A SADDLE POINT LEAST SQUARES METHOD FOR SYSTEMS OF LINEAR PDES. Klajdi Qirko

Transcription

1 A SADDLE POINT LEAST SQUARES METHOD FOR SYSTEMS OF LINEAR PDES by Klajdi Qirko A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics Summer 2017 c 2017 Klajdi Qirko All Rights Reserved

2 A SADDLE POINT LEAST SQUARES METHOD FOR SYSTEMS OF LINEAR PDES by Klajdi Qirko Approved: Louis Rossi, Ph.D. Chair of the Department of Mathematical Sciences Approved: George H. Watson, Ph.D. Dean of the College of Arts and Sciences Approved: Ann L. Ardis, Ph.D. Senior Vice Provost for Graduate and Professional Education

3 I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Constantin Bacuta, Ph.D. Professor in charge of dissertation I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Francisco J. Sayas, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Peter Monk, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Xiaoming Li, Ph.D. Member of dissertation committee

4 ACKNOWLEDGEMENTS I would like to thank my advisor, my mentor and friend, Dr. Constantin Bacuta, for his consistent support, his guidance and more than anything for being so patient and understanding! I would like to thank Dr. Fioralba Cakoni and her family for their unconditional love, support and company. Without Dr. Cakoni none of this would have happened. Special thanks go to our Graduate Director Dr. Francisco J. Sayas for being a friend, for always being supportive and for all the advice and constructive criticism. I would like to thank him for the guide and the base code upon which we built our numerical solver. I would like to thank Dr. Peter Monk for introducing me to the Finite Element Methods. I would like to express my gratitude to Ms. Deborah See for being so kind with me and always very helpful. I would also like to express my gratitude to the faculty and staff members at the Department of Mathematical Sciences at University of Delaware for their teaching and support throughout my graduate years. In particular, my sincere thanks go to Dr. Cristina Bacuta, Dr. David Colton, Dr. Dominique Guillot, Dr. Louis Rossi and Dr. Gilberto Schleiniger. I would like to deeply thank for the love and endless support, my family and friends, especially my mother Valentina, my father Vladimir, my wife Daniela, my sister Lona, my brothers Nertil and Besi, and my friends Andi x2 and Shao. Lastly I would like to thank my friends Irene, Tao, Pat, Bob for making my life at UD just better. The work was partially supported by NSF, DMS (Thank You!). iv

5 TABLE OF CONTENTS LIST OF TABLES vii LIST OF FIGURES ix ABSTRACT xi Chapter 1 INTRODUCTION SADDLE POINT PROBLEM THEORY AND DISCRETIZATION Notation and Background General Saddle-Point Problem Well-posedness SPP Approximation SPLS METHOD The General Mixed Formulation SPLS Discretization The SPLS Method Special Discrete Spaces No projection trial space Projection trial space Lumped inner product ITERATIVE SOLVER Algorithms v

6 5 DIV-CURL SYSTEM Div-Curl Background Preliminaries SPLS Discretization of the Div-Curl Problem SPLS Discretization, Construction of Special Discrete Spaces Approximability Numerical results SECOND ORDER ELLIPTIC PROBLEM Elliptic Problem Background Primal Mixed Variational Formulation SPLS Discretization for Second Order Elliptic Problems The no projection trial space leads to stability and approximability The projection trial space Numerical Examples A highly oscillatory coefficient example Interface problem examples Higher order flux recovery example CONCLUSIONS AND FUTURE WORK BIBLIOGRAPHY Appendix A FUNCTIONAL ANALYSIS RESULTS B PERMISSIONS vi

7 LIST OF TABLES 5.1 UCG for SPLS-P 1 -discretization Case 1) i) UCG for SPLS-P 1 -discretization Case 1) ii) UCG for SPLS-P 1 -discretization Case 1) iii) UCG for SPLS-P 1 -discretization Case 1) iv) Estimates for m h for different values of µ 0, Case I) v) UCG for SPLS-P 1 -discretization Case 2) i) UCG for SPLS-P 1 -discretization Case 2) ii) UCG for SPLS-P 1 -discretization Case 2) iii) UCG for SPLS-P 1 -discretization Case 2) iv) UCG for SPLS-P 1 -discretization Case 2) lump i) UCG for SPLS-P 1 -discretization Case 2) lump ii) UCG for SPLS-P 1 -discretization Case 2) lump iii) UCG for SPLS-P 1 -discretization Case 2) lump iv) UCG for SPLS-P 2 -discretization Case 1) i) UCG for SPLS-P 2 -discretization Case 1) ii) UCG for SPLS-P 2 -discretization Case 2) i) UCG for SPLS-P 2 -discretization Case 2) ii) Highly Oscillatory coefficients with no projection vii

8 6.2 Highly Oscillatory coefficients with orthogonal projection Interface problem with jump discontinuity c across x = Interface problem with singularity at (0, 0) Flux recovery on L-shaped domain viii

9 LIST OF FIGURES 6.1 Highly Oscillatory coefficients with n = 4: Approximated solution at each refinement level computed with no projection Highly Oscillatory coefficients with n = 16: Approximated solution at each refinement level computed with no projection Highly Oscillatory coefficients with n = 32: Approximated solution at each refinement level computed with no projection Interface problem: Kellog s; Graded refinement with κ = 0.022: Exact solution, computed approximation and zoomed mesh after each of the first 3 refinements Interface problem: Kellog s; Graded refinement with κ = 0.022: Exact solution, computed approximation and zoomed mesh after each of the last 3 refinements SPLS with adaptive mesh for Kellog s: Error rates Interface problem: Kellogs; Adaptive mesh refinement with BV estimator: Exact solution, computed approximation and zoomed mesh at 3 different levels of refinements (first 10 levels) Interface problem: Kellog s; Adaptive mesh refinement with BV estimator: Exact solution, computed approximation and zoomed mesh at 3 different levels of refinements (last 10 levels) SPLS with graded mesh for High Order Gradient Recovery on a L-shaped domain: Error rates Higher Order Gradient Recovery; Graded refinement with κ = 0.1: Exact solution, computed approximation and mesh after each of the first 3 refinements for the L-shaped domain ix

10 6.11 Higher Order Gradient Recovery; Graded refinement with κ = 0.1: Exact solution, computed approximation and mesh after each of the last 3 refinements for the L-shaped domain x

11 ABSTRACT We present a Saddle Point Least Squares (SPLS) method for solving variational formulations with different types of trial and test spaces. The general mixed formulation we consider assumes a stability LBB condition and a data compatibility condition at the continuous level. We expand on the Bramble-Pasciak s least square formulation for solving such problems by providing new ways to choose approximation spaces and new iterative processes to solve the discrete formulations. The proposed discretization method follows a general SPLS approach and has the advantage that a discrete inf sup condition is automatically satisfied for the standard choices of the test and trial spaces. For the proposed iterative processes a nodal basis for the trial space is not required and efficient preconditioning techniques that involve inversion only on the test space can be considered. Stability and approximation properties for two choices of discrete spaces are investigated. Applications of the new approach include discretization of first order systems of PDEs, such as div curl systems, second order problems with highly oscillatory coefficient, interface problems, and higher order approximation of the flux for elliptic problems with smooth coefficients. xi

12 Chapter 1 INTRODUCTION Over the last two decades, there have been many advances in applying finite element least squares methods to approximate first order systems of PDEs, [18, 19, 20, 29, 30, 31, 32, 33, 62, 63, 65]. However, when compared to the more established field of finite element methods for elliptic problems, a unified theoretical framework for least squares approximation of solutions of first order systems of PDEs is missing. Our proposed framework provides powerful preconditioning techniques, has efficient error estimators, is suitable to multilevel techniques, and leads to robust and easy to implement solvers. The method is related with the Bramble-Pasciak least squares approach introduced in [23], and it combines known theory and discretization techniques for approximating elliptic problems and symmetric saddle point problems, see e.g.,[2, 6, 7, 16, 24, 25, 26, 47, 66, 72, 77, 80]. For the applications we consider, the solution spaces are L 2 type spaces, and the data can reside in weak negative norm spaces. We require that the test spaces be H 1 type spaces with suitable boundary conditions, and the discrete test spaces be conforming finite element spaces built using the action of the continuous differential operator associated with a given problem. Among the advantages of the method are the following: the discretization leads to saddle point variational formulation with automatic discrete inf sup condition, and the assembly of a stiffness matrix for the trial space is avoided. The general abstract problem that we plan to discretize using a Saddle Point Least Squares Method (SPLS) is: Find p Q such that b(v, p) = f, v, for all v V or B p = f, (1.0.1) 1

13 where V and Q are infinite dimensional Hilbert spaces and b(, ) is a continuous bilinear form on V Q, that satisfies a standard inf sup condition, and f V belongs to the range of B. In the special case when the operator B associated with the form b(, ) is injective, our method can be viewed as a conforming Petrov-Galerkin method. From the point of view of choosing the discrete spaces, our method can be characterized as the dual of the Discontinuous Petrov-Galerkin (DPG) method introduced by Demkowicz and Gopalakrishnan in [41, 42]. While both methods have strong connections with the least squares and minimum residual techniques, the proposed discretization process stands apart from the DPG approach because of the opposite order and different ways in which the trial and test spaces are chosen. We apply the method to two very important classes of problems that fit in the first order systems of PDEs framework. The first one is the div-curl system. This system governs, for example, static electromagnetic fields, and incompressible irrotational fluid flows. The div-curl system is also fundamental from a theoretical point of view, since the Stokes equations and the incompressible Navier-Stokes equations written in the first-order velocitypressurevorticity formulation, as well as the Maxwell equations consist of two div-curl systems, see [55, 21], Chapter 4. The second one is the second order elliptic problem which appears in a variety of application in fields such as material science, fluid dynamics and biological processes, see [3, 49, 60, 12, 54, 51, 71, 78, 82, 83, 45, 58]. They are used in mathematical modeling of unidirectional composite materials, flow in porous media, heat conduction in heterogenous materials, electromagnetic field propagation on heterogeneous media, etc. We apply the method to approximate the solution of second order problems with highly oscillatory coefficient, interface problems, and higher order approximation of the flux for elliptic problems with smooth coefficients. The rest of the thesis is organized as follows: 2

