Chemnitz Scientific Computing Preprints

Transcription

1 G. Of G. J. Rodin O. Steinbac M. Taus Coupling Metods for Interior Penalty Discontinuous Galerkin Finite Element Metods and Boundary Element Metods CSC/11-02 Cemnitz Scientific Computing Preprints

2 Impressum: Cemnitz Scientific Computing Preprints ISSN ( : Preprintreie des Cemnitzer SFB393) Herausgeber: Professuren für Numerisce und Angewandte Matematik an der Fakultät für Matematik der Tecniscen Universität Cemnitz Postanscrift: TU Cemnitz, Fakultät für Matematik Cemnitz Sitz: Reicenainer Str. 41, Cemnitz ttp://

3 Cemnitz Scientific Computing Preprints G. Of G. J. Rodin O. Steinbac M. Taus Coupling Metods for Interior Penalty Discontinuous Galerkin Finite Element Metods and Boundary Element Metods CSC/11-02 Abstract Tis paper presents tree new coupling metods for interior penalty discontinuous Galerkin finite element metods and boundary element metods. Te new metods allow one to use discontinuous basis functions on te interface between te subdomains represented by te finite element and boundary element metods. Tis feature is particularly important wen discontinuous Galerkin finite element metods are used. Error and stability analysis is presented for some of te metods. Numerical examples suggest tat all tree metods exibit very similar convergence properties, consistent wit available teoretical results. CSC/11-02 ISSN September 2011

4 Contents 1. Introduction 1 2. Model Problem and Background Boundary Integral Equations Interior Penalty DG Finite Element Metods Coupling Metods New Coupling Metods First Metod: Non Symmetric Coupling Second Metod: Symmetric Tree Field Approac Tird Metod: Diriclet Based Coupling Stability and Error Analysis Numerical Examples Summary 18 A. Appendix 21 Autor s addresses: Günter Of Fakultät für Matematik, Tecnisce Universität Cemnitz, Reicenainer Str. 41, Cemnitz, Germany. of@tugraz.at Olaf Steinbac Institut für Numerisce Matematik, Tecnisce Universität Graz, Steyrergasse 30, 8010 Graz, Austria. o.steinbac@tugraz.at Gregory J. Rodin Institute for Computational Engineering and Sciences, Te University of Texas at Austin, Austin, TX, 78712, USA. gjr@ices.utexas.edu Mattias Taus Institute for Computational Engineering and Sciences, Te University of Texas at Austin, Austin, TX, 78712, USA. taus@ices.utexas.edu

5 1. Introduction Tis paper is concerned wit coupling metods for finite element and boundary element metods. Suc coupling metods are advantageous for problems wose domain involves an interior finite subdomain(s) embedded in an exterior unbounded subdomain, suc tat in te interior subdomain te governing partial differential equations are complex and require finite element metods, wereas in te exterior subdomain te governing partial differential equations are simple and can be solved using boundary element metods. Te coupling metods are well establised in te literature for classical (continuous) finite element and boundary element metods; we refer to [11] and te references given terein. Tis paper is motivated by applications tat require discontinuous Galerkin (DG) rater tan continuous finite element metods. Coupling metods involving DG finite element metods ave been analyzed by Bustinza, Gatica, Heuer, and Sayas [2, 3, 8, 9], wo establised tat essentially any boundary element metod can be combined wit any DG finite element metod, as long as one uses approximations continuous on te interface. For two dimensional problems, tis restriction can be removed if one combines DG finite element metods wit a particular Galerkin boundary element metod [8]. Te coupling metods considered by Bustinza, Gatica, Heuer, and Sayas are based on te symmetric formulation of boundary integral equations. In tis case, unique solvability of te coupling metod is a direct consequence of te unique solvability of te underlying finite element and boundary element systems. A disadvantage of symmetric boundary element metods is tat tey involve te ypersingular boundary integral operator tat not only precludes te use of basic collocation scemes but also requires functions continuous on te interface. Te latter restriction is particularly undesirable for coupling metods involving DG finite element metods in R 3 [8]. In tis paper, we present tree new metods tat allow for discontinuous functions on te interface, and terefore significantly simplify te coupling between DG finite element metods wit eiter Galerkin or collocation boundary element metods. Te first metod is based on te Jonson Nédélec coupling [13] extended to DG finite element metods. Tis metod admits bot collocation and Galerkin boundary element metods. However, te metod gives rise to non symmetric linear algebraic problems and its matematical foundations ave not been establised. Te second metod, wic combines a tree field approac [1] and a symmetric boundary integral formulation, addresses some of te drawbacks of te first metod, but it involves te ypersingular operator, and terefore it is limited to Galerkin boundary element metods. Tis metod gives rise to non symmetric but well structured linear algebraic problems tat can be solved almost as efficiently as symmetric ones. Following te coupling approac of DG and mixed finite element metods [10] te tird metod gives one two options. Te first option admits bot collocation and Galerkin scemes and results in non symmetric linear algebraic problems. Tis option as te advantage of aving a sound matematical foundation for te Galerkin sceme. Te second option is limited to Galerkin boundary element metods, but it results in symmetric 1

