A non-standard Finite Element Method based on boundary integral operators

Transcription

1 A non-standard Finite Element Method based on boundary integral operators Clemens Hofreither Ulrich Langer Clemens Pechstein June 30, 2010 supported by

2 Outline 1 Method description Motivation Variational formulation Discretization 2 A Céa-type estimate Bounding the mesh-dependant constants Estimating the approximation errors 3 A 3D numerical example Outlook

3 Motivation Variational formulation Discretization Motivation Let Ω R d a domain. We want to solve the PDE div(a(x) u(x)) = f (x), x Ω plus suitable boundary conditions on Γ = Ω. Discretize the domain by a mesh consisting of N polygonal/polyhedral elements T i : Ω = N i=1 T i

4 Motivation Variational formulation Discretization Meshes We want to treat general polygonal/polyhedral meshes: arbitrary and mixed element shapes hanging nodes supported naturally

5 Motivation Variational formulation Discretization DEEP LATTERBARROW N-S LATTERBARROW FAULTED TOP M-F BVG TOP M-F BVG FAULTED BLEAWATH BVG BLEAWATH BVG FAULTED F-H BVG Motivation These kinds of meshes arise, e.g., in reservoir simulation, F-H BVG FAULTED UNDIFF BVG UNDIFF BVG FAULTED N-S BVG N-S BVG FAULTED CARB LST CARB LST FAULTED COLLYHURST COLLYHURST FAULTED BROCKRAM BROCKRAM SHALES + EVAP FAULTED BNHM BOTTOM NHM FAULTED DEEP ST BEES DEEP ST BEES FAULTED N-S ST BEES N-S ST BEES FAULTED VN-S ST BEES VN-S ST BEES FAULTED DEEP CALDER DEEP CALDER FAULTED N-S CALDER N-S CALDER FAULTED VN-S CALDER VN-S CALDER MERCIA MUDSTONE QUATERNARY Image: rock permeability α(x) at Sellafield, UK. (c) NIREX UK Ltd. adaptive mesh refinement. Another advantage: automatic mesh manipulation (join, split, manipulate elements)

6 Motivation Variational formulation Discretization Other possible approaches Mimetic finite difference method Kuznetsov, Lipnikov, Shashkov: The mimetic finite difference method on polygonal meshes for diffusion-type problems, 2004 Brezzi, Lipnikov, Shashkov: Convergence of Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes, 2004 Discontinuous Galerkin method Dolejsi, Feistauer, Sobotikova: Analysis of the discontinuous Galerkin method for nonlinear convection-diffusion problems, 2005

7 Motivation Variational formulation Discretization Our approach Has its roots in symmetric BEM Domain Decomposition method. 1 Use piecewise PDE-harmonic ansatz functions in every element (local Trefftz method). 2 Reduce the variational formulation to the skeleton of the mesh. 3 Use boundary integral operators to solve local Dirichlet-to-Neumann problems.

8 Motivation Variational formulation Discretization Prior work A selection of previous work related to this approach: Hsiao, Wendland: Domain Decomposition in Boundary Element Methods, Hsiao, Steinbach, Wendland: Domain decomposition methods via boundary integral equations, Copeland, Langer, Pusch: From the boundary element method to local Trefftz finite element methods on polyhedral meshes, 2009.

9 Motivation Variational formulation Discretization Model problem Our method is designed to treat piecewise constant coefficients: a(x) a i for x T i For simplicity, we assume a(x) 1, f 0, i.e. Laplace equation: u = 0 in Ω, u = g on Γ = Ω.

10 Motivation Variational formulation Discretization Variational formulation For all v H0 1(Ω), 0 = Ω u v

11 Motivation Variational formulation Discretization Variational formulation For all v H0 1(Ω), 0 = u v Ω = u v i T i

12 Motivation Variational formulation Discretization Variational formulation For all v H0 1(Ω), 0 = = i = i Ω u v u v T i Γ i u v n i T i }{{} u v =0

13 Motivation Variational formulation Discretization Variational formulation For all v H0 1(Ω), 0 = = i = i = i Ω u v u v T i Γ i Γ i u v n i T i u n i v. }{{} u v =0 Is defined purely on the element boundaries (Γ i ) i=1,...,n.

