High-Order Extended Finite Element Methods for Solving Interface Problems

Transcription

1 Hig-Order Extended Finite Element Metods for Solving Interface Problems arxiv: v1 [mat.na] 21 Apr 2016 Fei Wang Yuanming Xiao Jincao Xu ey words. Elliptic interface problems, unfitted mes, extended finite element, ig order AMS subject classifications. 65N12, 65N15, 65N30 Abstract In tis paper, we study arbitrary order extended finite element (XFE) metods based on two discontinuous Galerkin (DG) scemes in order to solve elliptic interface problems in two and tree dimensions. Optimal error estimates in te piecewise H 1 -norm and in te L 2 -norm are rigorously proved for bot scemes. In particular, we ave devised a new parameter-friendly DG-XFEM metod, wic means tat no sufficiently large parameters are needed to ensure te optimal convergence of te sceme. To prove te stability of bilinear forms, we derive non-standard trace and inverse inequalities for ig-order polynomials on curved sub-elements divided by te interface. All te estimates are independent of te location of te interface relative to te meses. Numerical examples are given to support te teoretical results. Te first and tird autor is supported in part by te U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Researc as part of te Collaboratory on Matematics for Mesoscopic Modeling of Materials under contract number DE-SC Te second autor is supported in part by te Fundamental Researc Funds for te Central Universities under grant and Cina NSF under te grant Department of Matematics, Pennsylvania State University, State College, PA (feiwang.psu@gmail.com, ttp:// Department of Matematics, Nanjing University, Jiangsu, , P.R. Cina. (xym@nju.edu.cn). Department of Matematics, Pennsylvania State University, State College, PA (jincao@psu.edu, ttp: // 1

2 1 Introduction Many multi-pysics problems, including fluid structure interaction problems and multipase flow problems, involve coupling between different pysical systems troug te interface, wic separates two pases of matter, i.e., solid, liquid, or gaseous. In te endeavor to solve suc multi-pysics problems, one of te most callenging tasks is tat of devising an accurate numerical discretization of te interface problems. In tis paper, we consider te following elliptic interface problem: (α(x) u) = f, in Ω 1 Ω 2, [α(x) u] = g N, on Γ, (1.1) [u] = g D, on Γ, u = 0, on Ω. Te global regularity of te solution is low due to te nature of te interface. Here, domain Ω is a bounded and convex polygonal/polyedral domain in R d (d = 2 or 3) and an internal interface Γ divides Ω into two open sets, Ω 1 and Ω 2. We assume tat Γ = Ω 1 is C 2 -smoot (see Figure 1 for an illustration of a unit square tat contains a circle as an interface). We also assume tat Γ Ω =. Te jump [ ] is defined in (2.3) and (2.4), and te coefficient α(x) is bounded from below and above by some positive constants. Due to te discontinuity of te coefficient α(x), te standard numerical metods, wic are efficient for smoot solutions, usually lead to a loss of accuracy across te interface. One way to render te more accurate approximation is to use interface-fitted/resolved grids. Tis way, te non-smootness of te solution can be restricted to a narrow subdomain in respect to te grid near te interface, suc tat te approximation error caused by te grid-mismatc is reduced to some extent. In [34] (see also [35] for an Englis translation) and more recently in [11], te following error estimate is obtained for d = 2: u u L 2 (Ω) + u u H 1 (Ω 1 Ω 2 ) log 1/2 2 u H 2 (Ω 1 Ω 2 ). A sarper analysis is given in [6], werein te logaritm factor of te above estimate is removed for d = 2. Here, we use te notation H m (Ω 1 Ω 2 ) = {v L 2 (Ω), v Ω1 H m (Ω 1 ) and v Ω2 H m (Ω 2 )}, wic is equipped wit te norm H m (Ω 1 Ω 2 ) = ( 2 H m (Ω 1 ) + 2 H m (Ω 2 ) )1/2. Te interface-fitting assumption in te works referenced tus far can be loosened sligtly so tat te interface Γ is O( 2 )-resolved by te mes [21]. Furter, te sape-regularity restriction of te grid can be loosened to maximal-angle-bounded grids [9]. Te optimal approximation in regard to te accuracy of te linear element space can also be proved on tese grids. In an interface-fitted mes, te sides (d = 2) or te edges (d = 3) intersect wit te interface only troug teir vertices. Unfortunately, it is usually a nontrivial and time-consuming 2

3 Ω 2 Γ Ω 1 Figure 1: Domain Ω = Ω 1 Γ Ω 2 wit an unfitted mes. task to construct good interface-fitted meses for problems involving geometrically complicated interfaces. Wen te problem is time-dependent, te domain needs to be re-mesed at eac time step, wic introduces an interpolation error between two consecutive meses. Terefore, numerous unfitted mes metods, in wic te interface is allowed to cross te elements, ave been proposed in te literature. In te finite difference setting, we refer to te immersed boundary metod in [27], te immersed interface metod in [19, 20], te gost fluid metod in [23], and te references terein. In te finite element framework, we refer to te work of [12, 15, 22] for elliptic problems wit discontinuous coefficients in wic finite element basis functions are locally modified for elements tat intersect wit te interface were te coefficient jumps. In [10], te adaptive immersed interface finite element metod based on a posteriori error estimates is proposed for elliptic and Maxwell equations wit discontinuous coefficients. In te past decade, a combination of te extended finite element metod (XFEM) (sometimes also known as te unfitted finite element metod) wit te Nitsce sceme as become a popular discretization metod. XFEM was introduced in te context of crack formation in structure mecanics, and one of its benefits is te ability to model discontinuities independent of te mes structure [5, 25]. Te idea of XFEM is to enric te original finite element space by using specially designed basis functions tat reflect te local features (discontinuity, singularity, boundary layer, etc.) of te problem. We refer to [14] and te references terein for a istorical account of XFEM. Inspired by te simple idea for andling Diriclet boundary conditions described in [26], Hansbo [17] applied Nitsce s metod to reformulate te problem (1.1) in an XFE space. Hansbo [17] proved tat tis Nitsce-XFEM can acieve te optimal convergence rate for te linear element, tereby generalizing te results in [2, 3]. In 3