6 linear algebraic problems and as a sound matematical foundation. Te rest of te paper is organized as follows. In Section 2, we introduce a model problem and briefly outline relevant existing results necessary for presenting te new metods. In Section 3, we present te new coupling metods. In Section 4, we establis unique solvability and error estimates for one of te metods. In Section 5, we present numerical results indicating tat te proposed metods and teir variants ave very similar convergence properties, and tose properties are consistent wit available teoretical results. Te paper is concluded wit a brief summary. 2. Model Problem and Background For a bounded domain Ω R 3 wit a Lipscitz boundary Γ := Ω and a given volume density f L 2 (Ω), te model problem involves te partial differential equations te transmission conditions u i = f in Ω, u e = 0 in Ω c := R 3 \Ω, (2.1) u := u e u i = 0 and n u = n ( u e u i ) = 0 on Γ, (2.2) and te radiation condition ( ) 1 u e (x) = O x as x. (2.3) Here n denotes te outward unit normal vector on Γ. For our purposes, it is expedient to decouple te stated boundary value problem (2.1) (2.3) into an interior boundary value problem for te subdomain Ω and an exterior boundary value problem for te subdomain Ω c. We suppose tat te numerical treatment of te interior and exterior problems is based on DG finite element and boundary element metods, respectively. Our objective is to identify appropriate interior and exterior boundary value problems, boundary integral equations for te exterior problems, and discretization scemes on Γ Boundary Integral Equations Te Caucy data u e Γ and t e := (n u e ) Γ uniquely define a armonic function u e (x) for x Ω c via te representation formula, e.g. [24], u e (x) = U (x, y)t e (y)ds y + U (x, y)u e (y)ds y, (2.4) n y Γ Γ 2

7 wic satisfies te radiation condition (2.3), and were U (x, y) = 1 1 4π x y is te fundamental solution of te Laplace operator. Te Caucy data u e Γ and t e can be related to eac oter using boundary integral equations on Γ, all of wic follow from te exterior Calderon projection Here for x Γ (V t e )(x) = (K t e )(x) = Γ Γ ( ue t e ) ( 1 2 = I + K V ) ( ue 1 D I 2 K t e U (x, y)t e (y)ds y, (Ku e )(x) = n x U (x, y)t e (y)ds y, Γ ). (2.5) n y U (x, y)u e (y)ds y, (Du e )(x) = U (x, y)u e (y)ds y n x Γ n y denote te single layer, double layer, adjoint double layer, and te ypersingular boundary integral operators, respectively. Te mapping properties of tese operators are well establised, e.g. [7, 12, 14, 21, 24]. In particular, te single layer operator V : H 1/2 (Γ) H 1/2 (Γ) is bounded and H 1/2 (Γ) elliptic, and terefore invertible. Hence equations (2.5) imply te Diriclet to Neumann map t e = V 1 ( 1 2 I K)u e = [D + ( 12 I K )V 1 ( 12 ] I K) u e =: S ext u e. (2.6) Let us note tat bot representations of te Steklov Poincaré operator S ext : H 1/2 (Γ) H 1/2 (Γ) are self adjoint in te continuous setting. However, tese representations may ave different stability and symmetry properties in te discrete setting, e.g. [23]. Finally, let us mention tat te bilinear form induced by te ypersingular operator D allows for an alternative representation tat involves weakly singular surface integrals only [16]: Du, v Γ = 1 4π Γ Γ curl Γ u(y) curl Γ v(x) ds y ds x for all u, v H 1/2 (Γ) C(Γ), (2.7) x y were curl Γ is te surface curl operator. Tis representation is central to Galerkin boundary element metods involving te ypersingular operator D, as it allows one to represent te action of D in terms of te single layer operator V. Let us empasize tat (2.7) olds for continuous densities u and v only; oterwise (2.7) must include additional terms. 3

8 2.2. Interior Penalty DG Finite Element Metods In te proposed coupling metods, DG finite element metods are restricted to interior penalty metods [19, 20, 27] wic are well studied and widely applied. Let T = {T l } N Ω l=1 be a finite element mes of te interior domain Ω. For an element T T, we identify te boundary T, te diameter T, and te outward unit normal vector n T. We define te global mes size := max T T T. Te interior and exterior element faces of te finite element mes are defined as and E int := {e : T α, T β T : e = T α T β, α β}, E ext := {e : T T : e = T Γ}, respectively. For an interior face e = T α T β wit α < β, te jump and te average values of an element wise smoot function φ are defined as φ e := (φ Tα ) e (φ Tβ ) e and {φ} e := 1 ) ((φ Tα ) e + (φ Tβ ) e, 2 respectively, and te diameter of te face e is denoted by e. For s > 3 2 we introduce te broken Sobolev space V := { v L 2 (Ω) : v T H s (T ) T T }, and te semi discrete bilinear form for u, v V a DG (u, v) := u(x) v(x) dx T T {n u} e (x) v e (x) ds x (2.8) T e E int T e ξ {n v} e (x) u e (x) ds x + u e (x) v e (x) ds x, T e T e e e e E int e E int were ξ is a formulation parameter. In particular, te values ξ { 1, 0, 1} correspond to non symmetric, incomplete, and symmetric interior penalty DG finite element metods, respectively. Te parameters > 0 are required for stabilization. Te related energy norm is given by v 2 DG := v 2 L 2 (T ) + v 2 L 2 (e). (2.9) e T T Let e E int V := { v L 2 (Ω) : v T P p (T ) T T } = span{ϕi } M i=1 (2.10) denote te standard finite element space of local polynomials of degree p. For te coupling wit boundary element metods it is useful to consider a splitting of V = V Ω VΓ wit V Ω = span{ϕ i : ϕ i Γ = 0} M Ω i=1 and V Γ = span{ϕ i } M i=m Ω +1, (2.11) 4

