L 2 -Error Analysis of an Isoparametric Unfitted Finite Element Method for Elliptic Interface Problems

Transcription

1 A P R I L P R E P R I N T L 2 -Error Analysis of an Isoparametric Unfitted Finite Element Metod for Elliptic Interface Problems Cristop Lerenfeld * and Arnold Reusken Institut für Geometrie und Praktisce Matematik Templergraben 55, Aacen, Germany * Institut für Numerisce und Angewandte Matematik, WWU Münster, D Münster, Germany; lerenfeld@uni-muenster.de Institut für Geometrie und Praktisce Matematik, RWTH Aacen University, D Aacen, Germany; reusken@igpm.rwt-aacen.de

2 L 2 -ERROR ANALYSIS OF AN ISOPARAMETRIC UNFITTED FINITE ELEMENT METHOD FOR ELLIPTIC INTERFACE PROBLEMS CHRISTOPH LEHRENFELD AND ARNOLD REUSKEN Abstract. In te context of unfitted finite element discretizations te realization of ig order metods is callenging due to te fact tat te geometry approximation as to be sufficiently accurate. Recently a new unfitted finite element metod was introduced wic acieves a ig order approximation of te geometry for domains wic are implicitly described by smoot level set functions. Tis metod is based on a parametric mapping wic transforms a piecewise planar interface (or surface) reconstruction to a ig order approximation. In te paper [C. Lerenfeld, A. Reusken, Analysis of a Hig Order Finite Element Metod for Elliptic Interface Problems, arxiv , Accepted for publication in IMA J. Numer. Anal.] an a priori error analysis of te metod applied to an interface problem is presented. Te analysis reveals optimal order discretization error bounds in te H 1 -norm. In tis paper we extend tis analysis and derive optimal L 2 -error bounds. Key words. unfitted finite element metod, isoparametric finite element metod, ig order metods, geometry errors, interface problems, Nitsce s metod AMS subject classifications. 65M06, 65D05 1. Introduction. In recent years tere as been a strong increase in te researc on development and analysis of unfitted finite element metods, cf. for example [1, 5, 8, 15, 16, 17, 19, 27]. Te design, realization and analysis of unfitted finite elements metods wic are iger order accurate is a callenging task. Especially in a setting were geometries are only implicitly described, e.g. by a zero level of a level set function, accurate numerical integration is difficult. Different tecniques ave been developed to overcome tis difficulty [29, 26, 32, 30, 6, 9, 12, 3, 5, 14]. For a furter discussion of te relevant literature we refer to [24, Section 1.2]. Recently, in [22], we introduced a new ig order unfitted finite element metod for scalar interface problems based on isoparametric mappings wic is suitable for level set descriptions and iger order unfitted finite elements. Te metod is geometry based and is applicable to a wide range of problems, e.g. Stokes interface problems [21], fictitious domain problems [23] or surface PDEs [13]. A detailed a-priori error analysis of te metod applied to a scalar elliptic interface model problem is given in [24]. Optimal H 1 -norm discretization error bounds are derived for te case of a polygonal domain. In tis paper we consider te same elliptic interface problem as in [24] and extend te existing analysis in two directions. First, we allow for a piecewise smoot, not necessarily polygonal, domain, wic is approximated wit a iger order isoparametric boundary approximation. Secondly, we complement te existing analysis wit optimal L 2 -estimates. Te paper is organized as follows. In Section 2 we introduce te model interface problem. We assume a level set description of te interface and discuss a standard piecewise planar interface approximation in Section 3. A key ingredient in te iger order unfitted finite element metod is te isoparametric mapping used to obtain iger order geometry approximations. In Section 4 we present te construction of tis mapping and te isoparametric unfitted finite element metod for te elliptic Institut für Numerisce und Angewandte Matematik, University of Göttingen, D Göttingen, Germany; lerenfeld@mat.uni-goettingen.de Institut für Geometrie und Praktisce Matematik, RWTH Aacen University, D Aacen, Germany; reusken@igpm.rwt-aacen.de 1

3 interface problem. In particular it is explained ow te same extension tecnique, wic is a standard component in isoparametric finite element metods for ig order approximation of smoot boundaries, is applied bot for te boundary and interface mes transformation. A main contribution of tis paper is te error analysis of te metod, i.e. te derivation of optimal H 1 - and L 2 -error bounds. Te H 1 -error results obtained in [24] (for a polygonal domain) form te basis for tis analysis. In Section 5 we recall te most important results from [24] and adapt tese to te case of a iger order isoparametric boundary approximation. Te L 2 error analysis is presented in Section 6. Finally, in Section 7 we give results of a numerical experiment. We are aware of te fact tat parts of te paper contain results tat are essentially te same as in [24]. In particular te material given in te sections 3, 4.1, 4.2 and 4.4 can also be found in [24]. We decided to include tis material to make tis paper more self-contained and tus improve its readability. 2. Model interface problem. We consider te model problem div(α i u) = f i in Ω i, i = 1, 2, (2.1a) [α u] Γ n Γ = 0, [u] Γ = 0 on Γ, (2.1b) u = 0 on Ω. (2.1c) Here, Ω 1 Ω 2 = Ω R d, d = 2, 3, is a nonoverlapping partitioning of te domain, Γ = Ω 1 Ω 2 is te interface and [ ] Γ denotes te usual jump operator across Γ. Te source terms f i are given on eac subdomain Ω i, i = 1, 2. In te remainder we also use te source term f on Ω wic we define as f Ωi = f i, i = 1, 2. We assume Γ to be caracterized as te zero level of a level set function (not necessarily a distance function). Te diffusion coefficient α is assumed to be piecewise constant, i.e. it as a constant value α i > 0 on eac sub-domain Ω i. Te weak formulation of tis problem is as follows: determine u H0 1 (Ω) suc tat α u v dx = fv dx for all v H0 1 (Ω). Ω Ω In te error analysis we assume tat Ω 1 is strictly contained in Ω, i.e., Γ = Ω 1 and dist(γ, Ω) > 0. Furtermore, we assume Γ to be (globally) smoot and Ω to be piecewise smoot. 3. Boundary parametrization and level set representation of te interface. Te domain Ω, wic as a (piecewise) smoot boundary is approximated wit a family of polygonal approximations {Ω lin } >0 corresponding to a family {T } >0 of simplicial sape regular triangulations of {Ω lin } >0 wic are not fitted to Γ, cf. Fig In te analysis we assume quasi-uniformity of te triangulations, ence, T := diam(t ), T T. In te remainder we take a fixed Ω lin wit corresponding triangulation T and, to simplify te presentation, drop te index. Te standard finite element space of continuous piecewise polynomials up to degree k wit respect to te domain Ω lin is denoted by V k: V k := { v C(Ω lin ) v T P k for all T T, v Ω lin = 0 }. Te nodal interpolation operator in V k is denoted by I k. In te remainder we take a fixed k 1. We are particularly interested in k 2. Tis k denotes te polynomial degree of te finite element functions used in our finite element metod (explained 2