4 [8], te Nitsce-type weak boundary conditions are extended to a fictitious domain setting. A penalty term acting on te jumps of te gradients is added over te element faces, and optimal a priori error estimates are tereby derived for te linear element. In [24], an unfitted symmetric interior penalty discontinuous Galerkin metod for elliptic interface problems is considered, and optimal -convergence for arbitrary p is given for te two-dimensional case in te energy norm and in te L 2 -norm. In [33], an unfitted p-interface penalty finite element metod for elliptic interface problems is studied for bot two and tree dimensions. An extra flux penalty term is added to te bilinear forms. Tus te stability could be proved by applying local trace and inverse inequalities on regular sub-elements. In [7], te focus is a modified sceme for wic te error estimates are independent of te contrast between diffusion coefficients. In [31], a quadratic Nitsce-XFEM is studied for te interface problem, and a clear classification of te sape of interface intersecting elements is given. An overview of te ways in wic Nitsce s metod as been applied to interface problems is given in [18]. Te Nitsce-XFEM can be interpreted as applying interior penalty (IP) metods on te interface, and tecniques inspired by IP metods are used in [17, 24, 33]. In tis paper, we first extend IP-XFEM to ig-order XFE spaces, and ten we consider anoter new DG- XFEM for solving elliptic interface problems (1.1). We note tat in our first approac, te penalization is applied only to te jump of te solution values across te interface (compared wit te bilinear form in [33]), and te optimal -convergence rate for arbitrary p in te energy and L 2 -norm are proved regardless of te dimension. Te major and defining step in our variant is a delicate coice of te weigt in te average (see (2.5)), wic leads to an inverse estimate for possibly degenerated sub-elements (see (3.6)). Wereas Nitsce-type scemes are sometimes criticized for te inconvenient coice of stabilization parameters, we propose a second parameter-friendly sceme. In tis sceme a penalization based on a lifting operator is introduced locally along te interface, and parameters need not be sufficiently large in regard to establising te stability of te bilinear form. Furtermore, we derive a generalized version of Céa s lemma (see (4.19)) to retrieve te optimal convergence rate on ig-order XFE spaces, even toug te bilinear form is not bounded for functions in te continuous space in a normal sense. Te main results of te analysis are summarized in Teorems 4.4 and 4.5. Note tat te trace and inverse inequalities suc as (3.6) are pivotal bot in analyzing Nitsce-type metods, and in deriving approximate penalty parameters to stabilize tese scemes. Tey play an even more important role in te analysis of te unfitted mes approac were te interface is allowed to intersect elements in an arbitrary manner. Wen sub-elements degenerate, wic is not a rare case, te traditional tecnique by trace teorem and scaling argument is difficult to apply. Te lowest version of (3.6) is derived in [17], wic utilized te fact tat te gradient of te linear polynomial is constant. A similar inequality 4

5 as been proved for possibly degenerated sub-elements in [24] for two dimensions, wereas furter justification is required for tree dimensions. In Section 3, we prove te trace and inverse inequalities for polynomials of arbitrary order and for a general class of sub-elements, even toug some of tese may be very irregular in sape. Te stability and te optimal convergence rate of te scemes are tus obtained. Tis paper is organized as follows. In Section 2, we give some preliminary results, wic are used in subsequent sections, and ten we introduce te XFE spaces and reformulate te interface problem (1.1) in two types of DG scemes. In Section 3, we prove a special inequality (3.6) tis is te key step in proving te stability of DG-XFEM wit arbitrary polynomial order for bot te 2-d and 3-d interface problems. Te H 1 - and L 2 - error estimates of bot scemes wic attain te optimal order of te convergence rate in respect to mes size are given in Section 4. We also prove te parameter-friendly property of te second sceme in tis section. Numerical examples are provided in Section 5 to support te teoretical results. 2 XFE and DG scemes for interface problems 2.1 Notation and XFE space We begin by providing some of te notation used in tis paper. Given a bounded domain D R d and a positive integer m, H m (D) is te Sobolev space wit te corresponding usual norm and semi-norm, denoted, respectively, by H m (D) and H m (D). We use for te measure of domains, suc as te volume of a 3-d manifold, te area of a 2-d manifold, or te lengt of a 1-d manifold. In tis paper, d always denotes te dimension of domain Ω, unless stated oterwise. Trougout te paper, stands for C, te generic constant C is independent of bot mes size and te location of te interface relative to te meses. Denote by {T }, a family of conforming, quasi-uniform, and regular partitions of Ω into triangles and parallelograms/tetraedrons and parallelepipeds. For eac element T, we use for its diameter. Let = max{ : T }. As is of regular sape, tere is a constant γ 0 suc tat d γ 0, T. (2.1) We define te set of all elements intersected by Γ as T Γ = { T : Γ 0}. For an element in T Γ, let e = Γ be te part of Γ in. Eac T induces a partition of interface Γ, wic we denote by E Γ = {e : e = Γ, T Γ}. For any T, let i = Ω i denote te part of in Ω i and n i be te unit outward normal vector on 5