9 were V Ω is te space spanned by te degrees of freedom in te interior of Ω, and V Γ te space spanned by te degrees of freedom on te interface Γ. In addition, let is Ṽ Γ = span { ϕ i } M M Ω +1 VΓ be te subspace of boundary basis functions continuous on Γ Coupling Metods In tis section, we briefly describe te coupling metod of Gatica, Heuer, and Sayas [8]. We regard tis metod as te current state of te art for coupling DG finite element and boundary element metods. To tis end, we consider te interior Neumann boundary value problem u i (x) = f(x) for x Ω, u i (x) = t i (x) for x Γ. n x In te context of te interior penalty metods, tis problem results in te variational problem of finding u i, V suc tat a DG (u i,, v ) = Ω f(x)v (x)dx + Γ t i (x)v (x)ds x for all v V. (2.12) By using te Neumann transmission condition, t i = t e, and te Diriclet to Neumann map (2.6), t e = S ext u e, we obtain a DG (u i,, v ) + S ext u e, v Γ = f, v Ω, u i = u e. As in te symmetric coupling of classical finite element and boundary element metods [6, 23], we use te symmetric representation (2.6) of te Steklov Poincaré operator S ext to obtain a DG (u i,, v ) + Du e, v Γ ( 1 2 I K )t e, v Γ = f, v Ω (2.13) were t e = V 1 ( 1 2 I K)u e H 1/2 (Γ) is te unique solution of te variational problem V t e, w Γ + ( 1 2 I K)u e, w Γ = 0 for all w H 1/2 (Γ). (2.14) For a Galerkin discretization of te variational problem (2.13) and (2.14), we introduce a finite dimensional ansatz space W = span{ψ k } N Γ k=1 H 1/2 (Γ), 5

10 and approximate u e by te Diriclet trace of u i, on Γ. Tis results in te variational problem of finding (u i,, t e, ) (V Ω ṼΓ ) W suc tat for all v V Ω ṼΓ, and a DG (u i,, v ) + Du i,, v Γ ( 1 2 I K )t e,, v Γ = f, v Ω (2.15) V t e,, w Γ + ( 1 2 I K)u i,, w Γ = 0 for all w W. (2.16) Tis variational problem was first proposed and analyzed in [8]. Since te ypersingular operator D requires te use of continuous basis functions, one must use te subspace V Ω ṼΓ instead of te general space V. Altoug tis restriction guarantees unique solvability and leads to optimal error estimates, it is incompatible wit te spirit of DG finite element metods. Furtermore, constructing te restricted subspace poses significant practical difficulties, especially for tree-dimensional problems, and terefore te entire approac may not be appealing to practitioners. Tis issue may be addressed by introducing additional Lagrange multipliers tat ensure continuity of u on Γ [8]. However, tis modification seems to be also cumbersome for tree-dimensional problems. Te variational problem (2.15) (2.16) is equivalent to te system of linear algebraic equations were K DG ΩΩ K DG ΩΓ K DG ΓΩ K ΓΓ DG ( 1 M 2 K ) 1 2 K V u Ω i u Γ i t e = f Ω f Γ 0, (2.17) KΩΩ DG[j, = a DG(ϕ i, ϕ j ) for i, j = 1,..., M Ω, K ΓΩ DG[j, = a DG( ϕ MΩ +i, ϕ j ) for i = 1,..., M M Ω, j = 1,..., M Ω, K ΩΓ DG[j, = a DG(ϕ i, ϕ MΩ +j) for i = 1,..., M Ω, j = 1,..., M M Ω, K ΓΓ DG[j, = a DG( ϕ MΩ +i, ϕ MΩ +j) for i, j = 1,..., M M Ω are te blocks of te DG finite element matrix. Furter, D [j, i] = D ϕ MΩ +i, ϕ MΩ +j Γ, M [l, i] = ϕ MΩ +i, ψ l Γ, V [l, k] = V ψ k, ψ l Γ, K [l, i] = K ϕ MΩ +i, ψ l Γ, for i, j = 1,..., M M Ω, k, l = 1,..., N Γ are te Galerkin boundary element matrices, and te rigt and side is given by fj Ω = f(x)ϕ j (x)dx for j = 1,..., M Ω, Ω fj Γ = f(x)ϕ MΩ +j(x)dx for all j = 1,..., M M Ω. Ω 6

11 Since te discrete single layer integral operator V is symmetric and positive definite, we can eliminate te discrete exterior Neumann datum t e to obtain te Scur complement system were ( K DG ΩΩ K DG ΩΓ K DG ΓΓ K DG ΓΩ + S sym ) ( u Ω i u Γ i ) ( ) f Ω =, (2.18) S sym = D + ( 1 2 M K )V 1 (1 2 M K ) (2.19) is a symmetric Galerkin boundary element approximation of te exterior Steklov Poincaré sym operator (2.6). Note tat S is positive definite for any coice of admissible basis functions ϕ i and ψ k, e.g. [23]. f Γ 3. New Coupling Metods In tis section, we present tree new coupling metods for te interior penalty DG finite element metods and boundary element metods. All tree metods allow one to use te standard space V of globally discontinuous finite element functions, wic significantly simplifies te implementation in comparison to te coupling metods tat require functions continuous on Γ. Some of te coupling metods admit bot collocation and Galerkin boundary element metods, wic is particularly useful for practitioners First Metod: Non Symmetric Coupling Tis metod simply extends Jonson Nédélec s coupling metod [13] involving classical finite element metods to DG finite element metods. Tis metod uses only te first integral equation in (2.5), and terefore te corresponding Steklov Poincaré operator is represented as t e = S ext u e = V 1 ( 1 2 I K)u e. By combining tis equation wit te Diriclet transmission condition u i = u e, we obtain te variational problem of finding (u i,, t e, ) V W suc tat a DG (u i,, v ) t e,, v Γ = f, v Ω for all v V, (3.1) V t e,, w Γ + ( 1 2 I K)u i,, w Γ = 0 for all w W. (3.2) Since te ypersingular operator D does not appear in tese equations, we can solve tem using te standard discontinuous finite element space V. 7