4 in Section 4.4). To obtain a ig order finite element discretization of (2.1) we need sufficiently accurate representations of Ω and Γ. We now addresss tese representations. For simplicity we assume tat all boundary vertices of T = Ω lin are located on Ω (tis assumption is not essential, cf. [4], Sect. 4.7). On T we assume a given parametrization of te exact boundary Ω of te form g b (x) = x + χ b (x), (3.1) suc tat g b : T Ω is a bijection. We assume tat Ω as smootness properties suc tat te function χ b can be cosen piecewise smoot on T and satisfies χ b, T + Dχ b, T 2 and max F F( T ) Dl χ b,f 1, l = 2,.., k + 1. (3.2) olds. Here F( T ) denotes te set of all boundary edges (d = 2) or boundary faces (d = 3). Note tat χ b (x i ) = 0 at all boundary vertices x i of Ω lin. Te function χ b will be input for te parametric mapping used in our metod. We assume tat te smoot interface Γ is te zero level of a smoot level set function φ : Ω R, i.e., Γ = { x Ω φ(x) = 0 } and Ω i = {x Ω φ(x) 0}. Tis level set function is not necessarily close to a distance function, but as te usual properties of a level set function: φ(x) 1, D 2 φ(x) c for all x in a neigborood U of Γ. (3.3) In te error analysis we assume tat te level set function as te smootness property φ C k+2 (U). As input for te parametric mapping we need an approximation φ V k of φ, and we assume tat tis approximation satisfies te error estimate max φ φ m,,t U k+1 m, 0 m k + 1. (3.4) T T Here m,,t U denotes te usual semi-norm on te Sobolev space H m, (T U) and te constant used in depends on φ but is independent of. Note tat (3.4) implies te estimate φ φ,u + (φ φ),u k+1. (3.5) Te zero level of te finite element function φ (implicitly) caracterizes te discrete interface. Te piecewise linear nodal interpolation of φ is denoted by ˆφ = I 1 φ. Hence, ˆφ (x i ) = φ (x i ) at all vertices x i in te triangulation T. Te low order geometry approximation of te interface, wic is needed in our discretization metod, is te zero level of tis function: Γ lin := { ˆφ = 0}. Te subdomains corresponding to Γ lin are denoted by Ω lin i = {x Ω lin ˆφ (x) 0}. Note tat Ω lin 1 Ω lin 2 = Ω lin Ω. All elements in te triangulation T wic are cut by Γ lin are collected in te set T Γ := {T T T Γ lin }. Te corresponding domain is Ω Γ := {x T T T Γ }. We note tat Ω lin and Γ lin are (only) second order accurate approximations of Ω and Γ, respectively. To acieve iger order accuracy wit respect to te interface and te boundary approximation we consider a parametric mes transformation in section

5 4. Isoparametric mapping. Te isoparametric mapping Θ, defined on Ω lin, tat is used in te finite element metod is based on a local mapping Θ Γ on ΩΓ wic is combined wit a suitable extension tecnique. Te latter is taken suc tat Θ Γ is smootly extended and Θ yields a sufficiently accurate boundary approximation. In te error analysis we need a mapping Ψ : Ω lin Ω wic is sufficiently close (in a iger order sense) to Θ and as te properties Ψ(Γ lin ) = Γ, Ψ( Ω lin ) = Ω. Te construction of tis Ψ is also based on a local mapping Ψ Γ, tat is very similar to Θ Γ, wic is extended wit te same tecnique as used in te extension of Θ Γ. Te mappings Θ and Ψ for te case witout boundary approximation (i.e., Ω is polygonal) are explained in detail in [24]. In te subsections below we explain ow te construction of [24] can be extended to allow for isoparametric boundary approximation. In section 4.1 we recall te definitions of te local mappings Θ Γ, ΨΓ, wic are te same as in [24]. In section 4.2 we discuss a general boundary data extension tecnique, wic is essentially te same as te extension metod used in isoparametric finite elements [25, 2]. In section 4.3 tis extension tecnique is applied to bot Θ Γ and Ψ Γ resulting in te global mappings Θ and Ψ. We give some results, e.g., on te smootness of te extensions and on te approximation error in Θ Ψ, wic are derived in [24]. Tese results are used in te error analysis Local mappings. In te construction of te local mapping Θ Γ we need a projection step from a function wic is piecewise polynomial but possibly discontinuous (across element interfaces) to te space of continuous finite element functions. Let C(T Γ ) := C(T ) and V k(ωγ ) := V k ΩΓ. We introduce a projection operator T T Γ P Γ : C(T Γ ) d V k(ωγ ) d. Te projection operator relies on a nodal representation of te finite element space V k(ωγ ). Te set of finite element nodes x i in T Γ is denoted by N (T Γ ), and N (T ) denotes te set of finite element nodes associated to T T Γ. All elements T T Γ wic contain te same finite element node x i form te set denoted by ω(x i ) := { T T Γ x i N (T ) }, x i N (T Γ ). Let denote te cardinality of te set ω(x i ) and let ψ i be te nodal basis function corresponding to x i. We define te projection operator P Γ : C(T Γ ) d V k(ωγ ) d as P Γ v := x i N (T Γ ) A xi (v) ψ i, wit A xi (v) := 1 ω(x i ) T ω(x i) v T (x i ), x i N (T Γ ). (4.1) Tis is a simple and well-known projection operator considered also in e.g., [28, Eqs.(25)-(26)] and [11]. Te construction of Θ Γ is motivated by te following mapping ΨΓ C(Ω Γ ), defined in (4.3) below. We introduce te searc direction G := φ and a function d : Ω Γ R defined as follows: d(x) is te (in absolute value) smallest number suc tat φ(x + d(x)g(x)) = ˆφ (x) for x Ω Γ. (4.2) (Recall tat ˆφ is te piecewise linear nodal interpolation of φ.) Given te function dg C(Ω Γ ) d H 1, (Ω Γ ) d we define: Ψ Γ (x) := x + d(x)g(x), x Ω Γ. (4.3) Note tat te function d and mapping Ψ Γ depend on, troug ˆφ in (4.2). We do not sow tis dependence in our notation. By construction te mapping Ψ Γ as te 4