6 i wit i = 1, 2. As Γ is of class C 2, it is easy to prove tat (cf.[11, 35]) eac interface segment/patc e is contained in a strip of widt δ and satisfies δ γ 1 2 and n i (x) n i (y) γ 2, x, y e. (2.2) Now, let us simply introduce te XFE space. Let χ i be te caracteristic function on Ω i wit i = 1, 2. Given a mes T, let V be te continuous piecewise polynomial function space of degree p 1 on te mes. Let V 1 := V χ 1 and V 2 := V χ 2. Define te XFE space by V Γ = V 1 + V 2 Γ. Note tat te restrictions of te functions in V in eac sub-domain are standard continuous finite element functions, wereas discontinuity may occur only across Γ. Since te solution of problem (1.1) is non-smoot only in te vicinity of te interface, te XFE space is an appropriate coice for te discretization. Nitsce-XFEM, as noted in te introduction, can tus be regarded as relying on te application of te DG approac on te interface Γ instead of on te interelement edges. 2.2 DG scemes for interface problems For a scalar-valued function v, let v i = v i, and similarly, for a vector-valued function q, we denote q i = q i. We define te weigted average { } and te jump [ ] on e E Γ by {v} = κ 1 v 1 + κ 2 v 2, [v] = v 1 n 1 + v 2 n 2, (2.3) {q} = κ 1 q 1 + κ 2 q 2, [q] = q 1 n 1 + q 2 n 2. (2.4) For te stability analysis of our scemes, we define (κ 1, κ 2 ) on eac element as follows: 0, if i < c 0, κ i = 1, if i > 1 c 0, (2.5) i, oterwise. Clearly, 0 κ i 1 and κ 1 + κ 2 = 1 so tat { } is a convex combination along Γ. Rougly speaking, we adopt te weigt κ i = i suggested in [17] for general sub-elements and we set κ i = 0 for i < c d+1. Actually, we expect tat te contributions of functions wit very small support, say, O( d+1 ), can be eliminated witout influencing te approximation quality significantly. Here, te user-defined constant c 0 2γ 0 γ 1 represents tis tresold and γ 0, γ 1 are constants defined in (2.1) and (2.2), respectively. In Lemma 3.4, we already elaborate te dependence of c 0 on tese generic constants. For an alternative definition of κ i, we refer to [18] and te remarks presented after te proof of Lemma 3.4 in Section 3. For any scalar-valued function v and any vector-valued function w, we ave te following identity: (v 1 n 1 ) w 1 + (v 2 n 2 ) w 2 = [v] {w} + {v}[w] + (κ 2 κ 1 )(v 1 v 2 )[w]. 6

7 Testing te elliptic problem (1.1) by any v V Γ, using integration by parts and te above identity, we ave α(x) u v {α(x) u} [v] = f v + g N (κ 1 v 2 + κ 2 v 1 ). (2.6) Ω 1 Ω 2 Γ Ω Γ We propose two types of DG scemes for interface problem (1.1) on te XFE space V Γ. As te restrictions of te functions in V Γ on eac Ω i are standard continuous finite element functions, we introduce penalty terms for only tose elements cut by te interface in our bilinear forms. Te first sceme is inspired by te interior penalty (IP) metods. Let V = H 2 (Ω 1 Ω 2 ) and V () = V Γ + V. We define a bilinear form on V () V (): B (1) Ω (w, v) := α(x) w v {α(x) w} [v] 1 Ω 2 Γ β [w] {α(x) v} + η β [w] [v], (2.7) were η β 1 Γ [w] [v] is a penalty term acting on eac segment/patc of Γ and η Γ β is a parameter to be specified in Section 4. Here, β is a real number. Wen β = 1, B (1) (, ) is symmetric and corresponds to te symmetric interior penalty Galerkin (SIPG) metod [1, 32], wereas β = 1 gives a non-symmetric interior penalty Galerkin (NIPG) formulation [28]. T Γ To introduce te second type of penalization, for any T Γ a lifting operator r e : [L 2 (e)] d W : were e Γ and e = Γ, we define r e (q) α(x)w = q {α(x)w }, w W, (2.8) W = {w [L 2 (Ω)] d : w i [P p ( i )] d, i = 1, 2 and w Ω\ = 0}. By adding a penalization based on te operator r e, we propose a parameter-friendly DG sceme tat guarantees stability independent of a condition on te stabilization parameter: B (2) Ω (w, v) := α(x) w v {α(x) w} [v] [w] {α(x) v} 1 Ω 2 Γ Γ + η 1 [v] + Γ[w] ηα(x)r e ([w]) r e ([v]), T Γ e E Γ Ω 7