14 Motivation Variational formulation Discretization Function spaces on the skeleton Let Γ S := i Γ i denote the skeleton of the mesh. Let H 1/2 (Γ S ) be the trace space of H 1 (Ω) onto Γ S. Let W = {v H 1/2 (Γ S ) : v Γ = 0} be the space of skeletal functions which are zero on the boundary of Ω.

15 Motivation Variational formulation Discretization Function spaces on the skeleton Let Γ S := i Γ i denote the skeleton of the mesh. Let H 1/2 (Γ S ) be the trace space of H 1 (Ω) onto Γ S. Let W = {v H 1/2 (Γ S ) : v Γ = 0} be the space of skeletal functions which are zero on the boundary of Ω.

16 Motivation Variational formulation Discretization Function spaces on the skeleton Let Γ S := i Γ i denote the skeleton of the mesh. Let H 1/2 (Γ S ) be the trace space of H 1 (Ω) onto Γ S. Let W = {v H 1/2 (Γ S ) : v Γ = 0} be the space of skeletal functions which are zero on the boundary of Ω.

17 Motivation Variational formulation Discretization Harmonic extension For any (sufficiently regular) function on the boundary of a domain, there is a unique way to extend it to a harmonic function ( u = 0) in the domain: H i : H 1/2 (Γ i ) H 1 (T i ) We also define H S : H 1/2 (Γ S ) H 1 (Ω) by piecing together the local harmonic extensions.

18 Motivation Variational formulation Discretization Skeletal variational formulation Recall: Making use of the bijection we can rewrite (1) as i Γ i u v = 0 v H0 1 (Ω). (1) n i u i = γ Ti u u = H i u i on T i, i Γ i n i (H i u i )v i = 0 v W. (2) With the Dirichlet-to-Neumann map S i u i := n i (H i u i ), we get S i u i v i = 0 v W. (3) Γ i i

19 Motivation Variational formulation Discretization Relation to standard domain VF Standard variational formulation Find u Ω H 1 (Ω) with u Ω Γ = g such that for all v Ω H0 1(Ω), u Ω v Ω dx = 0. Ω Skeletal variational formulation Find u H 1/2 (Γ S ) with u Γ = g such that for all v W, N S i u i, v i Γi = 0. i=1 The formulations are equivalent: u Ω = H i u i on T i.

20 Motivation Variational formulation Discretization Evaluating the Dirichlet-to-Neumann map Evaluating S i u i = n i (H i u i ): 1 solve a Laplace problem on T i with Dirichlet data u i, 2 obtain Neumann data of solution. More directly: S i = D i + ( 1 2 I + K i 1 )Vi ( 1 2 I + K i) with the boundary integral operators V i, single layer potential operator, K i, double layer potential operator, D i, hypersingular operator. Defined in terms of the fundamental solution of the differential operator.

21 Motivation Variational formulation Discretization Evaluating the Dirichlet-to-Neumann map Evaluating S i u i = n i (H i u i ): 1 solve a Laplace problem on T i with Dirichlet data u i, 2 obtain Neumann data of solution. More directly: S i = D i + ( 1 2 I + K i 1 )Vi ( 1 2 I + K i) with the boundary integral operators V i, single layer potential operator, K i, double layer potential operator, D i, hypersingular operator. Defined in terms of the fundamental solution of the differential operator.

22 Motivation Variational formulation Discretization Discretization A Galerkin method is obtained by choosing a finite-dimensional subspace W h W. E.g.: piecewise linear, continuous functions (on the skeleton). Degrees of freedom are chosen in the vertices of the elements.

23 Motivation Variational formulation Discretization Discretizing the Dirichlet-to-Neumann map Symmetric representation of Steklov-Poincaré operator: S i u i = D i u i + ( 1 2 I + K i )w i, where V i w i = ( 1 2 I + K i)u i Problem: We cannot compute w i H 1/2 (Γ i ) exactly. Instead, perform a local Galerkin projection w h,i w i to some finite-dimensional space and compute S i u i = D i u i + ( 1 2 I + K i )w h,i. Natural choice: space of piecewise constant boundary functions, since w i represents a normal derivative.