6 property Ψ Γ (Γ lin ) = Γ. Furter properties of Ψ Γ are derived in [24, Corollary 3.2 and Lemma 3.4] and given in te following lemma. Lemma 4.1. Te following olds: d(x i ) k+1 for all vertices x i of T T Γ, (4.4a) Ψ Γ id,ω Γ + DΨ Γ I,Ω Γ 2, (4.4b) max D l Ψ Γ,T 1, l k + 1. (4.4c) T T Γ Te mapping Ψ Γ will be used in te error analysis, cf. section 4.3. We now explain te construction of Θ Γ, wic consists of two steps. In te first step we introduce a discrete analogon of Ψ Γ defined in (4.3), denoted by Ψ Γ. Based on tis Ψ Γ, wic can be discontinuous across element interfaces, we obtain a continuous transformation Θ Γ C(ΩΓ ) d by averaging wit te projection operator P Γ. For te construction of Ψ Γ we need an efficiently computable and accurate approximation of G = φ on T Γ. For tis we consider te following two options G (x) = φ (x), or G (x) = (P Γ φ )(x), x T T Γ. (4.5) Let E T φ be te polynomial extension of φ T. We define a function d : T Γ [ δ, δ], wit δ > 0 sufficiently small, as follows: d (x) is te (in absolute value) smallest number suc tat E T φ (x + d (x)g (x)) = ˆφ (x), for x T T Γ. (4.6) Clearly, tis d (x) is a reasonable approximation of te steplengt d(x) defined in (4.2). Given te function d G C(T Γ ) d we define Ψ Γ (x) := x + d (x)g (x) for x T T Γ, (4.7) wic approximates te function Ψ Γ defined in (4.3). To remove possible discontinuities of Ψ Γ in ΩΓ we apply te projection to obtain Θ Γ := P Γ Ψ Γ = id + P Γ (d G ). (4.8) Te approximation result in te following lemma is from Lemma 3.6 in [24]. Lemma 4.2. Te estimate olds. k+1 r max D r (Θ Γ T T Γ Ψ Γ ),T k+1 (4.9) r= Finite element boundary data extension metod. Let ˆT T be a subset of te triangulation suc tat ˆΩ lin := T ˆT T is a connected subdomain of Ω lin. We also need te notation ˆT = ˆΩ lin, ˆT bnd := { T ˆT T ˆT }, ˆT int = ˆT \ ˆT bnd, cf. Figure 4.1 (left). Given suc a connected triangulation ˆT, boundary data g C( ˆΩ lin ) and a polynomial degree k 1, a linear extension operator E k ( ˆT, g) is treated in [24], wic is essentially te same as te finite element extension tecnique studied in [25, 2]. Tis type of extension operator is a standard component in isoparametric finite element metods for ig order boundary approximations. 5

7 T ˆT T T ˆT ˆT int ˆT bnd T T Γ T T 1 T T 2 Γ lin Fig Connected subtriangulation ˆT T wit ˆT int, ˆT bnd and ˆT (left) and decomposition of a triangulation T into T Γ, T 1 and T 2 (rigt). Tis operator is local, meaning tat it is applied elementwise for T ˆT bnd. Its construction is ierarcical in te sense tat first an extension based on values at vertices lying on ˆT is determined, ten an extension from edges lying on ˆT and finally (for te tree-dimensional case) an extension of te data from faces lying on ˆT is determined. For a precise definition of tis operator we refer to [24]. In te following lemma we summarize some properties of tis operator wic are derived in [24, 25, 2]. Lemma 4.3. Let V( ˆT ) denote te set of vertices in ˆT and F( ˆT ) te set of all edges (d = 2) or faces (d = 3) in ˆT. For E(g) := E k ( ˆT, g) te following olds E(g) C(ˆΩ lin ), E(g) = 0 on ˆT int, E(g) ˆT = g. (4.10a) For all T ˆT bnd : if T ˆT V( ˆT ) ten E(g) P 1 (T ). (4.10b) For all T ˆT bnd : if g P k (T ˆT ) ten E(g) P k (T ). Furtermore, D n E(g),ˆΩlin max F F( ˆT ) r=n k+1 r n D r g,f (4.10c) + n max g(x i ), n = 0, 1, (4.10d) x i V( ˆT ) k+1 max D n E(g),T r n D r g,f, n = 2,..., k + 1, (4.10e) T ˆT r=n for all g C( ˆT ) suc tat g C k+1 (F ) for all F F( ˆT ). Te results in (4.10a)-(4.10c) give basic structural properties of te extension operator. In particular te result in (4.10c) yields tat piecewise polynomial boundary data result in piecewise polynomial extensions of te same degree. Te results in (4.10d), (4.10e) measure smootness properties of te extension in terms of te smootness of te boundary data Global isoparametric mapping. We use te local mappings Θ Γ, ΨΓ on Ω Γ and te boundary parametrization χ b as in (3.1), and combine tese wit te extension tecnique to obtain global mappings Θ and Ψ. For tis we decompose te triangulation T into disjoint subsets as follows T = T Γ T 1 T 2, wit T 1 := { T T ( ˆφ ) T < 0 }, T 2 := { T T ( ˆφ ) T > 0 }. 6