12 Te variational problem (3.1) and (3.2) is equivalent to te system of linear algebraic equations were K DG ΩΩ K DG ΩΓ K DG ΓΩ K DG ΓΓ M 1 2 M K V u Ω i u Γ i t e = K DG [j, i] = a DG (ϕ i, ϕ j ), f j = f, ϕ j Ω for i, j = 1,..., M. f Ω f Γ 0, (3.3) Te blocks of te stiffness matrix and te rigt and side vector are obtained according to te splitting (2.11). Te blocks M [l, i] = ϕ MΩ +i, ψ l Γ and K [l, i] = Kϕ MΩ +i, ψ l Γ for i = 1,..., M M Ω, l = 1,..., N Γ, and te block V as been already introduced in (2.17). Tis system of linear algebraic equations does not ave any apparent symmetry properties. By eliminating te discrete Neumann datum t e we obtain te Scur complement system ( ) K DG ( ΩΩ KΓΩ DG u Ω ) ( ) i f Ω KΩΓ DG KΓΓ DG + = u Γ Ŝns,G i f Γ, (3.4) were Ŝ ns,g = M V 1 (1 2 M K ) (3.5) is a non symmetric Galerkin boundary element approximation of te exterior Steklov Poincaré operator (2.6). In contrast to te symmetric approximation (2.19), Ŝns,G is in general not positive definite, and terefore a certain stability condition is necessary for positive definiteness. Tat condition can be satisfied wit a proper coice of te basis functions ϕ i Γ and ψ k, e.g. [23, 26]. Te stability condition is not necessary wen te coupling involves classical finite element metods [22, 25]. In tis paper, we simply conjecture tat te stability condition is also not necessary wen te coupling involves DG finite element metods. In Section 5, we present numerical examples supporting tis conjecture. Te singular boundary integral equation also admits collocation discretizations. Tis results in te approximate Steklov Poincaré operator were Ŝ ns,c = M V 1 ( 1 2 M K ), (3.6) V [l, k] = (V ψ k )(x l), M [l, i] = ϕ MΩ +i(x l), K [l, i] = (Kϕ MΩ +i)(x l) for i = 1,..., M M Ω, l = 1,..., N Γ are te entries of te collocation matrices, and x l are collocation nodes. 8

13 3.2. Second Metod: Symmetric Tree Field Approac Tis metod is based on coupling of te interior penalty DG finite element metods and symmetric boundary element metods. In contrast to te first metod, tis metod involves bot integral equations in (2.5). Te advantage of tis metod is tat its stability can be proved, and te resulting system of linear algebraic equations is block skew symmetric; tis algebraic structure can be advantageously exploited. Te drawback of te metod is tat it does not admit collocation scemes. Te metod as te structure similar to tat of te tree field domain decomposition metod of Brezzi and Marini [1]. Like in te first metod, we combine te variational problem (2.12) wit te Neumann transmission condition t i = t e : a DG (u i,, v ) t e, v Γ = f, v Ω for all v V. In contrast to te first metod, we insert te Diriclet transmission condition u i = u e into te first boundary integral equation in (2.5), but we do not use tis equation to eliminate u e : u i = u e = ( 1 2 I + K)u e V t e on Γ. To close te system of governing equations, we use te ypersingular boundary integral equation Du e + ( 1 2 I + K )t e = 0 on Γ. To proceed furter, we need to address te fact tat te ypersingular operator D and te double layer operator 1 I + K ave non trivial kernels. Terefore te exterior Diriclet 2 trace u e is not uniquely determined by te two boundary integral equations. To tis end we recall tat bot kernels coincide: u 0 (x) = 1, (Du 0 )(x) = 1 2 u 0(x) + (Ku 0 )(x) = 0 for x Γ a.e. Accordingly, we introduce te scaling condition u e, 1 Γ = 0, and te stabilized ypersingular operator D s defined as [17] D s u, v Γ = Du, v Γ + u, 1 Γ v, 1 Γ for all u, v H 1/2 (Γ). In addition to te trial space V, we introduce an ansatz space Q = span{φ i } M Γ i=1 H1/2 (Γ) C(Γ) 9

14 of continuous basis functions φ i. As a result we arrive at te variational problem of finding (u i,, t e,, u e, ) V W Q suc tat a DG (u i,, v ) t e,, v Γ = f, v Ω for all v V, (3.7) u i,, w Γ + V t e,, w Γ ( 1 2 I + K)u e,, w Γ = 0 for all w W, (3.8) ( 1 2 I + K )t e,, q Γ + D s u e,, q Γ = 0 for all q Q. (3.9) Tis variational problem is equivalent to te system of linear algebraic equations K DG ΩΩ K DG ΩΓ K DG ΓΩ K DG ΓΓ M M V 1M 2 K K D s, u Ω i u Γ i t e u e = were in addition to tose block matrices already used in (3.3) we ave f Ω f Γ 0 0, (3.10) D s, [j, i] = D s φ i, φ j Γ, M [l, i] = φ i, ψ l Γ, K [l, i] = Kφ i, ψ l Γ for i, j = 1,..., M Γ, l = 1,..., N Γ. By eliminating u e and t e from (3.10) we obtain te Scur complement system ( ) ( K DG ΩΩ KΓΩ DG u Ω ) ( ) i f Ω KΩΓ DG KΓΓ DG + = Stree u Γ i f Γ, (3.11) wit te symmetric tree field approximation of te exterior Steklov Poincaré operator S tree = M [ V + ( 1 2 M + K )D 1 s, (1 2 M + K )] 1 M. (3.12) 3.3. Tird Metod: Diriclet Based Coupling In te first two metods, te coupling involves passing te exterior Neumann data t e to te variational problem for te interior boundary value problem. In contrast, in te tird metod, te coupling is based on te Diriclet transmission condition u i = u e. Tis approac is similar to te coupling of DG and mixed finite element metods proposed by Girault, Sun, Weeler, and Yotov [10]. Te solution u i of te model problem (2.1) (2.3) coincides wit te solution of te Diriclet boundary value problem u i = f in Ω, u i = u e on Γ. 10