24 Motivation Variational formulation Discretization Discretizing the Dirichlet-to-Neumann map Symmetric representation of Steklov-Poincaré operator: S i u i = D i u i + ( 1 2 I + K i )w i, where V i w i = ( 1 2 I + K i)u i Problem: We cannot compute w i H 1/2 (Γ i ) exactly. Instead, perform a local Galerkin projection w h,i w i to some finite-dimensional space and compute S i u i = D i u i + ( 1 2 I + K i )w h,i. Natural choice: space of piecewise constant boundary functions, since w i represents a normal derivative.

25 Motivation Variational formulation Discretization Discretization of the variational problem N S i u i, v i = 0 v W i=1

26 Motivation Variational formulation Discretization Discretization of the variational problem N S i u i, v i = 0 v W i=1 Galerkin N S i u h,i, v h,i = 0 v h W h i=1

27 Motivation Variational formulation Discretization Discretization of the variational problem N S i u i, v i = 0 v W i=1 Galerkin N S i u h,i, v h,i = 0 v h W h i=1 D2N approximation N S i u h,i, v h,i = 0 v h W h i=1

28 Motivation Variational formulation Discretization Discretization of the variational problem N S i u i, v i = 0 v W i=1 Galerkin N S i u h,i, v h,i = 0 v h W h i=1 N S i u h,i, v h,i = 0 i=1 D2N approximation v h W h This creates a s.p.d. and sparse stiffness matrix. For purely tetrahedral mesh: identical to FEM.

29 A Céa-type estimate Bounding the mesh-dependant constants Estimating the approximation errors Skeletal energy norm We introduce the skeletal energy norm on H 1/2 (Γ S ), v S := ( N 1/2 S i v i, v i ) = i=1 ( N 1/2 H i v i 2 H 1 (T i )). i=1

30 A Céa-type estimate Bounding the mesh-dependant constants Estimating the approximation errors Céa-type error estimate Theorem u Ω H S u h H 1 (Ω) C S inf u v h S + N inf u Ω v h W h z h,i Z h,i n i i=1 z h,i 2 V i, where C S is a mesh-dependent constant. Proof: Using the first lemma of Strang. For a proper error estimate, we have to: bound the mesh constant C S, estimate the Dirichlet approximation error, estimate the Neumann approximation error.

31 A Céa-type estimate Bounding the mesh-dependant constants Estimating the approximation errors Céa-type error estimate Theorem u Ω H S u h H 1 (Ω) C S inf u v h S + N inf u Ω v h W h z h,i Z h,i n i i=1 z h,i 2 V i, where C S is a mesh-dependent constant. Proof: Using the first lemma of Strang. For a proper error estimate, we have to: bound the mesh constant C S, estimate the Dirichlet approximation error, estimate the Neumann approximation error.

32 A Céa-type estimate Bounding the mesh-dependant constants Estimating the approximation errors Mesh constants The constant C S depends only on c 0,i = D i v, v inf v H 1/2 (Γ i ) V 1 i v, v (0, 1 4 ) i {1,..., N}. Recent results by C. Pechstein: c 0,i may be bounded away from zero solely in terms of the Jones parameter C U (T i ), the Poincaré constant C P (T i ).

33 A Céa-type estimate Bounding the mesh-dependant constants Estimating the approximation errors Mesh constants The constant C S depends only on c 0,i = D i v, v inf v H 1/2 (Γ i ) V 1 i v, v (0, 1 4 ) i {1,..., N}. Recent results by C. Pechstein: c 0,i may be bounded away from zero solely in terms of the Jones parameter C U (T i ), the Poincaré constant C P (T i ).