14 In Chapter 2 we summarize the classical SPP theory. We introduce the notation and some important results on approximation. In Chapter 3 we formulate the SPLS method. We start with the general problem that the method solves and the necessary results that justify the method and its terminology. Secondly we discretize the problem using SPLS and summarize the main steps of the method. Lastly we elaborate on the test and trial spaces. In Chapter 4 we present the numerical methods used in the iterative process and their convergence estimates. In Chapter 5 we present the first application of the SPLS method on the div-curl system. We start with some background on the problem, the notation and the spaces and present wellposedness results. Then, we proceed with introducing the specific problem we are going to discretize using the the SPLS method. After that, we define the corresponding discrete spaces and present their approximation properties. To conclude, we show some numerical results we obtained. Chapter 6 follows with the second application, on the second order elliptic problem. As before, we start with some general background on the problem followed by the introduction of the specific problem we are going to apply the SPLS method. We proceed with defining the test and trial spaces, considering several cases for the latter. Next we prove some approximation results and present all the numerical experiments and corresponding results for the three main problems considered, second order elliptic with highly oscillatory coefficients, interface problem and higher order gradient flux recovery. The thesis ends with Chapter 7 which includes some conclusions and open projects left for the future. 3

15 Chapter 2 SADDLE POINT PROBLEM THEORY AND DISCRETIZATION In this chapter we formulate the general framework for the saddle point problem and introduce the notation to be used throughout the thesis. 2.1 Notation and Background In this section, we start with a review of the classical SPP theory and introduce the spaces, the operators and the norms for the general abstract case. We let V and Q be two Hilbert spaces with inner products a 0 (, ) and (, ) respectively, with the corresponding induced norms V = = a 0 (, ) 1/2 and Q = = (, ) 1/2. The dual pairings on V V and Q Q are denoted by,. Here, V and Q denote the duals of V and Q, respectively. With the inner products a 0 (, ) and (, ), we associate the operators A : V V and C : Q Q defined by Au, v = a 0 (u, v) for all u, v V and Cp, q = (p, q) for all p, q Q. The operators A 1 : V V and C 1 : Q Q are called the Riesz canonical isometries and satisfy the following properties: a 0 (A 1 u, v) = u, v, A 1 u V = u V, u V, v V, (2.1.1) (C 1 p, q) = p, q, C 1 p = p Q, p Q, q Q. (2.1.2) 4

16 Let L(V V; R) and L(V Q; R) be the sets of bounded, bilinear forms on the respective spaces. Next, we suppose that b(, ) L(V Q; R), satisfying the inf sup condition. More precisely, we assume that and inf sup p Q v V sup p Q sup v V b(v, p) p v = m > 0 (2.1.3) b(v, p) p v = M <. (2.1.4) Here, and throughout this thesis, the inf and sup are taken over nonzero vectors. With the form b, we associate the linear operators B : V Q and B : Q V defined by Bv, q = b(v, q) = B q, v for all v V, q Q. Let V 0 be the kernel of B or C 1 B, i.e., V 0 = Ker(B) = {v V Bv = 0} = {v V C 1 Bv = 0}. Due to (2.1.4), V 0 is a closed subspace of V. We notice here that C 1 B : V Q and A 1 B : Q V are dual to each other in the Hilbert sense, and consequently, the Schur complement S 0 := C 1 BA 1 B is a symmetric operator. In addition, S 0 is a positive definite operator with σ 0 (S 0 ) [m 2, M 2 ], see Lemma A.0.1 or [4]. 5

17 2.2 General Saddle-Point Problem In this section we review the general Saddle-Point problem and its approximation properties Well-posedness Let V and Q be two Hilbert Spaces and consider a 0 L(V V; R), symmetric positive definite and coercive, b L(V Q; R). The abstract general saddle point problem is: Given f V, g Q, find u V and p Q such that: a 0 (u, v) + b(v, p) = f, v for all v V, b(u, q) = g, q for all q Q. (2.2.1) Introducing the operators A, B and B defined as above, this problem is equivalent to: Find u V and p Q such that: Au + B p = f, Bu = g. (2.2.2) Under the above framework, problem (2.2.1) is well posed if and only if m > 0, inf sup p Q v V b(v, p) p v m. (2.2.3) For the proof see [50, 27, 73, 48] SPP Approximation This subsection introduces the conforming approximation of (2.2.1). Let V h be a subspace of V and let M h be a subspace of Q. Assume that V h and M h are finite dimensional. Then the approximate problem is: Find (u h, p h ) V h M h such that a(u h, v h ) + b(v h, p h ) = f h, v h for all v h V h, b(u h, q) = g h, q h for all q h M h, (2.2.4) 6

18 where f h V h and g h M h are defined by f h, v h = f, v h for all v h V h, g h, q h = g, q h for all q h M h. (2.2.5) Assume that the following discrete inf sup condition holds for the pair (V h, M h ), inf p h M h The matrix, or operator, form of (2.2.4) is: b(v h, p h ) sup v h V h p h v h = m h > 0. (2.2.6) A h u h + B h p h = f h, B h u h = g h. (2.2.7) It is well known from [22, 24, 77, 80] that, under the assumption (2.2.6), problem (2.2.4) has a unique solution (u h, p h ) and u u h + p p h C ( ) inf u v h + inf p q h m h v h V h q h M h where (u, p) is the solution of the continuous problem (2.2.1). See [50]. (2.2.8) 7

19 Chapter 3 SPLS METHOD In this chapter we formulate the SPLS method, its derivation and some general theoretical results. We assume the notation, spaces and operators introduced in the previous chapter. 3.1 The General Mixed Formulation Assume that b L(V Q; R), satisfying (2.1.3) and (2.1.4), and let f V be given. Many variational formulations of PDEs, including first order systems, can be written in the mixed form: Find p Q such that b(v, p) = f, v for all v V. (3.1.1) The operator form of (3.1.1) is to solve the following equation for p, B p = f. (3.1.2) The existence and the uniqueness of (3.1.1) was first studied by [69]. Then it appears in the notes of Aziz and Babuška in [2] and is known as the Babuška s Lemma. Lemma (Babuška) Let b L(V Q; R) be a bilinear form satisfying (2.1.3) and (2.1.4), and let f V. The problem: Find p Q such that b(v, p) = f, v for all v V (3.1.3) has a unique solution if and only if f, v = 0 for all v V 0. (3.1.4) 8

20 If (3.1.4) holds, and p is the solution of (3.1.3), then m p p S0 := (S 0 p, p) 1/2 = f V M p. (3.1.5) A proof of the Lemma can be found in [2, 4]. To justify our saddle point least squares terminology, we write (3.1.2) in the equivalent form A 1 B p = A 1 f. (3.1.6) Using that the Hilbert transpose of A 1 B : Q V is the operator C 1 B : V Q, the normal equation for solving (3.1.6) in the least square sense is C 1 BA 1 B p = C 1 BA 1 f. (3.1.7) Since S 0 := C 1 BA 1 B is a symmetric positive definite operator, the problem (3.1.7) has a unique solution p. Then, we consider the well posed saddle point problem: Find (u, p) (V, Q) such that a 0 (u, v) + b(v, p) = f, v for all v V, b(u, q) = 0 for all q Q. (3.1.8) One can immediately check that p is the unique solution of (3.1.7) if and only if (u, p) is the unique solution of (3.1.8), where u = A 1 (B p f). Proposition In the presence of the continuous inf sup condition (2.1.3) and the compatibility condition (3.1.4), we have that p is the unique solution of (3.1.1) if and only if (u = 0, p) is the unique solution of (3.1.8). Remark Even in the absence of the compatibility condition (3.1.4), but assuming the continuous inf sup condition (2.1.3), we have that (3.1.8) has a unique solution regardless of the fact that (3.1.1) is well posed or not, see [4]. The above arguments justify the name of the system (3.1.8) as the SPLS variational formulation of (3.1.1). 9

21 3.2 SPLS Discretization In light of Propsition 3.1.2, we are led to the corresponding SPLS discretization of (3.1.1). We let V h V and M h Q be finite dimensional approximation spaces and consider the restrictions of the forms a 0 (, ) and b(, ) to the discrete spaces V h and M h. Assume that the discrete inf sup condition, (2.2.6) holds for the pair (V h, M h ). We define the discrete operators A h, C h, f h, B h, and Bh corresponding to A, C, f, B, and B as follows: A h u h, v h = a 0 (u h, v h ) for all u h, v h V h, C h p h, q h = (p h, q h ) for all p h, q h M h, f h, v h = f, v h for all v h V h, B h u h, q h = b(u h, q h ) for all u h V h, q h M h, Bh p h, v h = b(v h, p h ) for all v h V h, p h M h. (3.2.1) We define V h,0 to be the kernel of B h, Furthermore, V h,0 := {v h V h b(v h, q h ) = 0, for all q h M h }. V h,1 := Vh,0 = A 1 h B hm h and S h,0 := C 1 h B ha 1 h B h. Remark If V h,0 V 0, then the compatibility condition (3.1.4) implies a discrete compatibility condition. (2.2.6), the problem: Find p h M h such that has a unique solution. Consequently, under the discrete stability assumption b(v h, p h ) = f, v h for all v h V h, In general, V h,0 is not contained in V 0. Thus, the compatibility condition (3.1.4) does not imply a discrete compatibility condition, and the problem: Find p h M h such that b(v h, p h ) = f, v h for all v h V h, or B h p h = f h, (3.2.2) 10

22 might not be well posed. Nevertheless, under the assumption (2.2.6), the following saddle point discrete variational problem: Find (u h, p h ) V h M h such that a 0 (u h, v h ) + b(v h, p h ) = f, v h for all v h V h, b(u h, q h ) = 0 for all q h M h, (3.2.3) always has a unique solution. Using the corresponding discrete operators, the problem (3.2.3) is equivalent to: Find (u h, p h ) V h M h such that u h = A 1 h (f h Bh p h), C 1 h B ha 1 h B h p h = C 1 h B ha 1 h f h. (3.2.4) Since the operator S h,0 = C 1 h B ha 1 h B h is symmetric positive definite on M h, it is invertible, and the second equation of (3.2.4) has a unique solution p h. Thus, one way to solve the system (3.2.3), would be to solve the second equation of (3.2.4) for p h, and then find u h from the first equation of (3.2.4). Since C 1 h B h and A 1 h B h are dual to each other in the Hilbert sense and the problem (3.2.2) is also equivalent to A 1 h B h p h = A 1 f h, we have that the solution of the second equation of (3.2.4) is in h fact the least squares solution of (3.2.2). The component u h of the solution of (3.2.3) or (3.2.4) becomes the representation on V h of the residual associated with the least squares solution of (3.2.2). Another reason why the saddle point least squares discretrization of (3.1.1) is the variational formulation (3.2.3) is that (3.2.3) is the natural discretization of the SPLS formulation (3.1.8) of (3.1.1). Using that (3.2.3) is the discrete variational formulation of (3.1.8), and based on the classical error analysis for SPP theory, we can find standard estimates for 0 u h and p p h, see [22, 24, 77, 80]. If we assume discrete stability, then the second component p h of the solution of (3.2.3) provides a good approximation to the least squares solution p of (3.1.1) even in the absence of the compatibility condition (3.1.4) from (2.2.8). In the case when the compatibility condition (3.1.4) 11