8 were η 1 and η are two positive parameters. Unlike B (1) (, ), in wic te selection of η β depends on te geometric property of te interface and triangulation, we prove tat te sceme (2.11) based on B (2) (, ) as a parameter-friendly feature. Furter, te sparsity of te stiffness matrix is not affected. In fact, te only requirement for te well-posedness of (2.11) is η 1 1 and η 2. Define furter te linear form F (i) ( ), i = 1, 2 on V (): F (1) (v) := f v + g N (κ 1 v 2 + κ 2 v 1 ) β g D {α(x) v} + Ω Γ Γ F (2) (v) :=F (1) (v) + e E Γ Ω T Γ η β Γ g D [v], (2.9) ηα(x)r e (g D ) r e ([v]), wit β = 1. (2.10) Ten, te DG-XFE metod for te interface problem (1.1) is: Find u V Γ B (u, v ) = F (v ), were B (, ) = B (i) (, ) and F ( ) = F (i) ( ) wit i = 1, 2. suc tat v V Γ, (2.11) As te solution u of (1.1) satisfies (2.6), it is easy to ceck tat (2.11) as become an identity for bot scemes if we replace u wit u. Furtermore, te Galerkin ortogonality olds true: B (u u, v ) = 0, v V Γ. (2.12) 2.3 Norm-equivalence property To end tis section, we derive a norm-equivalence result (Lemma 2.2) tat relates te L 2 - norm of any polynomial functions in a bounded convex domain to te L 2 -norm in a subset of comparable size. Tis property consists of te main step toward te proof of te trace and inverse inequalities in te next section. We start from a variant result of norm-equivalence in finite dimensional spaces. Lemma 2.1 Given an integer p 0 and λ (0, 1). For any v(x) P p [0, 1], tere exists a constant C dependent only on λ and p suc tat v L 2 (0,1) C(λ, p) v L 2 (0,λ), x 1 2 v L 2 (0,1) C(λ, p) x 1 2 v L 2 (0,λ). (2.13) For any domain T R d, we say tat T 0 is a omotetic image of T if T 0 = {λ(x x 0 ) + x 0 : x T } for suitable λ > 0 and x 0 R d. Here, x 0 is called te omotetic center, from wic eac point x in T is mapped to a corresponding x in T 0 on te ray x 0 x suc tat x 0 x = λ x 0 x. 8

9 O Σ T T O {}}{}{{} T T T O T Σ Figure 2: Te L 2 -norm in te wole convex domain T is dominated by te L 2 -norm in T, a subset of T, for any v P p (T ). Lemma 2.2 Given an integer p 0 and λ (0, 1). Let T be a closed convex domain in R d wit a (piecewise) smoot boundary. Assume tat T contains a omotetic subset of T wit te scaling factor λ. Ten, for any v P p (T ), we ave v L 2 (T ) C(λ, p + 1) v L 2 (T ), (2.14) were te upperbound constant C(λ, p + 1) is inerited from (2.13). Proof. We only need to consider case T as a omotetic subset of T. As λ < 1, by te fixed-point teorem, te omotetic center O T (Figure 2). Witout loss of generality, we take O as te origin suc tat T can be seen as a continuous contraction from (part of) its boundary Σ (Figure 2), tat is, T = {x : x = s r, s [0, 1], r Σ}, and T = λt. For d = 1, te result of (2.14) is a direct consequence of (2.13) by applying te scaling argument on eac segment of T separated by O. For iger dimensions, te result can be derived by reducing a multiple integral to single integrals. For example, wen d = 2, let Σ be parameterized by x = r(ξ), ξ I. Note tat s r(ξ) r (ξ) is te absolute value of te Jacobian determinant of te mapping x = s r(ξ). We can rewrite te double integrals of v 2 in a (s, ξ)-coordinate system and tereby obtain 1 v 2 L 2 (T ) = r(ξ) r (ξ) dξ v 2 (s r(ξ))sds I 0 λ C 2 (λ, p) r(ξ) r (ξ) dξ v 2 (s r(ξ))sds = C 2 (λ, p) v 2 L 2 (T ), I 9 0

10 were we used te fact of (2.13) as v 2 (s r) is a 1-d polynomial of s for any given ξ. Similarly, if d = 3, caracterized Σ by x = r(ξ, η), (ξ, η) U, ten we ave 1 v 2 L 2 (T ) = r(ξ, η) (r ξ (ξ, η) r η (ξ, η)) dξdη v 2 (s r(ξ, η))s 2 ds U 0 λ C 2 (λ, p + 1) r(ξ, η) (r ξ (ξ, η) r η (ξ, η)) dξdη v 2 (s r(ξ, η))s 2 ds =C 2 (λ, p + 1) v 2 L 2 (T ). Tis completes te proof of Lemma 2.2. U 0 3 Special trace and inverse inequalities In tis section, we give some special trace and inverse inequalities, wic are important in te stability analysis of DG-XFEMs (2.11) for interface problems. Lemma 3.1 For any v H 1 ( i ), te following trace inequality olds: v 2 L 2 (e ) v L 2 ( i ) v H 1 ( i ) + v 2 (s) (3.1) i \e if (0, 0 ]. Here, e = Γ and 0 is a constant independent of te location of Γ relative to. In fact, we can make 0 explicit wit 0 = 1 γ 2 were γ 2 is defined in (2.2). Proof. Let Γ be a line/plane passing at least d points in e. Denote n as te unit outward normal vector to Γ. Ten, we ave 2v v i n i = v 2 n = v 2 n n i = v 2 n n i + v 2 n n i. i e i \e Based on te assumptions tat Γ is C 2 smoot and tat mes size is small enoug (say, 1 γ 2, see (2.2)), we ave 1 2 n n i 1 on e. It follows tat ( v 2 2 2v v ) e i n v 2 n n i i \e wic completes te proof of Lemma 3.1. ( ) 4 v L 2 ( i ) v H 1 ( i ) + v 2 (s), i \e Te estimate of te interpolation error along Γ relies on te following variant of trace inequality, wic is a corollary of te above lemma. We also refer to [17, 33] for details of te proof. 10