15 Following [10], te discrete variational problem corresponding to tis boundary value problem is to find u i, V suc tat for all v V â DG (u i,, v ) η u e, v n Γ u e v ds = f, v Ω, (3.13) e e were â DG (u, v) := a DG (u, v) n u, v Γ + η n v, u Γ + u v ds x. (3.14) e e Here η { 1, 0, +1} is a formulation parameter, similar to te formulation parameter ξ in (2.8). Let us combine te Diriclet to Neumann map (2.6) and te Neumann transmission condition t e = t i = n u i, so tat we obtain S ext u e = n u i on Γ. By adding a stabilization term to tis equation [10] and approximating u e by a function u e, Q, we obtain te variational problem of finding (u i,, u e, ) V Q suc tat â DG (u i,, v ) η n v, u e, Γ u e, v ds x = f, v Γ (3.15) e for all v V, and Su e,, q Γ + n u i,, q Γ + e [u e, u i, ]q ds x = 0 (3.16) e e for all q Q, were S is a boundary element approximation of te exterior Steklov Poincaré operator (2.6). Te variational problem (3.15) (3.16) is equivalent to te system of linear algebraic equations were K DG ΩΩ K DG ΩΓ K DG ΓΩ K ΓΓ DG B 1, B η, S + C K DG [j, i] = â DG (ϕ i, ϕ j ) u Ω i u Γ i u e = f Ω f Γ 0 for i, j = 1,..., M, (3.17) is te stiffness matrix of te modified discontinous Galerkin finite element metod wic is obtained according to te splitting (2.11). Furter, B r, [j, i] = r n ϕ MΩ +i, φ j Γ ϕ MΩ +i φ j ds x e e 11

16 for i = 1,..., M M Ω, j = 1,..., M Γ, and C [j, i] = φ i φ j ds x for i, j = 1,..., M Γ. e e Te matrix S can represent any of te following approximations of te exterior Steklov Poincaré operator (2.6), namely eiter te symmetric Galerkin approximation S sym,g = D + ( 1 2 M K )V 1 (1 2 M K ), (3.18) or te non symmetric Galerkin approximation S ns,g or te non symmetric collocation approximation S ns,c = M V 1 (1 2 M K ), (3.19) = M V 1 ( 1 2 M K ). (3.20) Te Scur complement form of (3.17) is ( K DG ΩΩ KΓΩ DG K DG ΩΓ K DG ΓΓ + S Diriclet ) ( u Ω i u Γ i ) ( f Ω = f Γ ), (3.21) wit te discrete representation of te exterior Steklov Poincaré operator S Diriclet = B η, [ S + C ] 1 B1,. (3.22) As before, te matrix S can be represented by eiter sym,g S, or ns,g S, or S ns,c. 4. Stability and Error Analysis In tis section, we establis unique solvability and error estimates for te governing equation (3.15) (3.16) of te tird metod, wit te provision tat te approximation S of te Steklov Poincaré operator (2.6) is stable. Let us associate te variational problem (3.15) (3.16) wit te bilinear form A(u i, u e ; v, q) = â DG (u i, v i ) η n v, u e Γ u e vds x (4.1) + Su e, q Γ + n u i, q Γ + e e e [u e u i ]qds x. e 12

17 Also we define te energy norm (v, q) 2 A := T T v 2 L 2 (T ) + e E int v 2 L 2 (e) + e e v q 2 L 2 (e) + q 2 H 1/2 (Γ). (4.2) Teorem 4.1 Let S be a stable boundary element approximation of te exterior Steklov Poincaré operator (2.6), Sq, q Γ c S 1 q 2 H 1/2 (Γ) for all q Q, (4.3) and η { 1, 0, +1}. Ten for sufficiently large stability parameters, te bilinear form (4.1) is elliptic, A(v, q ; v, q ) c A 1 (v, q ) 2 A for all (v, q ) V Q, (4.4) and te variational problem (3.15) (3.16) as a unique solution. Proof. For sufficiently large, te bilinear form a DG (, ) defined in (2.8) is elliptic [19]. Tat is, tere exists a constant c A 1 > 0 independent of suc tat a DG (v, v ) c A 1 v 2 DG for all v V, (4.5) were v 2 DG := T T v 2 L 2 (T ) + e E int e v 2 L 2 (e). By inserting (3.14) in (4.1) and using (4.3) and (4.5), we obtain te inequality A(v, q ; v, q ) = a DG (v, v ) + Sq, q Γ (1 η) n v, v q Γ + One can sow (see Remark 3.1 in [10]) tat c A 1 v 2 DG + c S 1 q 2 H 1/2 (Γ) (1 η) n v, v q Γ + v q, n v Γ 1 v 2 L 8 2 (T ) T T [v q ] 2 ds x e e [v q ] 2 ds x. e e e v q 2 L 2 (e), 13