8 We assumed tat Ω 1 (were φ < 0 olds) is strictly contained in Ω. Hence (for sufficiently small) we ave tat T 1 is strictly contained in Ω. On T 1 we ave boundary data Θ and Ψ Γ, wic will be extended to T 1, cf. below. Te boundary of T 2 consists of two disjoint parts, namely T 2 = T ( T 2 \ T ), cf. Figure 4.1 (rigt). On ( T 2 \ T ) T Γ we ave boundary data Θ and Ψ Γ. On T we ave boundary data given by te parametrization χ b. Below, boundary data on te two parts T 2 \ T and T of T 2 are denoted as a pair of functions (g 1, g 2 ). Using te extension tecnique, te local mappings Ψ Γ and Θ Γ on ΩΓ we can define global mappings Ψ, Θ as follows: Ψ Γ ( on T Γ, Ψ := id +E k T1, (Ψ Γ id) ) on T 1, ( id +E k T2, (Ψ Γ id, χ b ) ) (4.11a) on T 2, Θ Γ ( on T Γ, Θ := id +E k T1, (Θ Γ ( id)) on T 1, id +E k T2, (Θ Γ id, I kχ b ) ) (4.11b) on T 2. Here I k χ b denotes te piecewise polynomial nodal interpolation of te boundary parametrization χ b on te edges (d = 2) or faces (d = 3) tat are on T. boundary approximation Ω Ω Θ Ω lin T Ωlin Ψ Ω Ω Ω 1, interface approximation Ω lin 1 Ω 1 Γ Θ Γ lin Ψ Γ Ω 2, Ω lin 2 Ω 2 Fig Sketc of geometries. Te basis is te polygonal approximation (center) of te boundary (top row) and te interface (bottom row). Te parametric mapping Θ (left) is used in te discretization metod and essential for te iger order accuracy. Te mapping Ψ (rigt) maps te discrete boundary and interface approximations to te exact geometries and is used in te error analysis below Isoparametric unfitted finite element metod. We introduce te isoparametric unfitted finite element metod, based on te isoparametric mapping Θ, for te discretization of te model elliptic interface problem (2.1). We define te isoparametric Nitsce unfitted FEM as a transformed version of te original Nitsce unfitted FE discretization [17] wit respect to te interface approximation Γ = Θ (Γ lin ). We introduce some furter notation. Te standard unfitted space w.r.t. Γ lin is denoted by V Γ := V k Ω lin,1 V k Ω lin,2. 7

9 To simplify te notation we do not explicitly express te polynomial degree k in V Γ. Te isoparametric unfitted FE space is defined as V Γ,Θ := { v Θ 1 v V Γ } = { ṽ ṽ Θ V Γ }. (4.12) Note tat functions from V,Θ Γ are defined on te isoparametrically transformed domain Ω = Ω 1, Ω 2,, wit Ω i, := Θ (Ω lin i ). Based on tis space we formulate a discretization of (2.1) using te Nitsce tecnique [17] wit Ω i,, i = 1, 2, as numerical approximation of te subdomains Ω i : determine u V,Θ Γ suc tat A (u, v ) := a (u, v ) + N (u, v ) = f (v ) for all v V Γ,Θ, (4.13) wit te bilinear and linear forms a (u, v) := 2 α i u v dx, (4.14a) Ω i, i=1 N (u, v) := N(u, c v) + N(v, c u) + N(u, s v), N(u, c v) := { α v } n[u] ds, N(u, s v) := ᾱ λ [u][[v ] ds, Γ Γ (4.14b) (4.14c) for u, v V,Θ Γ + V reg, wit V reg, := H 1 (Ω ) (H 2 (Ω 1, ) H 2 (Ω 2, )). Furtermore, n = n Γ denotes te outer normal on te boundary Γ of Ω 1, and ᾱ = 1 2 (α 1 + α 2 ) te mean diffusion coefficient. For te averaging operator { } tere are different possibilities. We use {w } := κ 1 w Ω1, + κ 2 w Ω2, wit a Heaviside coice were κ 1 + κ 2 = 1 wit κ 1 = 1 if T 1 > 1 2 T and κ 1 = 0 oterwise. Here, T i = T Ω lin i, i.e. te cut configuration on te undeformed mes is used. Tis coice in te averaging renders te sceme in (4.13) stable (for sufficiently large λ) for arbitrary polynomial degrees k, independent of te cut position of Γ (Lemma 5.2 in [24]). A different coice for te averaging wic also results in a stable sceme is κ i = T i / T. In order to define te rigt and side functional f we first assume tat te source term f i : Ω i R in (2.1a) is (smootly) extended to Ω i,. Tis extension is denoted by f i, and satisfies f i, = f i on Ω i. We define f on Ω by f Ωi, := f i,, i = 1, 2, ence, f (v) := f v dx = f i, vdx. (4.15) Ω Ω i, i=1,2 Remark 1. For te implementation of tis metod we use te same tecnique as in standard isoparametric finite element metods. In te integrals we apply a transformation of variables y := Θ 1 (x). For example, te bilinear form a (u, v) ten results in a (u, v) := α i DΘ T u DΘ T v det(dθ ) dy. (4.16) i=1,2 Ω lin i Based on tis transformation te implementation of integrals is carried out as for te case of te piecewise planar interface Γ lin. Te additional variable coefficients DΘ T, det(dθ ) are easily and efficiently computable using te property tat Θ is a finite element (vector) function. 8