11 Lemma 3.2 Tere exists a constant C tat is dependent on Γ but independent of te relative position of Γ to te mes, suc tat for any interface segment/patc e = Γ E Γ, v 2 L 2 (e ) C( 1 v 2 L 2 () + v 2 L 2 () ), v H1 (). (3.2) In te following lemma, we derive trace and inverse inequalities on arbitrary convex domains in R d. We are not aware of any study in wic te same or similar results relating to ig-order polynomial functions are reported. Lemma 3.3 For any convex domain T R d wit a (piecewise) smoot boundary and v P p (T ), te following estimates old: v L 2 (T ) 1 r v L 2 (T ), (3.3) v L 2 ( T ) 1 r 1/2 v L 2 (T ), (3.4) were r is te radius of te largest inscribed ball of T. Here, te idden constants in te inequalities depend only on p and d and are independent of te sape of T. Proof. It was sown in [16, 30] tat for any convex body T R d, tere exists a omotetic pair of boxes B 1 and B 2 suc tat B 1 T B 2. Here, by box we mean a parallelepiped generated by d ortogonal vectors. Furtermore, if we take te omotetic center as te origin suc tat B 2 = λb 1, ten λ is uniformly bounded from below in terms of d. By te scaling argument on te boxes and Lemma 2.2, we ave v L 2 (T ) v L 2 (B 1 ) 1 r v L 2 (B 1 ) 1 r v L 2 (B 2 ) 1 r v L 2 (T ), v P p (B 1 ), wic gives te result of (3.3). Concerning te second inequality, we perform an analysis only for 2-d convex domains. A similar argument can be made for 3-d convex bodies following te guideline for Lemma 2.2. Let T be parameterized (piecewise) by x = r(ξ), ξ I, and let P be te center of te largest inscribed circle in T (Figure 3). P is set as te origin, and T is caracterized by T = {x : x = s r(ξ), s [0, 1], ξ I}. Take v P p (T ) and consider its restriction on T as follows: v 2 (r(ξ)) = 1 0 ( (s 2 v 2 (s r(ξ)) ) 1 ds = 2 s 0 sv(sv) s ds. (3.5) 11

12 P 0 O B 2 T P B 1 T Figure 3: T is convex. Note tat for any fixed ξ, v is a polynomial wose degree does not exceeding p wit respect to s. Applying te following inverse inequality of 1-d polynomial in s (cf. [29]): w L 2 ([0,1]) p 2 w L 2 ([0,1]), w P p ([0, 1]), we ave v 2 p v2 s 2 ds p v2 s ds. By integrating v 2 along T, we find tat 1 v 2 L 2 ( T ) = v 2 r (ξ) dξ p 2 v 2 (s r(ξ)) s r (ξ) dsdξ I p 2 sup ξ I r (ξ) r(ξ) r (ξ) v 2 L 2 (T ). We observe tat G(ξ) := r(ξ) r (ξ) r (ξ) corresponds to te distance from P to te tangent line of T at point P 0 = r(ξ). As T is convex, it always resides on one side of te tangent line. Terefore, G(ξ) r for any ξ I. Ten, we derive tat wic yields te conclusion of (3.4). v 2 L 2 ( T ) p 2 inf ξ I G(ξ) v 2 L 2 (T ) p2 r v 2 L 2 (T ), Te crucial component in regard to establising te stability of bilinear forms is te control on te weigted normal derivatives, wic we state as a trace and inverse inequality in te following lemma. In [17], te validity of tis inequality for p = 1 leads to a euristic coice of weigt κ i = i. Based on a sligt modification of κ i defined in (2.5), we extend te result to arbitrary polynomial degree p. Lemma 3.4 Let γ 0 and γ 1 be constants defined in (2.1) and (2.2), respectively. If we coose c 0 2γ 0 γ 1 in te definition (2.5) of κ, tere exists a positive constant 0 suc tat for all (0, 0 ] and any interface segment/patc e = Γ E Γ, te following estimates old on bot sub-elements of : κ 1/2 i v i L 2 (e ) C 1/2 I 0 v i L 2 ( i ), v i P p ( i ), i = 1, 2. (3.6) 12

13 Proof. By te definition of te weigt (2.5), wen i < c 0 or i > 1 c 0, te result is eiter trivial or is reduced to a standard inverse inequality [33]. Tus, we need consider only te case were i is bounded from below and above by c 0 and 1 c 0. B C E δ E Γ 1 Γ 2 e D D A e Γ 2 Γ 1 Γ 2 Γ 1 e Figure 4: A 2-d simplex intersected by Γ. e = Γ. e is bounded by plane Γ 1 and Γ 2 Γ 2 e Γ 1 Γ 1 Γ 2 e (a) Te intersection is a curved triangle. (b) Te intersection is a curved quadrilateral. Figure 5: Intersection of Γ wit a 3-d simplex. = 1 or 2. We recall tat eac interface segment/patc e = Γ is contained in a strip of widt δ, wic is not greater tan γ 0 2. Denote by Γ 1 and Γ 2 te two boundaries of te strip, wic are parallel to a line/plane passing at least d distinct points in e (Figure 4). Let = 1 or 2 be a sub-element included in. Eac Γ i (i = 1, 2) divides into two polytopes. In tese four polytopes, T 1 denotes te one includes and T 2 denotes te one included in (Figure 4 sows a 2-d example were is te sub-element bounded by Γ, AB, and AC, and we set Γ 1 = DE, Γ 2 = D E, T 1 = ADE, and T 2 = AD E ). We know tat te area/volume of T 1 can be expressed as te integration of te lengt/area of cross-sections along any given direction τ. Take τ to be te normal vector to Γ 1. Let d Γ1 be te maximum distance from points in T 1 to Γ 1, and let δ denote te distance between Γ 1 13