34 A Céa-type estimate Bounding the mesh-dependant constants Estimating the approximation errors Estimating the approximation errors Dirichlet approximation error: Introduce a regular local triangulation of T i and use standard FEM approximation results. Neumann approximation error: recent results by C. Pechstein: bound Vi by C(C U (T i ), C P (T i )) H 1/2 (Γ i ) again use a local triangulation of T i composed of regular tetrahedra estimate error piecewise on each triangle on the surface of these tetrahedra

35 A Céa-type estimate Bounding the mesh-dependant constants Estimating the approximation errors Main result Theorem If every polyhedral element T i has a regular tetrahedral triangulation, has a uniform upper bound on the number of its surface triangles, has a uniform upper bound on its Jones and Poincaré constants, and if u Ω H 2 (Ω) is the exact solution, we have u Ω H S u h H 1 (Ω) C ( i h 2 i u Ω 2 H 2 (T i ) ) 1 2 Ch u Ω H 2 (Ω).

36 A 3D numerical example Outlook A 3D numerical example implementation in C++ based on PARMAX framework by Pechstein & Copeland generation of BEM matrix entries: OSTBEM (Steinbach et al.) solution of linear system by non-preconditioned CG method

37 A 3D numerical example Outlook Problem specification Solve the pure Dirichlet Laplace equation, u = 0 in Ω, u(x) = g(x) x Ω, on the unit cube, Ω = (0, 1) 3. Exact solution: u(x, y, z) = exp(x) cos(y)(1 + z) Two sets of meshes: a purely tetrahedral mesh, a mixed mesh consisting of tetrahedra and polyhedra Number of degrees of freedom is identical.

38 A 3D numerical example Outlook

39 A 3D numerical example Outlook

40 A 3D numerical example Outlook Tetrahedral mesh: mesh size h H 1 error L 2 error #tets e Mixed mesh: mesh size h H 1 error L 2 error #tets #polys e

41 A 3D numerical example Outlook error L 2 error (tets) L 2 error (mixed) H 1 error (tets) H 1 error (mixed) h (mesh size)

42 A 3D numerical example Outlook Outlook treat convection-diffusion-reaction problems: div(a(x) u(x)) + b(x) u(x) + c(x)u(x) = f (x), x Ω DG discretization develop optimal solvers higher order schemes: Levy functions (Wendland)

43 Bibliography Method description A 3D numerical example Outlook D. Copeland, U. Langer, and D. Pusch. From the boundary element method to local Trefftz finite element methods on polyhedral meshes. In M. Bercovier, M. J. Gander, R. Kornhuber, and O. Widlund, editors, Domain Decomposition Methods in Science and Engineering XVIII, volume 70 of Lecture Notes in Computational Science and Engineering. Springer, Heidelberg, G. C. Hsiao, O. Steinbach, and W. L. Wendland. Domain decomposition methods via boundary integral equations. Journal of Computational and Applied Mathematics, 125(1-2): , G. C. Hsiao and W. L. Wendland. Domain decomposition in boundary element methods. In R. Glowinski, Y. A. Kuznetsov, G. Meurant, J. Périaux, and O. B. Widlund, editors, Proceedings of the Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, Moscow, May 21-25, 1990, pages 41 49, Philadelphia, SIAM. C. Pechstein. Shape-explicit constants for some boundary integral operators. DK-Report 09-11, Doctoral Program in Computational Mathematics, Johannes Kepler University Linz, Austria, Submitted.

44 Boundary integral operators Fundamental solution of 3D Laplace operator: U (x, y) = 1 4π x y. V i : H 1/2 (Γ i ) H 1/2 (Γ i ) : (V i v)(y) = U (x, y)v(x) ds x Γ i K i : H 1/2 (Γ i ) H 1/2 (Γ i ) : U (K i u)(y) = Γi (x, y)u(x) ds x n x K i : H 1/2 (Γ i ) H 1/2 (Γ i ) : (K i v)(y) = D i : H 1/2 (Γ i ) H 1/2 (Γ i ) : Γi U (D i u)(y) = n y n y (x, y)v(x) ds x Γi U n x (x, y) ( u(x) u(y) ) ds x

45 Norms for error estimates We use harmonic extensions H i heavily. Natural norms to work in: v i H 1/2 (Γ i ) := H iv i H 1 (T i ) = inf φ H 1 (T i ) φ Γi =v i φ H1 (T i ), v i 2 H 1/2 (Γ i ) := 1 diam(t i ) 2 H iv i 2 L 2 (T i ) + H iv i 2 H 1 (T i ). Easily extended to the skeleton Γ S : v S := ( N 1/2 v i 2 H 1/2 (Γ i )) = HS v H 1 (Ω). i=1