10 5. Preliminaries for te error analysis. In tis section we collect results from [24] tat we need in te proof of te L 2 -estimate in Section 6. In [24] te case witout an isoparametric boundary approximation is considered. It turns out tat te results tat we need from [24] also old for te case wit isoparametric boundary approximation, i.e., wit te mapping Θ in (4.11b), and tat te proofs given in [24] require only minor modifications. In te proofs below we refer to te analysis given in [24] and explain only te modifications needed due to te additional isoparametric boundary approximation. In te error analysis we use te norm v 2 := v [v ] 2 1 2,,Γ + {α v } 2 1 2,,Γ, (5.1) wit v 2 ± 1 2,,Γ := (ᾱ/)±1 v 2 L 2 (Γ ) and v 2 1 := α i v 2 L 2 (Ω i, ). (5.2) i=1,2 Note tat te norms are formulated wit respect to Ω i, = Θ (Ω lin i ) and Γ = Θ (Γ lin ) and include a scaling depending on α. Te norm and te bilinear forms in (4.14) are well-defined on V Γ,Θ + V reg,. Lemma 5.1 (Ellipticity and continuity). For λ sufficiently large te estimates A (u, u) u 2 for all u V Γ,Θ, (5.3a) A (u, v) u v for all u, v V Γ,Θ + V reg, (5.3b) old. Proof. Te proof given in [24, Lemma 5.2] applies witout any modifications. We note tat a crucial component in te proof is te assessment tat L 2 and H 1 norms on Ω i, and Γ are equivalent to corresponding norms of mapped functions on Ω lin i and Γ lin, cf. [24, Lemma 3.12]. Due to tis te ellipticity and continuity can equivalently be sown on a piecewise linear interface approximation wit Γ lin. In te remainder we assume tat λ is taken sufficiently large suc tat te results in Lemma 5.1 old. Ten te discrete problem (4.13) as a unique solution. A key component of te error analysis in [24] is te mapping Φ := Ψ Θ 1. For te derivation of properties of tis mapping we first derive useful results for Ψ. Teorem 5.2. For Ψ te following olds: Ψ id,ω + DΨ I,Ω 2, (5.4) max T T Dl Ψ,T 1, 0 l k + 1. (5.5) Proof. On T Γ te results are a direct consequence of Lemma 4.1. On T 1 T 2 te result in (5.5) for l = 0, 1 follows from (5.4) and for l 2 it follows from (4.10e) combined wit (4.4c) and (3.2). It remains to derive te estimate as in (5.4) on T 1 T 2. We consider T 2 and write E(Ψ Γ id, χ b ) := E k ( T2, (Ψ Γ id, χ b ) ). Using (4.10d) and 9

11 χ b (x i ) = 0 at vertices x i T we get Ψ id,t2 + DΨ I,T2 = E(Ψ Γ id, χ b ),T2 + DE(Ψ Γ id, χ b ),T2 r( ) max F F( T 2\ T ) Dr (Ψ Γ id),f + max F F( T ) Dr χ b,f (4.10d) k+1 r=0 + max x i V( T 2\ T ) (ΨΓ id)(x i ) Ψ Γ id,t Γ + DΨ Γ I,T Γ + 2 max }{{} max D l Ψ Γ,T 2 l k+1 T T }{{ Γ } 2 (Lemma 4.1) 1 (Lemma 4.1) k+1 + r r=0 max F F( T ) Dr χ b,f + max }{{} 2 (3.2) For T 1 similar arguments can be applied. x i V(T Γ ) d(x i ) 2. }{{} 2 (Lemma 4.1) Te proof above generalizes te one given in [24] since it allows a boundary approximation via te function g b (x) = x + χ b (x). From (5.4) it follows tat, for sufficiently small, Ψ is a bijection on Ω. Furtermore tis mapping induces a family of (curved) finite elements tat is regular of order k, in te sense as defined in [2]. Te corresponding curved finite element space is given by V,Ψ := { v Ψ 1 v V k }. (5.6) Due to te results in Teorem 5.2 te analysis of te approximation error for tis finite element space as developed in [2] can be applied. Corollary 4.1 from tat paper yields tat tere exists an interpolation operator Π : H k+1 (Ω) V,Ψ suc tat u Π u L2 (Ω) + u Π u H1 (Ω) k+1 u H k+1 (Ω) for all u H k+1 (Ω). (5.7) For sufficiently small, te mapping Φ = Ψ Θ 1 : Ω Ω is a bijection and as te property Φ (Γ ) = Γ, cf. Fig In te remainder we assume tat is sufficiently small suc tat Φ is a bijection. It as te smootness property Φ C(Ω ) d C k+1 (Θ (T )) d. In te following lemma we derive furter properties of Φ tat will be needed in te error analysis. Lemma 5.3. Te following olds: Θ Ψ,Ω lin + D(Θ Ψ),Ω lin k+1, (5.8) Φ id,ω lin + DΦ I,Ω lin k+1. (5.9) Proof. Te estimate (5.8) follows by using te linearity of te extension operator E k, cf. (4.11), and ten using argmuments very similar to te ones used in te proof of Teorem 5.2. Note tat Φ id = (Ψ Θ )Θ 1. From te results in (5.4) and (5.8) it follows tat Θ 1,Ω 1, DΘ 1,Ω 1. Hence, te estimate in (5.9) follows from te one in (5.8). 10