23 holds, we are dealing with a special saddle point problem, and a sharp error estimate for p p h can be proved using the Xu-Zikatanov argument in [80]. Theorem Let b L(V Q; R) satisfy (2.1.3) and (2.1.4), and assume that f V is given and satisfies (3.1.4). Assume that p is the solution of (3.1.1) and V h V, M h Q are chosen such that the discrete inf sup condition (2.2.6) holds. Then, if (u h, p h ) is the solution of (3.2.3), the following error estimate holds: 1 M u h p p h M m h inf p q h. (3.2.5) q h M h Proof. Let p be the solution of (3.1.1) and assume that(u h, p h ) is the solution of (3.2.3). First, we notice that the operator T h : Q Q defined by T h p = p h is linear and idempotent. To prove that T h is idempotent we consider the problem: Find (ũ h, p h ) V h M h such that a 0 (ũ h, v h ) + b(v h, p h ) = b(v h, p h ) for all v h V h, b(ũ h, q h ) = 0 for all q h M h. (3.2.6) Due to the assumption (2.2.6), we get that (3.2.6) has a unique solution and by noticing that (ũ h, p h ) = (0, p h ) solves the problem, we can conclude that T h p h = p h and consequently, Th 2 = T h. According to Kato [56], and Xu and Zikatanov [80], we have I T h L(Q,Q) = T h L(Q,Q). Next, for any q h M h we have p p h = (I T h )p = (I T h )(p q h ) I T h p q h = T h p q h. (3.2.7) To estimate T h we use (2.2.6): T h p 1 b(v h, T h p) sup m h v h V h v h = 1 b(v h, p h ) sup. m h v h V h,1 v h 12

24 Solving for b(v h, p h ) from the first equation of (3.2.3), and using the fact that f, v h = b(v h, p) for all v h V h, we further get T h p 1 b(v h, p) a 0 (u h, v h ) sup m h v h V h,1 v h = 1 b(v h, p) sup m h v h V h,1 v h (3.2.8) M m h p. Thus, T h M m h, and from (3.2.7) we obtain the right inequality of (3.2.5). To prove the left inequality of (3.2.5), from (3.1.1) and the first equation of (3.2.3) we get b(v h, p p h ) = a 0 (u h, v h ) for all v h V h. Then, a 0 (u h, v h ) u h = sup v h V h v h b(v h, p p h ) = sup v h V h v h which concludes the validity of the estimate (3.2.5). M p p h, The right part of the estimate (3.2.5) is an improvement of the similar estimate presented in [23] that provides a bound that is linear in m 2. This improvement is significant if m h depends on h. h Remark All the considerations made so far in this section make sense if the form a 0 (, ), as an inner product on V h, is replaced by another inner product which gives rise to an independent of h equivalent norm on V h. Certainly, the definition of A h, S h,0, and m h, will change accordingly with the new inner product, but the error estimate (3.2.5) remains valid with different estimating constants that factor in the norm equivalence constants. This observation leads to an efficient preconditioning approach for SPLS discretization. More precisely, if the Uzawa algorithm is involved to solve (3.2.3), then the action of A 1 h can be replaced by the action of any equivalent preconditioner. 13

25 satisfies From the first equation of (3.2.4) and (3.2.5) we get that the solution (u h, p h ) u h = f h B hp h V h M 2 m h inf p q h, q h M h where f h is the functional v h f, v h on V h. This gives an estimate for the residual associated with the discrete equation (3.2.2). 3.3 The SPLS Method Now we introduce the five steps that define the saddle point least squares discretization method: Step 1) Write the general problem (3.1.1) as a saddle point least squares formulation (3.1.8), using the natural inner product a 0 (, ) on V V. Step 2) Choose a standard conforming approximation space V h for the variational space V. Step 3) Construct a discrete trial space M h Q using the operator B associated with the form b(, ). For example, take M h := C 1 BV h, or M h := Q h C 1 BV h, where C 1 is the Riesz representation operator for the space Q and Q h is an orthogonal projection from Q to a subspace M h Q. The pair (V h, M h ) will automatically satisfy a discrete inf sup condition. Step 4) Write (3.2.3) - the discrete version of the SPLS formulation and, if available, replace a 0 (, ) by an equivalent form a prec (, ) on V h V h. Step 5) Solve the new discrete SPLS problem using an Uzawa type iterative process that requires only the action of A 1 h of C 1 B or Q h C 1 B. (or the action of a preconditioner), and the action 3.4 Special Discrete Spaces Let V h be a finite element subspace of V. Assume that the action of C 1 at the continuous level is easy to obtain. 14

26 3.4.1 No projection trial space The first choice for M h Q is M h := C 1 BV h. (3.4.1) In this case we have that V h,0 V 0. A discrete inf sup condition holds: Indeed, using a generic function p h = C 1 Bw h M h, with w h V h,0 (a finite dimensional space), we have inf p h M h b(v h, p h ) sup v h V h p h v h = inf w h V h,0 (C 1 Bv h, C 1 Bw h ) sup v h V h p h v h C 1 Bw h inf w h Vh,0 w h := m h,0 > 0. Approximability: Using that V h,0 V 0 and Remark 3.2.1, the variational formulation (3.2.2) is well posed, has a unique solution p h M h, and using Proposition for the discrete pair (V h, M h ), we have that (u h = 0, p h ) is the solution of (3.2.3). In this case, if p is the solution of (3.1.1) and p h is the solution of (3.2.2) (or the SPLS solution of (3.2.3)), from (3.1.1) and (3.2.2) we obtain 0 = b(v h, p p h ) = (C 1 Bv h, p p h ) for all v h V h. Thus, we simply have that p h is the orthogonal projection of p onto M h, and consequently, p p h = inf p q h. (3.4.2) q h M h Using that C 1 B is onto Q we can represent p = C 1 Bw for some w V, and write q h = C 1 Bv h for some v h V h. Thus, we have p p h = inf v h V h C 1 Bw C 1 Bv h M inf v h V h w v h. (3.4.3) The estimate (3.4.2) gives optimal approximability and (see also (3.2.5)) it is independent of the inf sup constant m h. The estimate (3.4.3) reveals that even though the trial space M h is not a standard chosen approximation space, the approximability of the solution p Q with discrete functions in M h, reduces to approximability of the (best) representation w V of p by discrete functions in the standard approximation test space V h. 15

27 3.4.2 Projection trial space Next, we will present a method to address the lack of stability of the approximation spaces. Let M h be a finite dimensional subspace of Q that has good approximability properties. Typical examples of spaces M h are the spaces of piecewise polynomials. We equip M h with an inner product that could differ from the restriction of the Q inner product on M h, but induces an equivalent norm (independent of h), e.g. the lumping inner product we introduce below, in subsection For convenience, we denote the inner product on M h by (, ). If Qh : Q M h is the orthogonal projection onto M h, we simply define the space M h by M h := Q h C 1 BV h. We consider the restriction of the form a 0 (, ) to V h V h and the restriction of b(, ) to V h M h, and define the discrete operators A h, C h, B h, and B h for the pair (V h, M h ). For any q h M h, v h V h, we have b(v h, q h ) = B h v h, q h = (C 1 h B hv h, q h ). On the other hand, since q h M h Q, and v h V h V, we have b(v h, q h ) = (C 1 Bv h, q h ) = ( Q h C 1 Bv h, q h ). From the above identities, we obtain C 1 h B hv h = Q h C 1 Bv h for all v h V h. (3.4.4) This implies that C 1 h B h is onto M h and, using that V h and M h are finite dimensional spaces, a discrete inf sup condition (2.2.6) always holds in (V h, M h ), for some m h > 0 that might depend on h. 16

28 Stability: We investigate the stability of (V h, Q h C 1 BV h ) based on the stability of the no projection case (V h, C 1 BV h ). Theorem Assume that the following condition holds Q h q h c q h, for all q h C 1 BV h, (3.4.5) with a constant independent of h. Then, V h,0 V 0 and the variational formulation (3.2.2) has a unique solution p h M h. Using Proposition for the discrete pair (V h, M h ), we have that (u h = 0, p h ) is the solution of (3.2.3). The stability of {(V h, C 1 BV h )}, with m h,0 > c 0 > 0 independent of h, implies stability for the pair {(V h, Q h C 1 BV h )}. Proof. Using a generic function p h = Q h C 1 Bw h M h, with w h V h,0 we have and (3.4.5), m h := inf p h M h = inf w h V h,0 = c m h,0. b(v h, p h ) sup v h V h p h v h = inf w h V h,0 ( sup Q h C 1 Bv h, Q h C 1 Bw h ) v h V h p h v h Thus, problem (3.2.3) has a unique solution (u h, p h ). (C 1 Bv h, sup Q h C 1 Bw h ) v h V h p h v h c inf w h V h,0 C 1 Bw h w h Any of the three Uzawa type algorithms presented in Chapter 4 can be applied to approximate the aforementioned unique solution. In addition, in light of (3.4.4), we have that the residual q j for each Uzawa type algorithm satisfies q j = C 1 h B hu j = Q h C 1 Bu j. Consequently, the computation of q j involves the computation Bu j, the action of C 1 at the continuous level (often just a multiplication operator) to find C 1 Bu j, and a projection onto the space M h - that is usually a standard finite element space. Thus, for each of the Uzawa type algorithms, a basis for solving the variational problems associated with q j. M h is not needed for 17

29 Approximability: Due to Theorem 3.2.2, in order to expect small discretization error p p h, besides stability, one needs to investigate the minimization problem inf p q h in the special case when M h is a proper subspace of M h and might q h M h not be a standard approximation space for functions in Q. The continuous inf sup condition (2.1.3) that we assume, guaranties that A 1 B is injective and has closed range, see Lemma A.0.1 (iii). Thus, the dual operator C 1 B is onto Q, and, for any p Q we can represent p = C 1 Bw for some w V and write q h = Q h C 1 Bv h with v h V h. Next we have p q h = C 1 Bw Q h C 1 Bv h C 1 Bw Q h C 1 Bw + Q h C 1 Bw Q h C 1 Bv h (3.4.6) C 1 Bw Q h C 1 Bw + M w v h. Combining the above estimate with Theorem for p = C 1 Bw we get p p h M m h C 1 Bw Q h C 1 Bw + M 2 m h inf w v h. (3.4.7) v h V h Consequently, in order to obtain good SPLS approximation for the solution p of (3.1.1), it would be enough to ask for some regularity of a solution w of C 1 Bw = p, for approximation properties of the test space V h, and for approximation properties of the projection Q h : Q discrete stability presence. M h. Note that the argument remains valid regardless of the Lumped inner product A typical choice for the space M h used to define the projection Q h is the space of continuous piecewise polynomials of a given degree with respect to a mesh T h. On M h we can choose the standard inner product or we can consider the lumping inner product introduced in [8]. To define the lumping inner product, assume that {φ 1, φ 2,..., φ m } is a nodal basis for M h. We can define (, ) l by (φ i, φ j ) l := δ j i (1, φ i ), i, j = 1, 2,..., m, (3.4.8) 18