14 and Γ 2. Te measure of eac cross section is less tan d 1. Terefore, if d Γ1 < 2δ, we ave tat T 1 d Γ1 d 1 < 2δd 1 2γ 0 d+1 2γ 0γ 1 c 0. In oter words, te condition c 0 implies tat d Γ1 2δ. Tat is, we need to justify (3.6) under tis condition. Suppose tat d Γ1 is acieved at P 0, and let { } 1 T 0 = 2 (P + P 0) : P T 1. On tis basis, it is easy to verify tat T 0 is included in T 2 wen d Γ1 2δ. As T 0 is a omotetic copy of T 1 wit a scaling factor of λ = 1/2 and a omotetic center P 0, by Lemma 2.2, we ave v L 2 (T 1 ) C(1/2, p) v L 2 (T 0 ) C(1/2, p) v L 2 ( ), v P p(t 1 ). (3.7) In Figure 4-5, we illustrate various intersections wen a simplex is cut by a (d 1)- dimensional manifold Γ. For any v P p ( ), we simply apply (3.1) and (3.7) to obtain v 2 L 2 (e ) v L 2 ( ) v L 2 ( ) + v 2 L 2 ( \e ) v L 2 (T 1 ) v L 2 (T 1 ) + v 2 L 2 ( T 1 ) 1 r v 2 L 2 (T 0 ) 1 r v 2 L 2 ( ), were r is te radius of te largest ball inscribed in T 1 and (3.3) and (3.4) are used on T 1 in te last inequality. Since T 1 r d 1, we obtain κ 1/2 v L 2 (e ) ( r wic completes te proof of Lemma 3.4. ) 1/2 ( r d 1 ) 1/2 v L2( ) v r L2( ) 1 2 v L 2 ( ), Te key step in te above proof is, rougly speaking, tat of converting te original argument of (3.6) in i to a variant in a possibly convex domain T 0 (wit a straigt/planar boundary) included in i. Te estimate on T 0 is relatively easy to obtain wit te elp of Lemma 3.3. We note tat as an alternative, one simple coice of te element-wise defined average is to adopt κ i = 1 if i > 1 and κ 2 i = 0 if i < 1. Tus, for an intersected 2 element, we compute te numerical quantity only on te larger sub-element i wit i = 1 or 2. However, te proof of Lemma 3.4 is also applicable to tis weigting. We use te 14

15 specific definition (2.5) of κ i because te proof of te trace inequality (3.6), and likewise of te trace inequalties, (3.3) and (3.4) is of independent interest in its own rigt. Furtermore, te presence of κ i is essential to keeping te constant in (3.6) independent of te location of te interface relative to te mes. 4 Error Analysis 4.1 Boundedness and stability of B (, ) To consider te boundedness and stability of te primal forms B (i) (, ), we define te following semi-norms and norms for v V (): v 2 1,Ω 1 Ω 2 = α(x) 1/2 v 2 L 2 (Ω), = η β 1 [v] 2 L 2 (e ), v 2 0,E Γ v 2 B (1) v 2 B (2) Here, e = Γ. T Γ = v 2 1,Ω 1 Ω 2 + v 2 0,E Γ = v 2 1,Ω 1 Ω 2 + v 2 0,E Γ + T Γ v 2 0,T Γ Lemma 4.1 (Boundedness of B (1) (, )) We ave B (1) (w, v) C b w B (1) = T Γ η α(x) 1/2 r e ([v]) 2 L 2 (), η 1 β {α(x) v} 2 L 2 (e ), (4.1) + v 2 0,T, wit β = 1. (4.2) Γ v (1) B, w, v V (), (4.3) were C b is a positive constant dependent only on β. Actually, we can make tis upper-bound constant explicit wit C b = ( β 2 + 2β β 2 2β + 2)/2. Proof. Te inequality (4.3) is a direct consequence of te definitions (4.1) and te Caucy- Scwarz inequality. Notice tat norm (4.1) is te natural coice for obtaining te boundedness of te bilinear form B (1) (, ) in V (), wereas te similar continuity of B(2) (, ) in norm (4.2) is only valid in te discrete space V Γ, wic is also a simple result of te Caucy-Scwarz inequality. Te following lemma demonstrates te coercivity of B (i) (, ) in its respective norm 2. B (i) Note tat te second part of te results sows tat sceme (2.10) is parameter-friendly. 15

16 Lemma 4.2 (Stability of B (i) (, )) Tere exists a constant C(1) s > 0 suc tat B (1) (v, v) C(1) s v 2 B (1), v V Γ, (4.4) provided te penalty parameter η β is cosen sufficiently large. Moreover, if η 1 1 and η 2, tere exists a constant C s (2) > 0 suc tat B (2) (v, v) C(2) s v 2 B (2), v V Γ. (4.5) Here, C s (i) η 1 and η. is a positive constant dependent only on te parameter η β or on te parameters Proof. We perform an analysis of te symmetric and non-symmetric variants of B (1) (, ), tat is, β = ±1. For te alternative β in B (1) (, ), te only difference in te stability analysis is te selection of parameter η β and te determination of te corresponding C s (1). We omit te details. For β = 1, by te Caucy-Scwarz inequality, we know tat {α(x) v} [v] 1 2 {α(x) v} L 2 (e ) 1 2 [v] L 2 (e ). e Using te inequality 2ab ɛa ɛ b2, we deduce tat B (1) (v, v) = v 2 1,Ω 1 Ω 2 + v 2 0,E 2 {α(x) v} [v] Γ Γ v 2 1,Ω 1 Ω 2 + v 2 0,E 2 1 Γ 2 {α(x) v} L 2 (e ) 1 2 [v] L 2 (e ) T Γ v 2 1,Ω 1 Ω 2 + v 2 0,E 2 ɛ Γ = v 2 1,Ω 1 Ω 2 + (1 1 2ɛ ) v 2 0,E Γ T Γ η 1 1 {α(x) v} 2 L 2 (e ) + v 2 0,E Γ 4ɛ 2ɛ T Γ η 1 1 {α(x) v} 2 L 2 (e ), were ɛ > 0 is an arbitrary constant number. To estimate te last term, we draw on Lemma 3.4 and obtain {α(x) v} 2 L 2 (e ) 2C max x Ω {α(x)} v 2 1,Ω 1 Ω 2. (4.6) T Γ For any ɛ > 1, if we coose η 1 > 8ɛC max x Ω {α(x)}, we obtain B (1) (v, v) 1 2 ( v 2 1,Ω 1 Ω 2 + v 2 0,E ) C (1) Γ s v 2. (4.7) B (1) 16