12 We define V reg := H 1 (Ω) H 2 (Ω 1 Ω 2 ). Related to Φ we define te following transformed unfitted finite element space, cf. (4.12), V Γ,Φ := {v Φ 1, v V Γ,Θ} H 1 (Ω 1 Ω 2 ), and te linear and bilinear forms A(u, v) := a(u, v) + N(u, v), f(v) = Ω fv dx, u, v V Γ,Φ + V reg, (5.10) wit te bilinear forms a(, ), N(, ) as in (4.14) wit Ω i, and Γ replaced wit Ω i and Γ, respectively. In te following lemma a consistency result is given. Lemma 5.4 (Consistency). Let u V reg be a solution of (2.1). Te following olds: A(u, v) = f(v) for all v V Γ,Φ + V reg. (5.11) Proof. Proof is given in [24]. No modifications are needed. We introduce te subdomain U δ := { x Ω dist(x, Γ) δ or dist(x, Ω) δ }, wit δ > 0 (sufficiently small), consisting of a tubular neigborood of te interface Γ and a strip adjacent to Ω. To bound te consistency errors wit respect to A we use te following result. Lemma 5.5. For δ > 0 sufficiently small and u H 1 (Ω 1 Ω 2 ) tere olds Proof. See [10, Lemma 4.10]. u L2 (U δ ) cδ 1 2 u H1 (Ω 1 Ω 2). (5.12) Wit tis result a consistency bound from [24] can be improved. Lemma 5.6 (Consistency bounds). Let u V reg be a solution of (2.1). We assume f H 1, (Ω 1 Ω 2 ) and a data extension f, used in (4.15), tat satisfies f H 1, (Ω 1, Ω 2, ) f H 1, (Ω 1 Ω 2). Te following estimates old for w V Γ,Θ + V reg, : A(u, w Φ 1 ) A (u Φ, w ) k u H 2 (Ω 1 Ω 2) w, (5.13a) f(w Φ 1 ) f (w ) k+1 f H 1, (Ω 1 Ω 2) w. (5.13b) Proof. In [24, Lemma 5.13] equation (5.13a) is derived for te case witout isoparametric boundary approximation. Te proof applies, witout modifications, also to te case wit isoparametric boundary approximation. In [24, Lemma 5.13] a bound as in (5.13b) wit k+1 replaced by k is derived. We now sow ow one obtains te improved bound in (5.13b). Define := U δ, wic δ := c and c > 0 sufficiently large suc tat Φ 1 id only on. Note tat f(w Φ 1 ) f (w ) = f(w Φ 1 ) dx f w dx Ω Ω 2 f i (w Φ 1 ) dx f i, w dx Ω i Ω i,. i=1 11

13 For i = 1, 2 and using f i, = f i on Ω i we get: f i (w Φ 1 ) dx f i, w dx Ω i Ω i, = ( ) det(dφ )(f i, Φ ) f i, w dx Ω i, (det(dφ ) I)f i, w dx + det(dφ )((f i, Φ ) f i, )w dx Ω i, Ω i, 1 2 det(dφ ) I,Ω f,ω w L 2 ( ) det(dφ ),Ω f H 1, (Ω 1, Ω 2, ) Φ id,ω w L2 ( ) k+ 1 2 f H 1, (Ω 1, Ω 2, ) w L 2 ( ). In te last inequality we used te properties of te transformation Φ from (5.9). Next, we apply Lemma 5.5 to gain anoter factor 1 2 : w L 2 ( ) w Φ 1 L 2 (Φ ( )) w Φ 1 L 2 (U δ ) δ 1 2 w Φ 1 H 1 (Ω 1 Ω 2) 1 2 w H 1 (Ω 1, Ω 2, ) 1 2 w. Combining tese results completes te proof. (5.14) Te main result of [24] is given in te following teorem. Teorem 5.7 (Discretization error bound). Let u be te solution of (2.1) and u V Γ,Θ te solution of (4.13).We assume tat u Hk+1 (Ω 1 Ω 2 ) and te data extension f satisfies te condition in Lemma 5.6. Ten te following olds: u Φ u k ( u H k+1 (Ω 1 Ω 2) + f H 1, (Ω 1 Ω 2)), (5.15) u u Φ 1 H 1 (Ω 1 Ω 2) k ( u H k+1 (Ω 1 Ω 2) + f H 1, (Ω 1 Ω 2)). (5.16) Proof. Te result (5.16) easily follows from (5.15) using te definition of te norm and properties of Φ. Te proof of (5.15) in [24] applies wit only minor modifications. A key ingredient in te proof is te following approximation property, wic olds for arbitrary u H k+1 (Ω 1 Ω 2 ): inf u Φ v k u H k+1 (Ω 1 Ω 2), (5.17) v V,Θ Γ wic is based on te isoparametric interpolation error bound (5.7). 6. L 2 -error bound. In tis section we present an L 2 -norm discretization error bound for te unfitted finite element metod in (4.13). Te key ingredients are a duality argument, te already available H 1 -error bound given in Teorem 5.7 and te consistency error bound (5.13a)-(5.13b). In te remainder we assume tat te smootness conditions on te solution u and te rigt and side f as formulated in Teorem 5.7 are satisfied. We define te discretization error e := u u Φ 1 H 1 (Ω 1 Ω 2 ) and consider te (adjoint) problem: div(α i z) = e in Ω i, i = 1, 2, (6.1a) [α z ] Γ n Γ = 0, [z ] Γ = 0 on Γ, (6.1b) z = 0 on Ω. (6.1c) 12