30 and extend it to M h M h by ( m α i φ i, i=1 ) m β j φ j := j=1 l m α i β i (1, φ i ). (3.4.9) i=1 Using the identity (3.4.4), see [8], the computation of C 1 h B hv h becomes C 1 h B hv h = m i=1 (C 1 Bv h, φ i ) φ i = (1, φ i ) m i=1 b(v h, φ i ) (1, φ i ) φ i. We note here that by using the lumping inner product, one avoids mass matrix inversion at each iterative step. The meaning and importance of the previous statement will become more clear after the next chapter. 19

31 Chapter 4 ITERATIVE SOLVER In this chapter we introduce the iterative solvers we used to compute the numerical solutions and prove their convergence. 4.1 Algorithms We are going to use an iterative process to solve (3.2.3). Let S h : M h M h, be the discrete Schur complement defined by S h := B h A 1 h B h. It is easy to check that S h is a symmetric and positive definite operator on M h. We have that (, ) Sh := (S h, ) is another inner product on M h with the induced norm denoted by Sh. It is well known that the lowest and the largest eigenvalues of S h are m 2 h and M 2 h, respectively. Thus, m h q h q h Sh = (S h q h, q h ) 1/2 M h q h for all q h M h. (4.1.1) Remark On V h we consider the same norm as the norm on V. The inner product ((, )) on V h V h is not the restriction of the inner product a(, ), and is used only for defining the discrete operators A h and Bh. In what follows, we will need in fact only to work with A 1 h B h : M h V h and S h = B h A 1 h B h which are independent of the choice of the inner product ((, )). Indeed, if q h M h is arbitrary, then w h = A 1 h B h q h is the unique solution of the problem a(w h, v) = b(v, q h ) for all v V h, and S h q h = B h A 1 h B h q h = B h w h does not depend on the inner product ((, )). We also note that if r h M h is arbitrary and v h = A 1 h B h r h, then a(w h, v h ) = b(v h, q h ) = (B h A 1 h B hr h, q h ) = (S h q h, r h ) = (q h, r h ) Sh. (4.1.2) 20

32 In particular, we have w h 2 = a(w h, w h ) = q h 2 S h. (4.1.3) Using the Schur complement S h, the system (2.2.7) can be decoupled to S h p h = B h A 1 h f h g h, u h = A 1 h (f h Bh p h). (4.1.4) First, we present a unified variational form of the Uzawa, the Uzawa gradient, and the Uzawa Conjugate Gradient algorithms for solving the saddle point problem (2.2.4). The standard U and UG algorithms can be rewritten such that they differ only by the way the relaxation parameter α is chosen. For the Uzawa algorithm, we have to choose ( ) 2 α = α 0 a fixed number in the interval 0,. For the UG algorithm, the parameter Mh 2 α is chosen to impose the orthogonality of consecutive residuals associated with the second equation in (2.2.4), see [4]. The first step for Uzawa is identical with the first step of UG. We combine the two algorithms in: Algorithm (U-UG) Algorithms Step 1: Set u 0 = 0 V h, p 0 M h, compute u 1 V h, q 1 M h by a(u 1, v) = ((f h, v)) b(v, p 0 ) for all v V h, (q 1, q) = b(u 1, q) (g h, q) for all q M h. Step 2: For j = 1, 2,..., compute h j, α j, p j, u j+1, q j+1 by (U UG1) a(h j, v) = b(v, q j ), v V h, (Uα) α j = α 0 for the Uzawa algorithm or (UGα) α j = (q j, q j ) b(h j, q j ) = (q j, q j ) (q j, q j ) Sh for the UG algorithm, (U UG2) p j = p j 1 + α j q j, (U UG3) u j+1 = u j + α j h j, (U UG4) (q j+1, q) = b(u j+1, q) (g h, q) for all q M h. 21

33 In the second identity in (UGα), we involved Remark and (UG1). One can modify the UG algorithm to obtain the UCG algorithm, see e.g., [22, 79], as follows: First, we define d 1 := q 1 in Step 1, and then modify Step 2 by replacing b(, q j ) with b(, d j ), where {d j } is a sequence of conjugate directions: Algorithm (UCG) Algorithm Step 1: Set u 0 = 0 V h, p 0 M h. Compute u 1 V h, q 1, d 1 M h by a(u 1, v) = ((f h, v)) b(v, p 0 ), v V h, (q 1, q) = b(u 1, q) (g h, q) for all q M h, d 1 := q 1. Step 2: For j = 1, 2,..., compute h j, α j, p j, u j+1, q j+1, β j, d j+1 by (UCG1) a(h j, v) = b(v, d j ), v V h, (UCGα) α j = (q j, q j ) b(h j, q j ) = (q j, q j ) (d j, q j ) Sh, (UCG2) p j = p j 1 + α j d j, (UCG3) u j+1 = u j + α j h j, (UCG4) (q j+1, q) = b(u j+1, q) (g h, q) for all q M h, (UCGβ) β j = (q j+1, q j+1 ), (q j, q j ) (UCG6) d j+1 = q j+1 + β j d j. Remark It is not difficult to check that the UG and UCG algorithms produce the standard gradient and the standard conjugate gradient algorithms for solving the first equation in (4.1.4). We note that for the no projection trial space case the q j+1 from Step 2, (U- UG4), and (UCG4) can be computed by and in the projection trial space case we have q j+1 = C 1 Bu j+1, (4.1.5) q j+1 = Q h C 1 Bu j+1. (4.1.6) 22

34 At each iteration step only one inversion involving the form a 0 (, ) is needed. Theorem Let (u h, p h ) be the solution of (2.2.4), and let {(u j+1, p j )} j 0 be the iterations produced by a U, UG, or UCG algorithm. Then, for j 0, u j+1 u h = A 1 h B h(p h p j ), (4.1.7) and consequently, for j 1, q j+1 = S h (p h p j ), (4.1.8) 1 M 2 h q j p j 1 p h 1 q m 2 j, (4.1.9) h m h Mh 2 q j u j u h M h q m 2 j. (4.1.10) h The following proof was taken from [5]: Proof. By induction over j, it is easy to prove (for any of the U, UG, or UCG) that a(u j+1, v) + b(v, p j ) = ((f h, v)) for all v V h. (4.1.11) Combining the first equation in (2.2.4) and (4.1.11), we get a(u j+1 u h, v) = b(v, p h p j ) for all v V h, which gives (4.1.7). From (U4), (UG4), or (UCG4), the second equation of (2.2.7), and (4.1.7) we get q j+1 = B h u j+1 g h = B h (u j+1 u h ) = S h (p h p j ), which proves (4.1.8). As a consequence of (4.1.7), the estimate (4.1.1) and Remark 4.1.1, for j 1, we have m h p h p j 1 u j u h = p h p j 1 Sh M h p h p j 1. (4.1.12) 23

35 Using (4.1.8) and the fact that m 2 h and M 2 h are the extreme eigenvalues of S h, we get m 2 h p h p j 1 S h (p h p j 1 ) = q j M 2 h p h p j 1. (4.1.13) The estimates (4.1.9) and (4.1.10) are a direct consequence of(4.1.12)and (4.1.13). As a consequence of Theorem 4.1.5, we obtain 1 + m h M 2 h q j u h u j + p h p j M h q m 2 j. (4.1.14) h Besides convergence of the iteration processes, the result entitles q j+1 as a computable, robust, efficient, and uniform-modulo m h estimator for all three algorithms. Assuming stability, we can use Theorem and the estimates (4.1.9) and (4.1.10) to build adaptive or multilevel algorithms for SPLS discretization. A cascadic algorithm for solving symmetric SPPs was introduced in [5]. This would imply that, if a sequence of discrete pairs {(V hk, M hk )} k 1 is available, with M hk M hk+1, then, we can start the U-algorithm on M hk+1 M hk. using the best approximation p 0 from the previous level One major advantage of solving system (3.2.3) as the SPLS discretization of (3.1.1) using one of these three algorithms is that (4.1.9) and (4.1.10) remain valid, if the right correction of constant factors is made, when a 0 (, ) is replaced by a uniform equivalent form a prec (, ). An important observation here is that if the starting initial guess is p 0 = 0, then the p j - iterates remain in the space C 1 h B hv h and that {p j } approximates and represents the solution p h in the form C 1 h B hw h, with w h Vh,0. In addition, as presented in the next chapter, for certain spaces M h a basis is not needed for solving the variational problems associated with q j and at each step of the U-type iterative processes only the action of A 1 h or a preconditioner requires an inversion process. Compared to other methods for solving Saddle Point problems, our SPLS method has the advantage that a global matrix assembly is avoided. Even though the iteration process might take more time to approximate the discrete solution, the approach leads, in most cases, to a higher order of approximation. 24

36 Chapter 5 DIV-CURL SYSTEM In this chapter we introduce the Div-Curl system that is discretized using the SPLS discretization method described in the previous chapters. 5.1 Div-Curl Background This section will introduce some important background information and notation on the problem which we summarized based on the textbook on electromagnetics, [76]. More details can be found therein. The motivation comes from an area of electromagnetics, generally known as statics, in which the fields are time-invariant. We note at the outset that a static field is physically sensible only as a limiting case of a time-varying field as the latter approaches a time-invariant equilibrium, and then only in local regions. The static field equations we are going to study thus represent an idealized model of the physical fields. The static field Maxwell equations, which can be derived from the Maxwell - Minkowski equations by setting the time derivatives equal to zero, are: e = 0, d = ρ, h = j, b = 0, (5.1.1) where e, d, h and b are mediating fields, see Chapter 1 in [76]. The sources j and ρ must also be time-invariant. Under these conditions the dynamic coupling between the fields imposed by the time-varying nature of the fields, described by Maxwell s equations, 25