18 wic implies te estimate A(v, q ; v, q ) ( c A 1 1 η ) v 2 L 8 2 (T ) + T T ( η ) 8 {( min 1 1 η 8 e E ext ), e E int v 2 L 2 (e) e v q 2 L 2 (e) + c S 1 q 2 H 1/2 (Γ) e ( c A 1 1 η ) }, c S 8 1 (v, q ) 2 A. Tus A(v, q ; v, q ) is elliptic for η { 1, 0, 1}, if we coose sufficiently large to ensure c A 1 1 η 8 > 0. Remark 4.1 If S corresponds to te symmetric Galerkin approximation (3.18), ten te stability estimate (4.3) olds for any pair of boundary element spaces Q and W, e.g. [23]. For te non symmetric approximations (3.19) and (3.20), an additional stability condition becomes necessary. Tat condition requires properly cosen boundary element spaces Q and W. In particular, tis can be acieved by constructing W on a finer mes in comparison to tat used for constructing Q, e.g. [26]. Lemma 4.2 Let (u i, u e ) be te weak solution of te model boundary value problem (2.1) (2.3) suc tat u i H s (Ω) wit s > 3 2 and (u i,, u e, ) be a solution of te variational problem (3.15) and (3.16). Ten te perturbed Galerkin ortogonality condition takes te form A(u i u i,, u e u e, ; v, q ) = ( S S ext )u e, q Γ for all V Q. (4.6) Proof. Teorem 3.2 in [10] implies â DG (u i, v) η n v, u e Γ Te solution (u i, u e ) satisfies S ext u e, q Γ + n u i, q Γ + Hence we conclude tat A(u i, u e ; v, q) + (S ext S)u e, q, Γ = f, v Ω Te solution (u i,, u e, ) satisfies u e vds x = f, v Γ for all v V. e e [u e u i ]qds x = 0 for all q H 1/2 (Γ). e e for all (v, q) V H 1/2 (Γ). A(u i,, u e, ; v, q ) = f, v Ω for all (v, q ) V Q. For a conforming approac, V Q V H 1/2 (Γ), te last two equations imply (4.6). 14

19 Teorem 4.3 Let (u i, u e ) be te weak solution of te model boundary value problem (2.1) (2.3) suc tat u i H s (Ω) wit s > 3 2. Let (u i,, u e, ) be te unique solution of te variational problem (3.15) and (3.16), and let S satisfy te approximation property ( S S ext )u e H 1/2 (Γ) c S inf t e w H w W 1/2 (Γ) wit t e = S ext u e. (4.7) Ten tere exists a constant C > 0 suc tat te quasi optimal error estimate (u i u i,, u e u e, ) 2 A C u e q 2 H 1/2 (Γ) + u i v 2 DG + e (u i v ) n 2 L 2 (e) + olds for all (v, q ) V Q. Proof. See Appendix. 1 + (u i v ) (u e q ) 2 L 2 (e) + c S inf t e w 2 e τ W H 1/2 (Γ) Remark 4.2 Te approximation property (4.7) olds unconditionally for te Galerkin approximations (3.18) and (3.19) [23, 24]. In contrast, for te collocation approximation (3.20), (4.7) olds if te approximation is stable. At present, stability of te approximation under general conditions is an open problem. Corollary 4.4 Let all assumptions of Teorem 4.3 old, and in addition 3/2 < s 2. If V is te space of discontinuous linear finite element functions, Q is te space of piecewise continuous linear boundary element basis functions, and W is te space of piecewise constant boundary element basis functions, ten tere exists a constant C suc tat (u i u i,, u e u e, ) A C s 1 [ u i H s (Ω) + t i H s 3/2 (Γ)]. (4.8) Remark 4.3 Of course if u i H 2 (Ω), te error estimate implies linear convergence rate. Tis result is straigtforward to generalize to approximations based on iger order polynomial basis functions. Suc approximations are meaningful for sufficiently large s only. Remark 4.4 Te Scur complement systems (2.18), (3.4), (3.11) and of (3.21) allow for a unified treatment of te coupling metods. In particular, tis unified structure allows one to exploit standard results pertaining to stability and error analysis. However, stability analysis of te non symmetric formulation (3.1) (3.2) requires additional considerations. Wile it is likely tat a successful treatment of te non symmetric formulation is possible along te lines proposed in [22, 25], it is not pursued in tis paper. Here, we limit our study of tis issue to presenting numerical results confirming expected teoretical results.. 15

20 5. Numerical Examples In tis section, we present numerical results confirming te teoretical results establised for te tird metod and suggesting tat te first two metods are stable and exibit expected rates of convergence in te energy and L 2 norms. In all examples, Ω is a unit spere, and f(x) = f(r, φ, θ) = 4(cos φ + sin φ) sin θ, were r, φ, θ are te sperical coordinates wose origin is at te spere center. Te exact solution for tis problem is u(x) = ũ(r, φ, θ) = 1 3 (cos φ + sin φ) sin θ { (4 3r)r for r < 1, r 2 for r > 1. Tis function is piecewise analytic, and terefore te numerical solutions are expected to ave te optimal rates of convergence. Numerical solutions were obtained using piecewise linear discontinuous finite element basis functions for V, piecewise linear continuous basis functions for Q, and piecewise constant basis functions for W. Te interface Γ was approximated using piecewise linear continuous basis functions; errors associated wit tis approximation can be estimated and controlled by using standard tecniques, e.g. [5, 15]. Te collocation and Galerkin boundary element metods were accelerated using te fast multipole metod; again te errors associated wit te use of multipole and local expansions were estimated and controlled following [18]. Te DG formulation parameters were cosen to be ξ = η = 1, and te DG stabilisation parameters were cosen to be = 5. In presenting results, we denote te number of finite elements T in Ω by N Ω, te number of elements on Γ by N Γ, and te number of nodes on Γ by M Γ. First, let us present numerical results for te tird metod (3.15) (3.16) in wic te exterior Steklov Poincaré operator is discretized using eiter te symmetric Galerkin sceme (3.18), or te non symmetric Galerkin sceme (3.19), or te non symmetric collocation sceme (3.20). In all cases, te error norm was computed as te modified energy norm [ê DG ()] 2 := T T (u i u i, ) 2 L 2 (T ) + + e E int (u i u i, ) (u e u e, ) 2 L 2 (e) + e e u i u i, 2 L 2 (e) 1 e u e u e, 2 L 2 (e) were te H 1/2 (Γ) norm in (4.2) was replaced by a weigted L 2 (Γ) norm, e.g. [4]. Te estimated order of convergence was computed as ( ) êdg ( l ) eoc := log 2, ê DG ( l+1 ) 16