14 We assume tat Γ and Ω are sufficiently smoot suc tat te following regularity property olds: z H2 (Ω 1 Ω 2) e L2 (Ω), (6.2) cf. [18, 7, 20]. From Lemma 5.4 we obtain te identity e 2 L 2 (Ω) = A(z, e ). Te L 2 -error analysis is based on te following splitting, wit z V Γ,Θ : e 2 L 2 (Ω) = A(z, e ) = A(z, e ) A (z Φ, e Φ ) (6.3a) + A (z Φ z, e Φ ) (6.3b) + A (u Φ, z ) A(u, z Φ 1 ) (6.3c) + f(z Φ 1 ) f (z ). (6.3d) In te lemmas below we derive bounds for te terms in (6.3a)-(6.3d). Lemma 6.1. Te estimate A(z, e ) A (z Φ, e Φ ) 2k( u H k+1 (Ω 1 Ω 2) + f H 1, (Ω 1 Ω 2)) e L 2 (Ω) olds. Proof. Note tat e Φ V Γ,Θ + V reg,. We use (5.13a) wit w = e Φ, z instead of u and combine tis wit te regulariy estimate (6.2) and te result in Teorem 5.7. Tis yields A(z, e ) A (z Φ, e Φ ) k z H2 (Ω 1 Ω 2) e Φ wic completes te proof. 2k( u H k+1 (Ω 1 Ω 2) + f H 1, (Ω 1 Ω 2)) e L2 (Ω), In te terms (6.3b)-(6.3d) we need a suitable z V Γ,Θ. For tis we take z V Γ,Θ suc tat te following olds, cf. (5.17): z Φ z z H2 (Ω 1 Ω 2) e L2 (Ω). (6.4) Lemma 6.2. Let z be as in (6.4). Te estimate A (z Φ z, e Φ ) k+1( u H k+1 (Ω 1 Ω 2) + f H 1, (Ω 1 Ω 2)) e L 2 (Ω) olds. Proof. Using te continuity result in (5.3b), te result in Teorem 5.7 and te estimate (6.4) we obtain A (z Φ z, e Φ ) k ( ) z Φ z u H k+1 (Ω 1 Ω 2) + f H 1, (Ω 1 Ω 2) k+1( u H k+1 (Ω 1 Ω 2) + f H 1, (Ω 1 Ω 2)) e L2 (Ω). Lemma 6.3. Let z be as in (6.4). Te estimate A (u Φ, z ) A(u, z Φ 1 ) k+1 u H2 (Ω 1 Ω 2) e L2 (Ω) (6.5) 13

15 olds. Proof. We use te splitting A (u Φ, z ) A(u, z Φ 1 ) = A (u Φ, z z Φ ) A(u, z Φ 1 z) (6.6) + A (u Φ, z Φ ) A(u, z). (6.7) For te term in (6.6) we use te approximation result (5.13a) and (6.4): A (u Φ, z z Φ ) A(u, z Φ 1 k+1 u H 2 (Ω 1 Ω 2) e L 2 (Ω). For te term in (6.7) we use [[u] = [z ] = 0 and tus get: z) k u H 2 (Ω 1 Ω 2) z z Φ A (u Φ, z Φ ) A(u, z) 2 ) = α i( (u Φ ) (z Φ ) dx u z dx i=1 Ω i, Ω i 2 = α i C u z dx, Ω i i=1 wit te matrix C := det(b 1 )B T B I, B := DΦ. We ave C 0 only on := U δ, wit δ = c, for suitable c > 0. Furtermore, C,Ω k olds, cf. [24, Proof of Lemma 5.6]. Using tis and Lemma 5.5 we obtain A (u Φ, z Φ ) A(u, z) k u H 1 ( ) z H 1 ( ) Combing tese results completes te proof. k+1 u H 2 (Ω 1 Ω 2) z H 2 (Ω 1 Ω 2) k+1 u H 2 (Ω 1 Ω 2) e L 2 (Ω). Lemma 6.4. Let z be as in (6.4). Te estimate f(z Φ 1 ) f (z ) k+1 f H 1, (Ω 1 Ω 2) e L2 (Ω) (6.8) olds. Proof. Using (5.13b) we get Furtermore, using (6.4) we get f(z Φ 1 ) f (z ) k+1 f H 1, (Ω 1 Ω 2) z z z z Φ + z Φ e L 2 (Ω) + z H 2 (Ω 1 Ω 2) e L 2 (Ω). Tus we obtain te bound in (6.8). Combining te previous results we obtain te following main result. Teorem 6.5 (L 2 -error bound). We assume tat te smootness conditions on te solution u and on te data f, f as formulated in Teorem 5.7 are satisfied and tat Γ and Ω are sufficiently smoot suc tat we ave te regularity property (6.2) 14

16 for te solution of te problem (6.1). Te following bound olds for te discretization error: u u Φ 1 L 2 (Ω) u Φ u L2 (Ω ) k+1 ( u H k+1 (Ω 1 Ω 2) + f H 1, (Ω 1 Ω 2)). (6.9) 7. Numerical experiment. Results of numerical experiments tat confirm te optimal k+1 convergence in te L 2 -norm for te interface problem (2.1) are given in [22, 24]. However, In tese examples te domain boundary is polygonal. In te experiment presented in tis section we consider a case were te curved domain boundary is approximated by isoparametric elements. Te domain is a disk around te origin wit radius R = 2, Ω = B 2 ((0, 0)). Te domain is divided into an inner disk Ω 1 = B 1 ((0, 0)) and an outer ring Ω 2 = Ω \ Ω 1. Hence, te interface is a circle described wit te level set function φ(x) = x 2 1, Γ = {φ(x) = 0}. We approximate φ by φ V k by interpolation. We set (α 1, α 2 ) = (π, 2) and take te rigt and-side f suc tat te exact solution is given by (wit r(x) = x 2 ) u(x) = { 2 + (cos( π 2 r(x)) 1) x Ω 1, 2 r(x) x Ω 2. (7.1) Note tat u fulfills te interface conditions. For te Nitsce stabilization parameter we coose λ = 20k 2 and measure te L 2 error on te curved mes Ω wic is obtained by applying te isoparametric mapping Θ. To measure te errors we use domainwise te canonical extension of u leading to u e as in (7.1) wit Ω i replaced by Ω i,. Starting from a coarse initial mes, see Figure 7.1 (top left), we apply tree succesive mes refinements and use polynomial degrees k = 1,.., 5. To solve te arising linear systems we used a direct solver. Te metod as been implemented in te add-on library ngsxfem to te finite element library NGSolve [31]. Te results are sown in Figure 7.1. We observe te predicted optimal O( k+1 ) beavior. Acknowledgement. C. Lerenfeld gratefully acknowledges funding by te German Science Foundation (DFG) witin te project LE 3726/1-1. REFERENCES [1] P. Bastian and C. Engwer, An unfitted finite element metod using discontinuous Galerkin, International journal for numerical metods in engineering, 79 (2009), pp [2] C. Bernardi, Optimal finite-element interpolation on curved domains, SIAM J. Numer. Anal., 26 (1989), pp [3] T. Boiveau, E. Burman, S. Claus, and M. G. Larson, Fictitious domain metod wit boundary value correction using penalty-free nitsce metod, arxiv preprint , (2016). [4] S. Brenner and L. Scott, Te Matematical Teory of Finite Element Metods, Springer New York, [5] E. Burman, P. Hansbo, and M. G. Larson, A cut finite element metod wit boundary value correction, arxiv preprint , (2015). [6] K. W. Ceng and T.-P. Fries, Higer-order XFEM for curved strong and weak discontinuities, Int. J. Num. Met. Eng., 82 (2010), pp [7] C.-C. Cu, I. Graam, and T.-Y. Hou, A new multiscale finite element metod for igcontrast elliptic interface problems, Mat. of Computation, 79 (2010), pp [8] K. Deckelnick, C. Elliott, and T. Ranner, Unfitted Finite Element Metods using bulk meses for surface partial differential equations, SIAM Journal on Numerical Analysis, 52 (2014), pp