37 disappears. For static fields we also require that any dynamic coupling between fields in the constitutive relations vanish. For the remainder of the chapter we will assume that there is no coupling between e and h or between d and b in the constitutive relations. Then, the static equations decouple into two independent sets of equations in terms of two independent sets of fields. The static electric field set (e, d) is described by the equations e = 0, d = ρ. (5.1.2) The static magnetic field set (b, h) is described by h = j, b = 0. (5.1.3) It is important to note that any separation of the electromagnetic field into independent static electric and magnetic portions is illusory. The electric and magnetic components of the EM field depend on the motion of the observer. An observer stationary with respect to a single charge measures only a static electric field, while an observer in uniform motion with respect to the charge measures both electric and magnetic fields. Although the equations describing the electrostatic and magnetostatic field sets decouple, the phenomena themselves remain linked. Since current is moving charge, electrical phenomena are associated with the establishment of the current that supports a magnetostatic field. Adding the boundary condition b n = 0 which corresponds to the boundary of a perfect conductor we end up with the div-curl system: Let Ω R 3 be a polyhedral domain. Find the vector function h L 2 (Ω) such that h = j in Ω, b = 0 in Ω, (5.1.4) b n = 0 on Γ := Ω, where h is the magnetic field intensity (force), b is the magnetic flux density (induction), the vector field j is the current density, and n is the outward unit normal on Γ. 26

38 Assuming a linear, isotropic region, we can augment the equations with the relation b = µh, where µ is a given parametric scalar L function that is strictly positive on Ω. In the presence of material discontinuities (jumps in µ), the continuity properties of the fields are given by [b n] = 0 on an interface between materials with normal n. Here [ ] denotes the jump across the interface. 5.2 Preliminaries In this section we provide the notation, spaces, norms and some results we will need for solving the div-curl systtem. We will consider Ω, a simply connected bounded polyhedral domain, contained in R 3. Its boundary will be denoted Γ. We will have to deal with vector and scalar functions, therefore we will use bold face to indicate vector functions. We shall use (, ) to denote both the inner product in L 2 (Ω) and L 2 (Ω) = (L 2 (Ω) 3 ), while will denote the norms. The function spaces we will use: H 1 0(Ω) = (H0(Ω)) 1 3, H(curl) = {v L 2 (Ω) : v L 2 (Ω)}, H 0 (curl) = {v H(curl) : v n = 0}, H(div) = {v L 2 (Ω) : v L 2 (Ω)}. (5.2.1) The norms for all spaces are the usual ones. As mentioned above, the problem will involve a parameter µ which corresponds to the magnetic permeability in a magnetostatic field problem. We shall assume there are constants µ 0 and µ 1 satisfying 0 < µ 0 µ(x) µ 1 for all x Ω. Following [23], we have the following decomposition results: Proposition Let u be in L 2 (Ω). Then there exists a decomposition u = v + µ ϕ, (5.2.2) where v H 0 (curl) and ϕ H 1 (Ω). Proposition Let v be in H 0 (curl). Then there exists a decomposition v = w + ϕ, 27

39 where ϕ H 1 (Ω) and w H 1 0(Ω), satisfies w := w H 1 0 C v = C w. By combining these two properties we obtain the following lemma. Lemma Let u be in L 2 (Ω). Then there exists a decomposition u = w + µ ϕ, (5.2.3) where w H 1 0(Ω) and ϕ H 1 (Ω)/R (isomorphic to the space functions in H 1 with zero mean value). Furthermore w H 1 0 C w. 5.3 SPLS Discretization of the Div-Curl Problem The problem we are going to consider is a more general one: Let Ω R 3, find the vector function h L 2 (Ω) such that h = j in Ω, (µh) = g in Ω, (µh) n = σ on Γ := Ω. (5.3.1) The variational formulation we adopt for (5.3.1) is similar to the approach of [23]. Since C0 (Ω) is dense in H 1 0(Ω), we multiply the first equation in (5.3.1) by w H 1 0(Ω) and, after we integrate by parts, we obtain (h, w) = (j, w) for all w H 1 0(Ω). (5.3.2) If h L 2 (Ω) and g L 2 (Ω), then h H(div). Assuming σ H 1/2 (Γ) and integrating by parts the second equation in (5.3.1) we obtain (h, µ ϕ) = (µh n, ϕ) Γ ( (µh), ϕ) = (σ, ϕ) Γ (g, ϕ) G, ϕ for all ϕ H 1 (Ω)/R. (5.3.3) 28

40 Here G is the unique functional in (H 1 (Ω)/R) defined by (5.3.3). If we define V := H 1 0(Ω) H 1 (Ω)/R, Q := L 2 (Ω), and the form b(, ) on (V, Q), by b((w, ϕ), h) := (h, w + µ ϕ) for all (w, ϕ) V, h Q, the variational formulation for (5.3.1) becomes: Find h Q such that b((w, ϕ), h) = F, (w, ϕ) := (j, w) + G, ϕ for all (w, ϕ) V. (5.3.4) The choice of inner products on the spaces V and Q is essential in building robust solvers with respect to the function µ. We choose the following weighted inner products on Q = L 2 (Ω): (u, v) µ = (µu, v), (u, v) µ 1 = (µ 1 u, v), with the corresponding norms denoted by µ and µ 1 respectively. Let the weighted inner product on V := H 1 0(Ω) H 1 (Ω)/R be induced by the norm (w, ϕ) 2 V := a µ 1(w, w) + a µ (ϕ, ϕ) := µ 1 w 2 + µ ϕ 2. Ω Ω Using the above we are going to include the proof that the from b(, ) satisfies a continuous inf sup condition with a constant independent of µ as done by Bramble- Pasciak in [23]. From Lemma 5.2.3, for u L 2 (Ω), let u = w + µ ϕ be the decomposition. Since w H 1 0(Ω) and ϕ H 1 (Ω), we have ( w, µ ϕ) µ 1 = 0. Hence, u 2 µ 1 w 2 µ 1 + µ ϕ 2 µ 1 = (µ 1 u, w) 2 w 2 µ 1 + (u, ϕ)2. ϕ 2 µ Using the inequality in Lemma 5.2.3, we conclude that u 2 µ 1 Cµ( sup (µ 1 u, w) 2 w H 1 0 (Ω) w 2 H sup ϕ H 1 /R (u, ϕ) 2 ). (5.3.5) ϕ 2 µ 29

41 Using the Generalized Poincaré Inequality it follows that where u 2 µ 1 Cµ1/2 sup (w,ϕ) V b((w, ϕ), u) (w, ϕ) V, (w, ϕ) 2 V = ( w 2 H ϕ 1 H 1)1/2. The C 1 B operator that appears in the SPLS discretization is C 1 B (w, ϕ) = µ 1 curl w + ϕ. (5.3.6) By using a Helmholtz decomposition, see [23], we obtain that C 1 B is onto Q = L 2 (Ω). If the data (j, g, σ) is such that the compatibility condition (3.1.4) is satisfied, then the problem consisting of (5.3.2) and (5.3.3) is well-posed and our SPLS discretization can be applied. 5.4 SPLS Discretization, Construction of Special Discrete Spaces In Section 3.4 we introduced the general theory on the SPLS discretization and the respective approximation spaces. Now we define the corresponding discrete spaces for the specific Div-Curl problem at hand. We can choose W h Φ h =: V h V := H 1 0(Ω) H 1 (Ω)/R to be the standard vector space of continuous piecewise functions of degree m with the appropriate boundary conditions for each component of V. For M h we consider two choices: Case 1) M h := C 1 BV h = {µ 1 curl w h + ϕ h w h W h, ϕ h Φ h }, Case 2) M h := Q h C 1 BV h = { Q h (µ 1 curl w h + ϕ h ) w h W h, ϕ h Φ h }, where Q h is the orthogonal projection onto M h, the space that componentwise consists of continuous piecewise polynomials of degree m. On M h for Case 2 we can consider using: a) the standard inner product, b) the lumped inner product, given in subsection

42 5.4.1 Approximability If Q h C 1 Bv h = 0 implies C 1 Bv h = 0, for any v h V h, (i.e., Qh is injective on C 1 BV h ), then V h,0 V 0 and, see Remark 3.2, (3.2.2), has a unique solution p h M h. Using Proposition for the discrete pair (V h, M h ), we have that (u h = 0, p h ) is the solution of (3.2.3). This will always happen for Case 1), where M h = C 1 BV h, and Q h is the identity operator on C 1 BV h. In this special case we do have M h = M h = C 1 BV h and, if p is the solution of (3.1.1) and p h is the solution of (3.2.2), from (3.1.1) and (3.2.2) we obtain that 0 = b(v h, p p h ) = (C 1 Bv h, p p h ) for all v h V h. Thus, we simply have that p h is the orthogonal projection of p onto M h, and consequently, using the representation p = C 1 Bv for some v V, we have h h h = inf h m h = inf C 1 Bv C 1 Bv h m h M h v h V h = inf (w h,ϕ h ) W h Φ h µ 1 curl w + ϕ (µ 1 curl w h + ϕ h ) (5.4.1) 1 inf w w h + inf ϕ ϕ h. µ 0 w h W h ϕ h Φ h Compared with (3.2.5), the discretization error estimate (5.4.1) has the advantage that it is independent of the stability constant m h and, in addition, it reduces to approximability of functions in V by discrete functions in V h. If one of the Uzawa type iterative methods of Chapter 4 is applied to find the solution (0, h h ) of (3.2.3) then, according to Theorem 4.1.5, the iteration error satisfies h j h h 1 q m 2 j+1. h If the discretization error order is available, say h h h = O(h α ), and an estimate about m h is also available, the iteration error can match the discretization error just by imposing the stopping criteria q j+1 m 2 h O(h α ). (5.4.2) 31

43 Consequently, using one of the Uzawa algorithms, we can approximate the solution h by an iterate h j C 1 BV h up to optimal discretization error order, regardless of (discrete B-) stability presence. The numerical experiments we performed for concrete SPLS discretization problems, projecting on smooth spaces the continuous solution p. Projecting on coarser spaces M h, i.e. Case 2), lead to better approximation of M h could lead to stability, see section It is worth noticing here that, if one chooses M h satisfying (3.4.5), then stability of (V h, C 1 BV h ) implies stability for the pair (V h, Q ) h C 1 BV h Numerical results We performed several numerical experiments for approximating (5.3.1) with Uzawa Conjugate Gradient (UCG). We implemented two instances for V h : I. Conforming P 1 elements II. Conforming P 2 elements For M h in I we consider both cases described in Section 5.4 above. In Case 2) we project on M h, the vector space of continuous P 1 functions and consider both inner products: a) the standard L 2 and b) the lumped inner product given by (3.4.9). For M h in II we still consider both Cases 1 and 2 but for Case 2) we consider only the standard L 2 -inner product. The domain Ω was chosen to be the unit cube, and the function µ was chosen to be the restriction to Ω of 1 if x < 1 µ =, 2 (5.4.3) µ 0 if x 1, 2 for various values of the constant µ 0. The numerical experiments below are going to be ordered in the following way: ( Note that the capital Roman numerals representing the items in the list correspond with the appropriate instance chosen for V h above.) 32