21 were l refers to te refinement level, and l = 2 l+1. Table 1 contains numerical results for six meses (five refinement levels) for all tree scemes. Clearly te results support te notion tat te asymptotic order of convergence is linear. Table 2 mimics Table 1 except it is based on te L 2 norm e 2 () = u i u i, L2 (Ω) rater tan te energy norm. As expected, te asymptotic order of convergence is quadratic. Galerkin (3.18) Galerkin (3.19) Collocation (3.20) N Ω N Γ M Γ ê DG eoc ê DG eoc ê DG eoc Table 1: Errors and rates of convergence for te tird metod measured using te energy norm. Galerkin (3.18) Galerkin (3.19) Collocation (3.20) N Ω N Γ M Γ e 2 eoc e 2 eoc e 2 eoc Table 2: Errors and rates of convergence for te tird metod measured using te L 2 norm. Numerical results for te tree field formulation (3.7) (3.9) are given in Table 3, were we use te DG energy norm e DG () := u i u i, DG and te L 2 norm. Again we observe te expected linear and quadratic orders of convergence, respectively. Finally, Table 4 contains numerical results for te non symmetric approac (3.1) (3.2) for te Galerkin (3.5) and collocation (3.6) scemes. Numerical results indicate tat all versions of te tree metods performed similarly to eac oter. In particular, all of tem exibit linear order of convergence in te pertinent 17

22 Galerkin (3.10) N Ω N Γ M Γ e DG eoc e 2 eoc Table 3: Errors and rates of convergence for te second metod measured using te energy norm and L 2 norms. Galerkin (3.5) Collocation (3.6) N Ω N Γ M Γ e DG eoc e 2 eoc e DG eoc e 2 eoc Table 4: Errors and rates of convergence for te first metod measured using te energy norm and L 2 norms. energy norm and quadratic order of convergence in te L 2 (Ω) norm. Furtermore, te differences in te absolute values of te corresponding errors appear to be minimal. 6. Summary Tis paper introduces tree new coupling metods for te interior penalty DG finite element metods and boundary element metods. Te key advantage of te new metods is tat tey allow one to use discontinuous basis functions on te interface, wic is particularly important for coupling metods involving DG finite element metods. Te new coupling metods ave six variations associated wit different approximations of te Steklov Poincaré operator of te underlying boundary integral equations. We presented teoretical results pertaining to stability and error analysis for some of te versions of te metods, wereas establising suc results for oter versions can be done in a similar way. Numerical results suggest tat all versions perform very similar to eac oter, and exibit expected rates of convergence in bot energy and L 2 norms. 18

23 Acknowledgement Te autors would like to tank M. Neumüller for providing te DG solver wic was used for te DG part of all computations reported in tis paper. GJR acknowledges te financial support from te Department of Energy (grant DOE DE-FG02-05ER25701) and by te W. A. Tex Moncrief, Jr. Endowment In Simulation Based Engineering Sciences troug a Grand Callenge Faculty Fellowsip. MT acknowledges te financial support from te Fulbrigt Foundation. References [1] F. Brezzi, L. D. Marini: A tree field domain decomposition metod. Contemp. Mat. 157 (1994) [2] R. Bustinza, G. N. Gatica, F. J. Sayas: A LDG BEM coupling for a class of nonlinear exterior transmission problems. In: Numerical Matematics and Advanced Applications, Springer, Berlin, pp , [3] R. Bustinza, G. N. Gatica, F. J. Sayas: On te coupling of local discontinuous Galerkin and boundary element metods for non linear exterior transmission problems. IMA J. Numer. Anal. 28 (2008) [4] C. Carstensen, M. Maiscak, D. Praetorius, E. P. Stepan: Residual based a posteriori error estimate for ypersingular equation on surfaces. Numer. Mat. 97 (2004) [5] P. G. Ciarlet: Te Finite Element Metod for Elliptic Problems. Nort Holland, [6] M. Costabel: Symmetric metods for te coupling of finite elements and boundary elements. In: Boundary Elements IX (C. A. Brebbia, G. Kun, W. L. Wendland eds.), Springer, Berlin, pp , [7] M. Costabel: Boundary integral operators on Lipscitz domains: elementary results. SIAM J. Mat. Anal. 19 (1988) [8] G. N. Gatica, N. Heuer, F. J. Sayas: A direct coupling of local discontinuous Galerkin and boundary element metods. Mat. Comp. 79 (2010) [9] G. N. Gatica, F. J. Sayas: An a priori error analysis for te coupling of local discontinuous Galerkin and boundary element metods. Mat. Comp. 75 (2006) [10] V. Girault, S. Sun, M. F. Weeler, I. Yotov: Coupling discontinuous Galerkin and mixed finite element discretizations using mortar finite elements. SIAM J. Numer. Anal. 46 (2008)