17 k L u e u L 2 (Ω ) ( eoc ) ( - ) (1.95) (2.01) (2.00) ( - ) (3.24) (3.20) (3.05) ( - ) (4.63) (4.29) (4.22) ( - ) (6.18) (5.61) (5.40) ( - ) (7.74) (6.68) (6.22) Fig Initial mes wit approximated geometry configuration Ω lin, Ω lin i (top left) and isoparametrically mapped domain for k = 2 (bottom left). Te table sows L 2 -norm discretization errors on successively refined meses and estimated orders of convergence. [9] K. Dréau, N. Cevaugeon, and N. Moës, Studied X-FEM enricment to andle material interfaces wit iger order finite element, Comput. Met. Appl. Mec. Eng., 199 (2010), pp [10] C. M. Elliott and T. Ranner, Finite element analysis for a coupled bulk surface partial differential equation, IMA Journal of Numerical Analysis, 33 (2013), pp [11] A. Ern and J.-L. Guermond, Finite element quasi-interpolation and best approximation, ESAIM:M2AN, 51 (2017), pp [12] T.-P. Fries and S. Omerovi, Higer-order accurate integration of implicit geometries, Int. J. Num. Met. Eng., (2015). [13] J. Grande, C. Lerenfeld, and A. Reusken, Analysis of a ig order trace finite element metod for PDEs on level set surfaces, Preprint arxiv: , (2016). NAM preprint Accepted for publication in SIAM J. Numer. Anal. (2017). [14] J. Grande and A. Reusken, A iger order finite element metod for partial differential equations on surfaces, SIAM J. Numer. Anal., 54 (2016), pp [15] S. Groß, V. Reicelt, and A. Reusken, A finite element based level set metod for two-pase incompressible flows, Comput. Visual. Sci., 9 (2006), pp [16] S. Groß and A. Reusken, An extended pressure finite element space for two-pase incompressible flows, J. Comput. Pys., 224 (2007), pp [17] A. Hansbo and P. Hansbo, An unfitted finite element metod, based on Nitsces metod for elliptic interface problems, Computer Metods in Applied Mecanics and Engineering, 191 (2002), pp [18] J. Huang and J. Zou, Some new a priori estimates for second-order elliptic and parabolic interface problems, Journal of Differential Equations, 184 (2002), pp [19] F. Kummer and M. Oberlack, An extension of te discontinuous Galerkin metod for te singular poisson equation, SIAM Journal on Scientific Computing, 35 (2013), pp. A603 A622. [20] O. Ladyzenskaya and N. Ural tseva, Linear and quasilinear equations of elliptic type, Nauka, Moscow, [21] P. L. Lederer, C.-M. Pfeiler, C. Wintersteiger, and C. Lerenfeld, Higer order un- 16

18 fitted FEM for Stokes interface problems, PAMM, 16 (2016), pp [22] C. Lerenfeld, Hig order unfitted finite element metods on level set domains using isoparametric mappings, Computer Metods in Applied Mecanics and Engineering, 300 (2016), pp [23] C. Lerenfeld, A iger order isoparametric fictitious domain metod for level set domains, arxiv preprint arxiv: , (2016). [24] C. Lerenfeld and A. Reusken, Analysis of a ig order unfitted finite element metod for elliptic interface problems, IMA J. Numer. Anal., (2017). drx041, ttps://doi.org/ /imanum/drx04. [25] M. Lenoir, Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM Journal on Numerical Analysis, 23 (1986), pp [26] B. Müller, F. Kummer, and M. Oberlack, Higly accurate surface and volume integration on implicit domains by means of moment-fitting, Int. J. Num. Met. Eng., 96 (2013), pp [27] M. A. Olsanskii, A. Reusken, and J. Grande, A finite element metod for elliptic equations on surfaces, SIAM Journal on Numerical Analysis, 47 (2009), pp [28] P. Oswald, On a BPX-preconditioner for P 1 elements, Computing, 51 (1993), pp [29] J. Parvizian, A. Düster, and E. Rank, Finite cell metod, Comput. Mec., 41 (2007), pp [30] R. Saye, Hig-order quadrature metod for implicitly defined surfaces and volumes in yperrectangles, SIAM J. Sci. Comput., 37 (2015), pp. A993 A1019. [31] J. Scöberl, C++11 implementation of finite elements in NGSolve, Tec. Report ASC , Institute for Analysis and Scientific Computing, September [32] Y. Sudakar and W. A. Wall, Quadrature scemes for arbitrary convex/concave volumes and integration of weak form in enriced partition of unity metods, Computer Metods in Applied Mecanics and Engineering, 258 (2013), pp