44 I. For each case of M h, (three cases here), we will report the results for the following problems: i) h = curl w + ϕ, where w and ϕ in (5.3.6) are smooth functions ii) h = curl w + ϕ, where h is smooth but w and ϕ might not be iii) µh is a smooth function, no discontinuity iv) µh is discontinuous with µ as described above v) results on m h with varying values of µ 0, the jump in µ as defined above. II. For each case of M h, (only two cases here) as described above we report the results on the following problems: i) h = curl w + ϕ, where w and ϕ are smooth functions ii) h = curl w + ϕ, where h is smooth but w and ϕ might not be Note: The problem v) in instance I will be reported only for Case 1) of M h and q h denotes the residual in the last iteration within each level of refinement, given by: q h = q j+1 = µ 1 curl w j+1 + ϕ j+1. We would like to thank Francisco J. Sayas for the FEM code we used as a base for our code. For I, Case 1) of M h, we use a fixed tolerance with c 0 = and α = 1, where c 0 denotes the constant that includes m 2 h and α the order in (5.4.2). In I-i) the data is chosen such that the exact solution is: h = (z + (x 1)x(y 1)y(2z 1), 0, x (2x 1)(y 1)y(z 1)z) (5.4.4) with the components of the Helmholtz decomposition being w = (0, x(1 x)y(1 y)z(1 z), 0), ϕ = xz. 33

45 Level µ 0 = 1 P 1 C 1 BP 1 h h comp Order # of it. q h e e e e e-5 Table 5.1: UCG for SPLS-P 1 -discretization Case 1) i) In I-ii) the we chose the data such that the exact solution is: h = (x 2 yz, xy 2 z, xyz 2 ) (5.4.5) Level µ 0 = 1 P 1 C 1 BP 1 h h comp Order # of it. q h e e e e e-5 Table 5.2: UCG for SPLS-P 1 -discretization Case 1) ii) In I-iii) we use the data corresponding to the following exact solution: h = (x(1 x)(1/2 x)yz, x(1/2 x)y(1 y)z, x(1/2 x)yz(1 z)) (5.4.6) Level µ 0 = 1 P 1 C 1 BP 1 h h comp Order # of it. q h e e e e e-05 Table 5.3: UCG for SPLS-P 1 -discretization Case 1) iii) 34

46 Next, for I-iv) we consider µ discontinuous defined as in (5.4.3) with µ 0 = 50 and data chosen such that the exact solution h is as above, in (5.4.6). Level µ 0 = 50 P 1 C 1 BP 1 h h comp Order # of it. q h e e e e e-05 Table 5.4: UCG for SPLS-P 1 -discretization Case 1) iv) In case I-v), using a power method for the discrete Schur complement S h,0, we estimated m h, in (5.4.2). While we can notice instability with respect to h (or Level), we can also notice numerical stability with respect to µ 0. The estimates for the discrete inf sup constant m h are presented in the table below. Level µ Table 5.5: Estimates for m h for different values of µ 0, Case I) v) For Case 1), with µ having 2D checkerboard jump discontinuities, 1 on white squares, µ 0 on the black squares and constant 1 in the z-direction, we obtained similar results. 35

47 Remark We note that for Case 1), the discretization error h h h satisfies (5.4.1) and for the conforming choice P 1 for V h we have h h h = O(h). Then, since h h comp h h h + h h h comp, and the iteration error h h h comp matches the discretization error (see Theorem 4.1.5) we also expect O(h) for h h comp. This is numerically observed in Table 5.1 where the components are smooth functions. In the other cases, Tables 5.2, 5.3, 5.4, we observe that it takes more refinement for the method to achieve the expected error. One explanation for these observations is the fact that the components in these cases might be non-smooth. Worth noting is the fact that the order of convergence remains unaffected when µ, thus µh, has a jump dicontinuity, observed in Table

48 Next, for I, Case 2) of M h, we consider two choices for the inner product as explained in section 5.4. First we are going to show the results obtained using the standard L 2 -inner product. We use a fixed tolerance with c 0 = 0.1 and α = 2. For I-i) the exact solution is as in (5.4.4). Level µ 0 = 1 P 1 Q h C 1 BP 1 h h comp Order # of it. q h e e-5 Table 5.6: UCG for SPLS-P 1 -discretization Case 2) i) For I-ii) the exact solution is as in (5.4.5). Level µ 0 = 1 P 1 Q h C 1 BP 1 h h comp Order # of it. q h e e-5 Table 5.7: UCG for SPLS-P 1 -discretization Case 2) ii) 37

49 Similarly as in Case 1) iii), iv), above, now we take a look at the approximation of the exact solution of (5.1.3) using the SPLS method for the cases where µ is defined as in (5.4.3) for µ 0 = 1 and µ 0 = 50 reported in the tables below, respectively. The exact solution is chosen as in (5.4.6) and the data taken appropriately. Level µ 0 = 1 P 1 Q h C 1 BP 1 h h comp Order # of it. q h e e e-5 Table 5.8: UCG for SPLS-P 1 -discretization Case 2) iii) Level µ 0 = 50 P 1 Q h C 1 BP 1 h h comp Order # of it. q h e e e-5 Table 5.9: UCG for SPLS-P 1 -discretization Case 2) iv) Remark For Case 2), the discretization error h h h satisfies an estimate of type (3.4.6). In this case, our estimate for the discretization error is not optimal. The maximum order of convergence expected in this case provided the necessary regularity of the solution is O(h 2 ). There are several reasons why we observe this behavior including the regularity of the components of the Helmholtz decomposition and the fact that when we project we have only a subspace of M h. Nonetheless, the convergence behavior in this case remains to be theoretically and numerically investigated. These results motivated the implementation of the orthogonal projection with the lumped inner product. 38

50 Now we present the results for I, Case 2) of M h, with the lumped inner product defined in (3.4.8). We use the same fixed tolerance of with c 0 = 0.1 and α = 2. The first case, as previously, corresponds to an exact solution given by (5.4.4) with smooth components w, ϕ of the decomposition (5.3.6). The data is chosen to satisfy (5.1.3) with the given h. Level µ 0 = 1 P 1 Q h C 1 BP 1 h h comp Order # of it. q h e e e-5 Table 5.10: UCG for SPLS-P 1 -discretization Case 2) lump i) Again, this second table corresponds to an exact solution given by (5.4.5). Level µ 0 = 1 P 1 Q h C 1 BP 1 h h comp Order # of it. q h e e-5 Table 5.11: UCG for SPLS-P 1 -discretization Case 2) lump ii) 39

51 The last two tables below will present the results of approximating the exact solution of (5.1.3) for the cases where µ is continuous and discontinuous, respectively. The exact solution is chosen as in (5.4.6). Level µ 0 = 1 P 1 Q h C 1 BP 1 h h comp Order # of it. q h e e e-5 Table 5.12: UCG for SPLS-P 1 -discretization Case 2) lump iii) Level µ 0 = 50 P 1 Q h C 1 BP 1 h h comp Order # of it. q h e e-5 Table 5.13: UCG for SPLS-P 1 -discretization Case 2) lump iv) Remark For Case 2), with the lumped inner product the same argument as above applies. Again, the most we can expect from Q h C 1 Bv h C 1 Bv h is O(h 2 ). Nevertheless, the results of the four tables above, Tables 5.10, 5.11, 5.12 and 5.13, show that we get a significant improvement over the projection case with the standard L 2 inner product. The theoretical arguments to back up these results are interesting and worth investigating in future work. A key observation that needs to be pointed out is the fact that it takes a minimal number of iterations to achieve the reported convergence rate in all the cases and iterating more does not give better results. This is extremely relevant since the computational time is reduced significantly. 40

52 Next we report the results obtained from the numerical experiments with V h being the conforming P 2 elements. As in the linear case, we first report for Case 1) of M h. We use a fixed tolerance with c 0 = and α = 2. II-i) the data is chosen such that the exact solution is given by (5.4.4). Level P 2 C 1 BP 2 h h comp Order # of it. q h e e e e-6 Table 5.14: UCG for SPLS-P 2 -discretization Case 1) i) II-ii) the exact solution is given by (5.4.5). Level P 2 C 1 BP 2 h h comp Order # of it. q h e e e e-6 Table 5.15: UCG for SPLS-P 2 -discretization Case 1) ii) Remark Looking at the theory again, the discretization error h h h satisfies (5.4.1) and for the conforming choice P 2 for V h we have h h h = O(h 2 ). Then, since the iteration error h h h comp matches the discretization error (see Theorem 4.1.5) we also expect O(h 2 ) for h h comp. This is numerically observed in Table 5.14 where the components w and ϕ of the decomposition of the exact solution, given by (5.3.6), are smooth functions. In the other case, Table 5.15, the rate of convergence drops to O(h) due to the non-smoothness of the solution components. 41

53 In the last two tables for this chapter we will present the results for Case 2) of M h. Keeping in mind that it involves only the case of the standard L 2 -inner product. The exact solutions are given by (5.4.4), (5.4.5) respectively. Level P 2 Q h C 1 BP 2 h h comp Order # of it. q h e e e e e e-5 Table 5.16: UCG for SPLS-P 2 -discretization Case 2) i) Level P 2 Q h C 1 BP 2 h h comp Order # of it. q h e e e e-5 Table 5.17: UCG for SPLS-P 2 -discretization Case 2) ii) Remark In Table 5.16, with smooth Helmholtz decomposition components, the expected order of convergence is O(h 3 ) but we observe a higher order for the refinement levels presented here. This higher order of convergence can be attributed to the use of Gaussian quadrature points to compute the error. It is known that this choice of interpolation points yields, in some cases, to superconvergence. Again, as in the linear case, we observe a decaying trend in the rate of convergence when we use the orthogonal projection case for M h, as can be noted more clearly from Table A key difference here is that, nonetheless, we recover O(h 2 ) of convergence in the case where the components of the decomposition are non-smooth for which we were getting O(h) in the linear case. As a concluding note, we want to point out, again, the significantly small number of iteration required to achieve these results. 42

54 Chapter 6 SECOND ORDER ELLIPTIC PROBLEM In this chapter we consider another set of boundary value problems where the SPLS method can be applied to numerically approximate their solutions. We will use the proposed technique to discretize second order elliptic problems with oscillatory coefficients, interface problems and for high order approximation of the flux for elliptic problems with smooth coefficients. 6.1 Elliptic Problem Background Let Ω be a bounded, open, connected subset of R n with a Lipschitz continuous boundary Ω. Denote n the outward unit normal vector to the boundary. Let the boundary of the domain Ω be partitioned into two relatively open subsets Γ D and Γ N such that Ω = Γ D Γ N and Γ D Γ N =. The general second order elliptic problem with homogeneous boundary conditions can be formulated as: div(a u) + b u + cu = f in Ω, u = 0 on Γ D, (6.1.1) n A u = 0 on Γ N, where b and f are given vector-valued functions, c is a given scalar function. The matrix A of (piecewise continuous) coefficients is uniformly symmetric positive definite: a min ξ 2 A(x)ξ, ξ a max ξ 2 for all x Ω, ξ R n, (6.1.2) for some positive constants a min a max. Here and in what follows, for vectors in R n, we denote by, and the standard Euclidean inner product and norm, respectively. 43