24 [11] G. C. Hsiao, E. Scnack, W. L. Wendland: Hybrid coupled finite boundary element metods for elliptic systems of second order. Comput. Metods Appl. Mec. Engrg. 190 (2000) [12] G. C. Hsiao, W. L. Wendland: Boundary integral equations. Applied Matematical Sciences, vol. 164, Springer, Berlin, [13] C. Jonson, J. C. Nédélec: On te coupling of boundary integral and finite element metods. Mat. Comp. 35 (1980) [14] W. McLean: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, [15] J. C. Nédélec: Curved finite element metods for te solution of singular integral equations on surfaces in R 3. Comp. Met. Appl. Mec. Engrg. 8 (1976) [16] J. C. Nédélec: Integral equations wit nonintegrable kernels. Int. Eq. Operator T. 5 (1982) [17] G. Of, O. Steinbac: A fast multipole boundary element metod for a modified ypersingular boundary integral equation. In: Analysis and Simulation of Multifield Problems (M. Efendiev, W. L. Wendland eds.), Lecture Notes in Applied and Computational Mecanics, vol. 12, Springer, Berlin, pp , [18] G. Of, O. Steinbac, W. L. Wendland: Te fast multipole metod for te symmetric boundary integral formulation. IMA J. Numer. Anal. 26 (2006) [19] B. Rivière: Discontinuous Galerkin metods for solving elliptic and parabolic equations. Frontiers in Applied Matematics, vol. 35, SIAM, Piladelpia, [20] B. Rivière, M. F. Weeler, V. Girault: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin metods for elliptic problems. I. Comput. Geosci. 3 (2000) [21] S. A. Sauter, C. Scwab: Boundary Element Metods. Springer Series in Computational Matematics, vol. 39. Springer, Berlin, [22] F. J. Sayas: Te validity of Jonson Nedelec s BEM FEM coupling on polygonal interfaces. SIAM J. Numer. Anal. 47 (2009) [23] O. Steinbac: Stability estimates for ybrid coupled domain decomposition metods. Lecture Notes in Matematics, vol. 1809, Springer, Heidelberg, [24] O. Steinbac: Numerical Approximation Metods for Elliptic Boundary Value Problems. Finite and Boundary Elements. Springer, New York, [25] O. Steinbac: A note on te stable one equation coupling of finite and boundary elements. SIAM J. Numer. Mat. 49 (2011)

25 [26] W. L. Wendland: On asymptotic error estimates for te combined BEM an FEM. In: In Finite Element and Boundary Element Tecniques from Matematical and Engineering Point of View (E. Stein, W. L. Wendland eds.), CISM Lecture Notes, vol. 301, Springer, Wien, pp , [27] M. F. Weeler: An elliptic collocation finite element metod wit interior penalties. SIAM J. Numer. Anal. 15 (1978) A. Appendix In tis appendix we prove te error estimate given in Teorem 4.3. For arbitrary (v, q ) V Q, te triangle inequality implies (u i u i,, u e u e, ) A (u i v, u e q ) A + (u i, v, u e, q ) A. Te first term can be furter estimated by using standard tecniques, ence it remains to bound te second term. We use te ellipticity estimate (4.4) and te perturbed Galerkin ortogonality (4.6) to obtain c A 1 (u i, v, u e, q ) 2 A A(u i, v, u e, q ; u i, v, u e, q ) = A(u i, u i, u e, u e ; u i, v, u e, q ) + A(u i v, u e q ; u i, v, u e, q ) = (S ext S)u e, u e, q Γ + A(u i v, u e q ; u i, v, u e, q ). By combining tis equation wit (4.1) we obtain c A 1 (u i, v, u e, q ) 2 A (S ext S)u e, u e, q Γ + A(u i v, u e q ; u i, v, u e, q ) = (S ext S)u e, u e, q Γ + S(u e q ), u e, q Γ + a DG (u i v, u i, v ) n (u i v ), (u i, v ) (u e, q ) Γ + η n (u i, v ), (u i v ) (u e q ) Γ + [(u i v ) (u e q )][(u i, v ) (u e, q )]ds x e e 21

26 (S ext S)u e H 1/2 (Γ) u e, q H 1/2 (Γ) + c S 2 u e q H 1/2 (Γ) u e, q H 1/2 (Γ) +c A 2 u i v DG u i, v DG + n (u i v ) L 2 (e) (u i, v ) (u e, q ) L 2 (e) + + n (u i, v ) L 2 (e) (u i v ) (u e q ) L 2 (e) e (u i v ) (u e q ) L 2 (e) (u i, v ) (u e, q ) L 2 (e). By applying te weigted Hölder inequality we obtain c A 1 (u i, v, u e, q ) 2 A [ c (S ext S)u e 2 H 1/2 (Γ) + u e q 2 H 1/2 (Γ) + u i v 2 DG + e E ext e n (u i v ) 2 L 2 (e) e (u i v ) (u e q ) 2 L 2 (e) u e, q 2 H 1/2 (Γ) + u i, v 2 DG + e n (u i, v ) 2 L 2 (e) e (u i, v ) (u e, q ) 2 L 2 (e) Let T e T be a finite element containing a face e E ext. Ten for a finite element function v V we obtain n v L 2 (e) c e 1/2 v L 2 (T e), and Hence we obtain 1/2. 1/2 e n (u i, v ) 2 L 2 (e) c T T (u i, v ) 2 L 2 (T ). u e, q 2 H 1/2 (Γ) + u i, v 2 DG e (u i, v ) (u e, q ) 2 L 2 (e) e n (u i, v ) 2 L 2 (e) 1/2 c (u i, v, u e, q ) A, 22

27 and so c A 1 (u i, v, u e, q ) A [ c (S ext S)u e 2 H 1/2 (Γ) + u e q 2 H 1/2 (Γ) + u i v 2 DG + e n (u i v ) 2 L 2 (e) e Tis inequality implies te error estimate given in Teorem 4.3. (u i v ) (u e q ) 2 L 2 (e) 1/2. 23

28 Cemnitz Scientific Computing Preprints ISSN