17 Tis completes te proof for te case β = 1. For β = 1, te result of (4.4) follows from te identity B (1) (v, v) = v 2 1,Ω 1 Ω 2 + v 2. 0,E Γ Concerning te second formulation, we observe tat {α(x) v} [v] = α(x) v r e ([v]), for any e = Γ and tat B (2) (v, v) = v 2 B (2) e + 2 e E Γ Ω α(x) v r e ([v]) (1 ɛ) v 2 1,Ω 1 Ω 2 + (1 1 ɛη ) v 2 0,T Γ + v 2 0,E. Γ Ten, (4.5) olds wit C s (2) = min(1 ɛ, 1 1 ). In particular, by coosing ɛ = 1 ɛη 2, and η 2, we ave C s (2) = Approximation capability of V Γ We want to sow tat te XFE space as optimal approximation quality for piecewise smoot functions w H p (Ω 1 Ω 2 ). For tis purpose, we construct an interpolant of w by te nodal interpolants of H s -extensions of w 1 and w 2 as follows. Let s 2 be an integer and coose extension operators E i : H s (Ω i ) H s (Ω) suc tat (E i w) Ωi = w and E i w H s (Ω) w H s (Ω i ), i = 1, 2. Let I be te standard nodal interpolation tat is associated wit V and tat satisfies ([29]) v I v H j (Ω) C µ j v H s (Ω), j = 0, 1, 2, (4.8) were v H s (Ω) H0(Ω) 1 wit s 2 and µ = min{p + 1, s}. Denote E i w by w i, and define an interpolation of w V to V Γ by We present an approximation error bound for te XFE space: Π w = χ 1 I w 1 + χ 2 I w 2. (4.9) w Π w 2 B (1) + w Π w 2 B (2) 2(µ 1) w 2 H s (Ω 1 Ω 2 ). (4.10) 17

18 For te proof of tis result, we need to address te interpolation error along te interface. Indeed, we can apply Lemma 3.2 and obtain {α(x) (w Π w)} 2 L 2 (e ) χ i (w I w) 2 L 2 (e ) = ( w i I w i ) 2 L 2 (e ) i=1,2 i=1,2 i=1,2 ( ) w i I w i 2 H 1 () + 2 w i I w i 2 H 2 () (4.11) and 1 [w Π w] 2 L 2 (e ) i=1,2 i=1,2 1 χ i(w I w) 2 L 2 (e ) = i=1,2 1 w i I w i 2 L 2 (e ) ( ) w i I w i 2 L 2 () + 2 w i I w i 2 H 1 (). (4.12) Furtermore, we need te following property of te local lifting operator r e in order to address w Π w 0,E Γ. We note tat te reverse of te following inequality is not generally true, in particular, wen one of sub-elements of degenerates. Lemma 4.3 Tere exists a positive constant C independent of te relative position of Γ respect to suc tat for eac e = Γ E Γ. r e (q) L 2 () C 1 2 q L 2 (e), q [L 2 (e)] 2 (4.13) Proof. We take w = r e (q) in (2.8) and find α(x) 1/2 r e (q) 2 L 2 () q L 2 (e) {α(x)r e (q)} L 2 (e) C 1 2 q L 2 (e) r e (q) L 2 (), were te last inequality follows from (3.6). From (4.11), (4.12), and (4.13), te estimates of te edge terms are reduced to tose of bulk terms, wic ten follow te standard interpolation arguments (4.8). We point out tat (3.2) can be modified by replacing in te rigt-and side by its larger sub-element i wit i = 1 or 2. Tus, te alternative definition of κ i from (2.5) is possible [18, 33] for many oter coices. We empasize tat (3.2) leads to a uniform constant idden in of te interpolation estimates (4.10). 18

19 4.3 Error estimates To summarize, we ave te following error estimate for eac sceme in its respective norm. Teorem 4.4 Assume tat te interface Γ is C 2 smoot and tat te solution of te elliptic interface problem (1.1) satisfies u H s (Ω 1 Ω 2 ), were s 2 is an integer. Let µ = min{p + 1, s}. Te following error estimates old for any (0, 0 ]. (i) If η β is cosen sufficiently large (see (4.4)) and u is te solution to te first sceme of (2.11), ten u u B (1) µ 1 u H s (Ω 1 Ω 2 ), 0 < 0. (4.14) (ii) For any given η 1 1 and η 2 wit u as te solution to te second sceme of (2.11), we ave u u (2) B µ 1 u H s (Ω 1 Ω 2 ), 0 < 0. (4.15) Te idden constants in te above estimates are dependent on te angle condition of te mes T, te degree of te polynomials, te parameter in te sceme, and α(x), but are independent of te location of te interface relative to te mes. Here, te constant 0 is from Lemma 3.4. Proof. Let Π u V Γ be te interpolant of u as defined in (4.9). We recall te stability (4.4) and (4.5) of te bilinear form B (i) (, ). Denote by B (, ) = B (i) (, ) and C s = C s (i) wit i = 1, 2, and we ave C s Π u u 2 B B (Π u u, Π u u ) = B (Π u u, Π u u ), (4.16) were we use te Galerkin ortogonality (2.12) to derive te last identity. Te error estimate for te first sceme follows from te boundedness (4.3) of B (1) (, ) and te triangle inequality u u B (1) u Π u B (1) + Π u u B (1) (1 + C b /C s (1) ) u Π u (1) B. (4.17) Tus, (4.14) is te consequence of (4.10). 19