55 Elliptic problems arise in a vast number of applications. We are just going to give a very brief introduction to the set of problems mentioned at the beginning of this chapter. Note that Gradient Recovery, Interface problems and Highly Oscillatory Coefficients are much more general problems. Even the restriction to elliptic problems is very broad, thus we will briefly mention some possible applications for each type of problem. As mentioned above, we will focus only on three types of problems, which are very similar in nature but have very different applications and in most of the cases require very different approaches to be solved numerically. Gradient Recovery is an effective and widely used post-processing technique in scientific and engineering computations, see [52, 51, 71, 78, 82, 83, 84]. The main purpose is to reconstruct a better numerical gradient from a finite element solution. It can be applied in different instances, see the introduction in [51] and the references therein. In our case, we are more concerned in applying the SPLS method to recover the flux, σ = A u, which is an important physical quantity, often the primary concern in practice. For more details on flux recovery techniques used see [35]. Elliptic Interface problems have a variety of applications in different fields. In fluid dynamics, for instance, they model several layers of fluids with different viscosities, [17], or diffusion through heterogenous porous media, [45]. In material science they arise in the design and study of composite materials built from essentially different components, see [3, 49, 60, 12]. The elliptic interface problem is used in biological systems, [58], and to model stationary heat conduction problems with a conduction coefficient which is discontinuous across a smooth internal interface, [53]. Highly Oscillatory Coefficients appear in partial differential equations that describe a large class of multiscale problems. The coefficients may characterize the heterogeneity of a medium, as in porous media flows, or they may represent the random velocity field in a turbulent transport problem, see [54]. 44

56 6.2 Primal Mixed Variational Formulation We will apply the SPLS discretization method to a simpler version of the second order elliptic problem that can be formulated as: div(a u) = f in Ω, (6.2.1) where A satisfies (6.1.2). We also consider that (6.2.1) could model second order elliptic interface problems with a smooth interface Γ Ω. In this case, the entries of the matrix A could be discontinuous across Γ, and, we require that the normal component of the solution flux n (A u) be continuous across Γ. The primal mixed variational formulation that we consider is: Find σ = A u with u H0(Ω) 1 such that: (σ, v) = (A u, v) = (f, v) for all v H0(Ω). 1 (6.2.2) The formulation (6.2.2) is known as primal mixed formulation for second order elliptic problems. For the Laplace operator it is described in Chapter 5 of Braess book, [22]. Other related works using primal mixed formulation can be found in [13, 14, 15, 59]. In what follows, (, ) and denote the standard L 2 inner product and norm for scalar or vector functions, and v := v for any v H0(Ω). 1 We notice that for this problem, the representation σ = C 1 Bu = A u with u H0(Ω) 1 is unique. To fit the variational formulation (6.2.2) into the abstract formulation (3.1.1), we take V := H0(Ω), 1 Q := A V. Define b L(V Q; R) by b(σ, v) := (σ, v), and let f, v := (f, v) for all v V = H 1 0(Ω). We keep the original notation for test spaces that was introduced in [11] with the bold face notation for the test space V, (and the corresponding discrete space V h ) 45

57 even though, in what follows, V is a space of scalar functions. On V we consider the standard inner product a 0 (u, v) := ( u, v) for all u, v V. On Q = A V, we define the inner product (σ, τ) Q = (A u, A v) Q := (A u, A v) A 1 := (A u, v). (6.2.3) One can immediately check that B : V Q and C 1 B : V Q become Bv = v and C 1 Bv = A v for all v V. We further note that V 0 := Ker(B) = {v V Bv = 0} = {v H0(Ω) 1 v = 0} = {0}, and the compatibility condition (3.1.4) is trivially satisfied. continuity and inf sup constants satisfy: We also note that the and M = sup sup σ=a u Q v V sup sup u V v V a max <, b(v, σ) σ v = sup sup u V v V A u v (A u, v) (A u, u) 1/2 v (A u, u) 1/2 v = sup A u (A u, u) 1/2 m = inf sup b(v, σ) σ=a u Q v V σ v = inf (A u, u) inf u V u V sup u V v V (A u, u) 1/2 u a min > 0. (A u, v) (A u, u) 1/2 v (6.2.4) (6.2.5) Consequently, the mixed variational formulation (6.2.2) is well posed and suitable for the SPLS formulation and discretization. 6.3 SPLS Discretization for Second Order Elliptic Problems We take V h V = H 1 0(Ω) to be the space of continuous piecewise polynomials of degree m with respect to a regular mesh T h. 46

58 6.3.1 The no projection trial space leads to stability and approximability The corresponding trial space M h Q is M h := C 1 BV h = A V h. (6.3.1) We do have stability in this case. More precisely, similar arguments used to establish (6.2.5) give m h := inf σ h =A u h M h b(v h, σ h ) sup v h V h σ h v h a min > 0. The discrete mixed variational formulation in this case is: Find σ h = A u h, with u h V h uniquely representing σ h, such that (σ h, v h ) = (A u h, v h ) = (f, v h ) for all v h V h. (6.3.2) The SPLS discretization (3.2.3) (to be solved by an Uzawa type algorithm) is: Find (w h = 0, σ h = A u h ) such that ( w h, v h ) + (A u h, v h ) = (f, v h ) for all v h V h, A w h = 0, (6.3.3) where the bilinear form (, ) can be replaced by a prec (, ). If σ = A u is the solution of (6.2.2) and σ h = A u h is the solution of (6.3.2) or the SPLS formulation (6.3.3), using also (6.2.4), the general estimate (3.4.2) gives σ σ h Q = (A (u u h ), (u u h )) 1/2 a max inf (u v h ). v h V h The projection trial space Guided by the general theory of Section 3.4.2, we define M h as a finite dimensional subspace of Q = A V (L 2 (Ω)) n, to be M h = A M h,0 were each component of M h,0 consists of all continuous piecewise polynomials of degree m with respect to the mesh T h (used to define V h ) with no restrictions on Ω. We equip M h with the restriction of the inner product on Q that is defined in (6.2.3). We let Q h : Q M h be the orthogonal projection onto M h and define the space M h by M h := Q h C 1 BV h = Q h A V h. (6.3.4) 47

59 As discussed in Section 3.4.2, we have a discrete inf sup condition for the pair (V h, M h ), and approximability properties on M h that depend on the approximability of the space V h and the approximation quality of the orthogonal projection Q h : A V h M h. The discrete mixed variational formulation in this case is: Find σ h = Q h A u h with u h V h such that (σ h, v h ) = ( Q h A u h, v h ) = (f, v h ) for all v h V h. (6.3.5) The SPLS discretization (3.2.3) (to be solved by an Uzawa type algorithm) is: Find (w h, σ h = Q h A u h ) such that ( w h, v h ) + ( Q h A u h, v h ) = (f, v h ) for all v h V h, Q h A w h = 0, (6.3.6) where, again, the bilinear form (, ) can be replaced by a prec (, ). In order to discuss stability for families of pairs {(V h, M h )}, we will make further assumptions. We assume that Ω is a polygonal domain in R 2 and the (triangular) mesh T h is locally quasi-uniform. The result could be easily extended to polyhedral domains in R 3, but, to simplify the presentation, we will focus on the case Ω R 2. We let {z 1, z 2,, z N } be the set of all nodes of T h and assume that all triangles adjacent to z j are of regular shape and their area is of order h 2 j. The mesh size is then h := max{h 1, h 2,, h N }. We further assume that m = 1, i.e., V h consists of all continuous piecewise linear functions with respect to the mesh T h that vanish on Ω, and each component of M h,0 consists of all continuous piecewise linear functions with respect to the mesh T h (with no restrictions on the boundary). We let {Φ 1, Φ 2, Φ 2N } be the nodal basis for M h,0 and assume that Φ j = (φ j, 0) t and Φ N+j = (0, φ j ) t, for j = 1, 2,, N, where {φ 1, φ 2,, φ N } is the nodal basis for the space of scalar continuous piecewise linear functions with respect to the mesh T h. We note that {AΦ 1, AΦ 2, AΦ 2N } is a basis for M h and denote by M A the Gramm (or mass) matrix of this basis with respect to the (, ) Q inner product. We further denote D 1 the diagonal matrix with diagonal entries h 2 1, h 2 2,, h 2 N and by D the 2N 2N 48

60 diagonal matrix, which is the 2 2 block diagonal matrix with D 1 repeated as the diagonal blocks. In order to prove (3.4.5) for our pair (V h, M h ) we will first need the following: Lemma Under the above assumptions we have consequently: M A γ, γ c a max Dγ, γ for all γ R 2N, (6.3.7) M 1 c A γ, γ D 1 γ, γ for all γ R 2N. (6.3.8) a max Here, and in what follows, c is a generic constant that does not depend on h and can be different at different occurrences. Proof. For any γ R 2N, we define q h := 2N i=1 γ i Φ i and q A h := Aq h. Next, we note that M A γ, γ = (q A h, q A h ) Q = (Aq h, q h ) a max q h 2 = a max If τ = [z 1τ, z 2τ, z 3τ ], then q τ h = 3 γ jτ φ jτ j=1 3 j=1 γ jτ +Nφ jτ ( 3 q h 2 τ c τ γτ 2 j + j=1, and we have τ T h q h 2 τ. (6.3.9) ) 3 γτ 2 j+n. (6.3.10) Using (6.3.9), (6.3.10), and the fact that each coefficient γ k can appear at most three times, we get: ( N M A γ, γ c a max h 2 jγj 2 + h 2 j j=1 The estimate (6.3.8) follows from (6.3.7). N j=1 j=1 γ 2 j+n ) = c a max Dγ, γ. Lemma Under the above assumptions, there exists a constant c independent of h such that Q h A v h Q c a min a max A v h Q for all v h V h. (6.3.11) 49

