 e e 0 0 e e 0 0 e e e 3e e d d d d d d d d 0 0 e e e e e e e e were te components denoted by are specified as follows:, m II 5,5 = λ + µ, m II m II 5,7 = dµ eµ, m II m II 5,9 = dµ eµ, m II m II 5, = λ, m II m II 5,3 = (d + e)λ + (d + 3e)µ, m II m II 5,5 = λ + (d e )µ, m II m II 6,5 = (d e)λ, m II m II 6,7 = µ, m II m II 6,9 = µ, m II m II 6, = (d e)(λ + µ ), m II m II 6,3 = (d e )λ µ, m II m II 6,5 = (d e)λ (d 3e)µ, m II 5,6 = (λ + + µ + ), 5,8 = dµ + + eµ +, 5,0 = dµ + + eµ +, 5, = λ +, 5,4 = (d + e)λ + (d + 3e)µ +, 5,6 = λ + (d e )µ +, 6,6 = (d e)λ +, 6,8 = µ +, 6,0 = µ +, 6, = (d e)λ + (d e)µ +, 6,4 = (d e )λ + + µ +, 6,6 = (d e)λ + + (d 3e)µ +. Te procedure of finding vector-valued nonconforming rotated Q IFE basis functions on te interface element T wose geometry is specified in (5.6) and (5.7) is similar to constructing scalar-valued IFE basis functions in Section.3 and.4. We let V = V j R 6, j =,, 8 be te j-t canonical unit vector suc tat v j =, and v i = 0 if i j. For eac vector V j, we can solve for C = C j from (5.3) wic contains te coefficients of an IFE function. Ten we use te values of C j in (5.8) to form te j-t vector-valued nonconforming rotated Q IFE local basis function, denoted by Φ j,t, for eiter Type I or Type II interface element. A typical vector-valued nonconforming rotated Q finite element local basis function Ψ 4,T on a non-interface element is plotted in Figure 5.. Te left plot is for te first component, and te one on te rigt is for te second component. We note tat te first component of Ψ 4,T is te scalar rotated Q finite element local basis function ψ4,t I as formulated in

124 Xu Zang Capter 5. Elasticity Interface Problems (.d), and te second component of Ψ 4,T is completely zero. As a comparison, te vectorvalued nonconforming rotated Q local IFE basis functions Φ 4,T on Type I and Type II interface elements are plotted in Figure 5.3 and Figure 5.4, respectively. Note tat, te second component of Φ 4,T is not completely zero because te IFE basis function Φ 4,T is constructed to satisfy te interface jump conditions (5.9) - (5.). Since te traction continuity (5.) involves bot components of a vector-valued IFE function, in general neiter of te two components in tese vector-valued IFE functions is zero. Figure 5.: A vector-valued nonconforming rotated Q finite element local basis function. Figure 5.3: A vector-valued nonconforming rotated Q IFE local basis function on a Type I interface element. On eac interface element T T i, we define te local IFE space Si (T ) by S i (T ) = Span {Φ j,t : j =,, 8}. (5.7)

125 Xu Zang Capter 5. Elasticity Interface Problems 3 Figure 5.4: A vector-valued nonconforming rotated Q IFE local basis function on a Type II interface element. Te global IFE space on Ω is ten defined as follows { S (Ω) = Φ (L (Ω)) : Φ T S n (T ) if T T n, Φ T S i (T ) if T T i ; } if T T = b, ten Φ T ds = Φ T ds. (5.8) 5.. Properties of Nonconforming Rotated Q IFE Spaces b b In tis subsection, we discuss basic properties of vector-valued nonconforming rotated Q IFE basis functions and te corresponding IFE spaces. Lemma 5.. (Continuity) On eac interface element T T i, te local IFE space Si (T ) is a subspace of C(T ). Proof. Note tat every IFE function Φ T S i (T ) is a piecewise vector-valued polynomial; ence, it suffices to sow Φ T is continuous across te line segment DE. Note tat te jump of te function [Φ T ] = Φ + T Φ T is linear because Φ T is made to satisfy te condition (5.). Ten, [Φ T ] = 0 follows from (5.0). Lemma 5.. (Partition of Unity) On eac interface element T T i, te vector-valued nonconforming rotated Q IFE basis functions Φ j,t S i (T ), j =,, 8, satisfy te following partition of unity property: 4 Φ j,t = j= ( 0 ), 8 Φ j,t = j=5 ( 0 ). (5.9)