20 To derive te error estimate for te second sceme, we observe tat for w V and v V Γ, B (2) (w, v ) = α(x) w v [v ] + Ω 1 Ω 2 Γ{α(x) w} α(x) v r e ([w]) e E Γ Ω + η 1 [v ] + Γ[w] ηα(x)r e ([w]) r e ([v ]) T Γ e E Γ Ω ( ) 1 C w 2 + w 2 2 B (1) 0,T v Γ (2) B. (4.18) Instead of using te boundedness of te bilinear form on V (), wic is not generally true for B (2) (, ) in te norm, we substitute w = Π B (2) u u and v = Π u u in (4.18) and obtain tat ( ) u u (2) B C u Π u (1) B + u Π u (2) B. (4.19) Ten, te proof is completed from te interpolation error bound (4.10). We derive te optimal order L 2 -error estimate for te first sceme wen β = 1 by using Nitsce s duality argument (cf. [13]). Te L 2 -error estimate for te second sceme follows a similar procedure plus a variant of (4.18). We omit te details. Consider an auxiliary function w as te solution to te adjoint problem (α(x) w ) = u u, in Ω 1 Ω 2, [w] = 0, [α(x) w] = 0, on Γ, w = 0, on Ω. As Ω is convex, elliptic regularity gives (cf. [2]) (4.20) w H 2 (Ω 1 Ω 2 ) u u L 2 (Ω). (4.21) Teorem 4.5 Under te conditions of Teorem 4.4, te following estimate olds for te first sceme wen β = 1: u u L 2 (Ω) µ u H s (Ω 1 Ω 2 ), 0 < 0. Proof. Let θ = u u. Testing (4.20) by θ and using (2.12), we obtain θ 2 L 2 (Ω) = B(1) (w, θ) = B(1) (θ, w) = B(1) (θ, w Π w), (4.22) 20

21 were Π w V Γ satisfies te estimate (cf. (4.10)) Terefore, from (4.3) and (4.21), w Π w B (1) θ 2 L 2 (Ω) C b w Π w (1) B θ (1) B w H 2 (Ω 1 Ω 2 ). (4.23) θ L 2 (Ω) θ (1) B, tat is θ L 2 (Ω) θ (1) B, wic by using (4.14) completes te proof of Teorem Numerical examples To test te numerical metods, we consider te following example. Let domain Ω be te unit square (0, 1) (0, 1) and interface Γ be te zero level set of te function φ(x) = (x 1 0.5) 2 + (x 2 0.5) 2 1/8 so tat te subdomain Ω 1 is caracterized by φ(x) < 0 and Ω 2 by φ(x) > 0. We use te Cartesian grids to partition te domain Ω into squares of te same size. Let te exact solution be { 1/α1 exp(x u(x) = 1 x 2 ), x Ω 1, 1/α 2 sin(πx 1 ) sin(πx 2 ), x Ω 2. Te rigt-and side can be computed accordingly. We examine te -convergence rate of te first numerical sceme wit β = 1, tat is, te symmetric case, and coose te parameter η 1 = 20 in all cases. Teorems 4.4 and 4.5 imply tat ( 2 1/2 u u 1,Ω1 Ω 2 = α(x) 1/2 (u u ) 2 L 2 (Ω i )) C p, u u L 2 (Ω) C p+1. i=1 Figure 6 (left) plots log 10 ( u u 1,Ω1 Ω 2 / u 1,Ω1 Ω 2 ) versus log 10 (1/) wit = 1/4, 1/8, 1/16, 1/32 for α 1 = 10, α 2 = 1, and p = 1, 2, 3, respectively. Figure 6 (rigt) gives corresponding plots for α 1 = 1, α 2 = 10. Te dotted lines give reference lines of slopes 1, 2, and 3, respectively. Figure 7 sows te results on te relative errors in te L 2 -norm for bot coices of te coefficient α(x). Te convergence rate of O( p ) and O( p+1 ) are observed, respectively, in tese cases, wic confirms our teoretical results. References [1] D. N. Arnold, An interior penalty finite element metod wit discontinuous elements, SIAM J. Numer. Anal. 19 (1982),

22 p =1 p =2 p = p = p = p = Figure 6: Error Reduction of log 10 ( u u 1,Ω1 Ω 2 / u 1,Ω1 Ω 2 ) to log 10 (1/). Left: α 1 = 10, α 2 = 1. Rigt: α 1 = 1, α 2 = p =1 3 4 p =2 6 4 p = p =1 3 6 p =2 4 8 p = Figure 7: Error Reduction of log 10 ( u u L 2 (Ω)/ u L 2 (Ω)) to log10 (1/). Left: α 1 = 10, α 2 = 1. Rigt: α 1 = 1, α 2 =

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