61 Proof. For a fixed A v h with v h V h we define the dual vector G h R 2N by (G h ) i := (A v h, AΦ i ) Q = (A v h, Φ i ), and let Q h A v h = Thus, α = (α 1, α 2,..., α 2N ) t is the solution of and, using (6.3.8) we have Q h A v h 2 Q = 2N i,j=1 1 a max ( D 1 G h, G h ) 1 = 1 a max = 1 a max ( ) a2 min a max 2N i=1 2N i=1 2N h 2 i h 2 i 2N i=1 M A α = G h, α i AΦ i. α i α j (AΦ i, Φ j ) = M 1 A G h, G h a max 2N i=1 h 2 i (G h, G h ) [ ( ) 2 ( ) ] 2 v h a 11 x + a v h 12 y, φ v h i + a 21 x + a v h 22 y, φ i (a v 11, φ i ) τ (a 12, φ i ) h τ 2 τ x (a 21, φ i ) τ (a 22, φ i ) τ τ supp(φ j ) i=1 τ supp(φ j ) v h τ y h 2 i v h 2 τ = c a2 min v h 2 c a2 min A v h 2 a Q. max a 2 max Here, τ denotes the L 2 norm on τ. To justify the inequality ( ), we note that the lowest eigenvalue of the matrix (a 11, φ i ) τ (a 12, φ i ) τ is bounded below by c h 2 i a 2 min (a 21, φ i ) τ (a 22, φ i ) τ with a constant c independent of τ and h. For the last estimate we used that A v h 2 Q = (A v h, v h ) a max v h 2. As a direct consequence of Remark and Lemma we obtain: Theorem Let Ω R 2 be a polygonal domain and let {T h } be a family of locally quasi-uniform triangular meshes for Ω. For each h, let V h be the space of continuous piecewise linear functions with respect to the mesh T h that vanish on Ω, and let M h be the corresponding projection space (defined in this section). Then, the family of spaces {(V h, M h )} associated with the SPLS discretization (6.3.6) is stable. 50

62 6.4 Numerical Examples In this section we apply the SPLS discretization method to second order PDEs on polygonal domains Ω R 2. For all examples we choose the test space V h H0(Ω) 1 to be the P 1 conforming space, i.e. the space of all continuous piecewise linear functions with respect to quasi uniform or locally quasi uniform meshes T h, and use the Uzawa conjugate gradient without preconditioning to obtain the numerical approximation of the flux A u. The stopping criterion we imposed was based on (5.4.2) and the stability Theorem The maximum possible order, O( A u σ h ), was taken to be h 1 for the no projection case (σ h = A u h ) and h 2 for the projection case (σ h = Q h A u h ). For the non-uniform refinement cases, the parameter h is defined as h := DoF 1/2, where DoF is the number of degrees of freedom associated with the discrete test space V h A highly oscillatory coefficient example We consider the problem (6.2.1) on Ω = [ 1, 1] 2. We define f such that the exact solution is given by u(x, y) = sin(nπx) sin(nπy) nπ 2 + sin(πx) sin(πy) π 2, and the matrix A is given by A(x, y) = 2 + cos(nπy) cos(nπx) Next, we use SPLS discretization with both no projection and projection cases for M h. For the numerical results shown on Table 6.1, with the no projection trial space, we notice optimal rate of convergence (for n not too large). Here, A u c (the computed A u) is defined to be A u j where j represents the maximum number of iterations performed due to the imposed stopping criterion for each level k. For the projection case on Table 6.2, we observe higher order than h 1 approximation of A u, using a small number of iterations.. 51

63 The following table shows the results for three levels of oscillation, as determined by the value of n in the example above. The first table shows the solutions computed with a no-projection trial space for all three cases. h = 2 k k P 1 C 1 BP 1, error = A u A u c n = 4 n = 16 n = 32 error rate it error rate it error rate it Table 6.1: Highly Oscillatory coefficients with no projection space. Table 6.2, below, shows the respective results for the orthogonal projection trial The rate of convergence improved by a factor of two compared to the no projection case, while the number of iterations still remains relatively small. h = 2 k k P 1 Q h C 1 B(P 1 ), error = A u Q h A u c n = 4 n = 16 n = 32 error rate it error rate it error rate it Table 6.2: Highly Oscillatory coefficients with orthogonal projection Following we have the solution computed for the no projection trial space at each refinement level. The images are in increasing order of oscillation, starting with n = 4 and ending with n =

64 Figure 6.1: Highly Oscillatory coefficients with n = 4: Approximated solution at each refinement level computed with no projection. Figure 6.2: Highly Oscillatory coefficients with n = 16: Approximated solution at each refinement level computed with no projection. 53

65 Figure 6.3: Highly Oscillatory coefficients with n = 32: Approximated solution at each refinement level computed with no projection Interface problem examples Many approaches have been designed to efficiently approximate interface problems, [28, 38, 44, 46, 61, 67, 68, 70]. We would like to illustrate next, with two examples, that the SPLS discretization leads to simple and efficient discretization too. For both examples we implemented the no projection choice for M h. For the first example, we solve (6.2.1) with Ω = [0, 1] [0, 1] and a family of quasi uniform meshes {T h } obtained by a standard uniform refinement strategy, starting with a uniform coarse mesh. We defined f such that for c if x 1 A(x, y) =, 2 1 if x < 1, 2 the exact solution is cx(x 1 u(x, y) = )y(y 1) if x < 1, 2 2 (x 1)(x 1)y(y 1) if x

66 h = 2 k k c = 4 c = 64 c = 1024 error rate it error rate it error rate it Table 6.3: Interface problem with jump discontinuity c across x = 1 2 In Table 6.3 we compute the error := A u A u c. The experiments show an optimal convergence rate of O(h) in only 3 iterations and independent of the jump discontinuity c. As a second example of interface problems we consider Kellog s example [57], a standard benchmark problem with intersecting interfaces. In this example we solve again the (interface) problem (6.2.1), on Ω = [ 1, 1] [ 1, 1]. The exact solution given in polar coordinates is u(r, θ) = r β µ(θ), with: For cos(β( π σ)) cos(β(θ π + ρ)) if θ [ ] 0, π 2 2 2, cos(βρ) cos(β(θ π + σ)) if θ [ π µ(θ) =, π], 2 cos(βσ) cos(β(θ π ρ)) if θ [ ] π, 3π 2, cos(β( π ρ)) cos(β(θ 3π σ)) if θ [ 3π, 2π] R if (x, y) [0, 1] 2 [ 1, 0] 2, A(x, y) = 1 if (x, y) Ω\([0, 1] 2 [ 1, 0] 2 ), and the scalars R, β, ρ, σ satisfying nonlinear relations designed by Kellogg, [57], we have f = 0. For the numerical experiments we test with β = 0.1, R = , ρ = π/4, σ

67 Special adaptive methods to deal with this (very) singular problem have been designed to obtain optimal approximation with piecewise linear functions, see e.g. [34, 36, 39]. Our SPLS approach produces good approximations of the solution for the simple refinement we consider. The family of locally quasi uniform meshes {T h } is obtained by a graded refinement strategy, which depends on a refining parameter κ, [9, 10]. We refine by dividing all the edges that contain the singular point (the origin) under a fixed ratio κ such that the segment containing the singular point is κ -times the other segment. Numerical results using graded meshes with κ = are shown in Table 6.4 below. Here h = DoF 1/2, where DoF is the number of degrees of freedom for V h on level k, as defined above.. level k u u c A u A u c L 2 -err rate H 1 -err rate L 2 rate it Table 6.4: Interface problem with singularity at (0, 0) The number of iterations needed is larger here because, by using the stopping criterion given by (5.4.2) to match the (maximum possible order of the) discretization error, we imposed q j c 0 h m 2 h. From Remark and Lemma we have m h 1/R, which leads to a very small iteration error. We note also that an optimal rate of convergence for the flux was not achieved. This is because we used a simple way to refine (graded meshes) when changing levels. This technique takes care of the singularity caused by r β, and can find quasi optimal approximation for u, which in this case is at most h 1/2. In other words, by using graded refinement, the best we can expect for the discretization error is order h 1/2. In the following pages, we include the figures obtained for the initial 6 levels of refinement for the above example. 56

68 Figure 6.4: Interface problem: Kellog s; Graded refinement with κ = 0.022: Exact solution, computed approximation and zoomed mesh after each of the first 3 refinements 57

69 Figure 6.5: Interface problem: Kellog s; Graded refinement with κ = 0.022: Exact solution, computed approximation and zoomed mesh after each of the last 3 refinements 58

70 Next, we combined the SPLS discretization approach with an adaptive method to obtain a quasi-optimal approximation for the flux A u, i.e. close to order h 1. We employed the MATLAB package of adaptive finite element methods (AFEM), see [37]. The AFEM package uses the newest vertex bisection method, see [64]. It is really important here to mention a few facts about the way AFEM works. At every refinement level, it marks elements that will be refined via a bulk marking strategy, see [43]. The marking depends on the error estimator used. The AFEM package has the Zienkiewicz - Zhu (ZZ) recovery based error estimator [83, 84]. We modified it and implemented the Bernardi - Verfürth (BV) error estimator, see [17]. The following table shows the convergence results obtained for the aforementioned Kellog s example. Figure 6.6: SPLS with adaptive mesh for Kellog s: Error rates. As we can see, there is a significant improvement in the convergence rate. Keeping in mind that h DoF 1/2, we see that we get order h 2 for the L 2 error and the intended order h 1 for the flux. Following, the approximated solution and the refined mesh at specific refinement stages. 59

71 Figure 6.7: Interface problem: Kellogs; Adaptive mesh refinement with BV estimator: Exact solution, computed approximation and zoomed mesh at 3 different levels of refinements (first 10 levels) 60

72 Figure 6.8: Interface problem: Kellog s; Adaptive mesh refinement with BV estimator: Exact solution, computed approximation and zoomed mesh at 3 different levels of refinements (last 10 levels) 61

73 6.4.3 Higher order flux recovery example We solved (6.2.1) on Ω = [ 1, 1] 2 \ ((0, 1) (0, 1)) with the data f computed such that the exact solution is given in polar coordinates by: u = r a sin(aθ)(1 r 2 ), where a = 2 3, for A = I. Using graded meshes with κ = 0.1 we got the following results: Figure 6.9: SPLS with graded mesh for High Order Gradient Recovery on a L-shaped domain: Error rates. In the table below, we can see the numerical values of the errors and rates of convergence for the plot above. Level u u c u Q h u c L 2 err rate H 1 err rate it L 2 rate it Table 6.5: Flux recovery on L-shaped domain Before we analyze the results we got, the following images show the exact solution, the approximated solution computed and the refined mesh at each refinement for the first 6 levels. 62

74 Figure 6.10: Higher Order Gradient Recovery; Graded refinement with κ = 0.1: Exact solution, computed approximation and mesh after each of the first 3 refinements for the L-shaped domain 63

75 Figure 6.11: Higher Order Gradient Recovery; Graded refinement with κ = 0.1: Exact solution, computed approximation and mesh after each of the last 3 refinements for the L-shaped domain 64