46 Jones parameter Definition (Uniform domain) A bounded and connected set D R d is called a uniform domain if there exists a constant C U (D) such that any two points x 1, x 2 D can be joined by a rectifiable curve γ(t) : [0, 1] D with γ(0) = x 1 and γ(1) = x 2 such that l(γ) C U (D) x 1 x 2, min x i γ(t) C U (D) dist(γ(t), D) t [0, 1]. i=1,2 We call the smallest such C U (D) the Jones parameter of D. Any Lipschitz domain is a uniform domain.

47 Poincaré constant Definition (Poincaré constant) Let D be a uniform domain. The Poincaré constant C P (D) is the smallest constant such that inf u c L c R 2 (D) C P (D) diam(d) u H 1 (D) u H 1 (D). C P (D) can be bounded using the constant in an isoperimetric inequality. For convex domains, C P (D) 1 π (Payne & Weinberger 1960; Bebendorf 2003).

48 Mesh constants The constant C S depends only on c 0,i = D i v, v inf v H 1/2 (Γ i ) V 1 i v, v (0, 1 4 ) i {1,..., N}. Recent results by C. Pechstein: For each element, fix a ball B i enclosing it: B i T i, dist( B i, T i ) 1 2 diam(t i). c 0,i may be bounded away from zero solely in terms of the Jones parameters C U (T i ) and C U (B i \ T i ), the Poincaré constants C P (T i ) and C P (B i \ T i ).

49 Mesh constants The constant C S depends only on c 0,i = D i v, v inf v H 1/2 (Γ i ) V 1 i v, v (0, 1 4 ) i {1,..., N}. Recent results by C. Pechstein: For each element, fix a ball B i enclosing it: B i T i, dist( B i, T i ) 1 2 diam(t i). c 0,i may be bounded away from zero solely in terms of the Jones parameters C U (T i ) and C U (B i \ T i ), the Poincaré constants C P (T i ) and C P (B i \ T i ).

50 Mesh regularity (3D) Standard regularity assumptions on every tetrahedron τ Ξ i : with J τ Jacobian of affine mapping from unit tetrahedron to τ, c 1 (diam τ) 3 det J τ c 1 (diam τ) 3, J τ l2 c 2 diam τ, J 1 τ l2 (c 2 diam τ) 1. This implies regularity of the boundary faces: for all f F i and the tetrahedron τ on which f lies, c 2 diam τ diam f diam τ, c 1 2c 2 (diam τ) 2 f 1 2 (diam τ)2. Furthermore, we assume that the number of boundary faces per element is uniformly small.

51 Mesh regularity (3D) Standard regularity assumptions on every tetrahedron τ Ξ i : with J τ Jacobian of affine mapping from unit tetrahedron to τ, c 1 (diam τ) 3 det J τ c 1 (diam τ) 3, J τ l2 c 2 diam τ, J 1 τ l2 (c 2 diam τ) 1. This implies regularity of the boundary faces: for all f F i and the tetrahedron τ on which f lies, c 2 diam τ diam f diam τ, c 1 2c 2 (diam τ) 2 f 1 2 (diam τ)2. Furthermore, we assume that the number of boundary faces per element is uniformly small.

52 Mesh regularity (3D) Standard regularity assumptions on every tetrahedron τ Ξ i : with J τ Jacobian of affine mapping from unit tetrahedron to τ, c 1 (diam τ) 3 det J τ c 1 (diam τ) 3, J τ l2 c 2 diam τ, J 1 τ l2 (c 2 diam τ) 1. This implies regularity of the boundary faces: for all f F i and the tetrahedron τ on which f lies, c 2 diam τ diam f diam τ, c 1 2c 2 (diam τ) 2 f 1 2 (diam τ)2. Furthermore, we assume that the number of boundary faces per element is uniformly small.

53 Estimating the Dirichlet approximation error Conforming global triangulation Ξ = i Ξ i. Introduce globally cont., pw. linear FE space V h over Ξ. Choose suitable FE interpolant I Vh u Ω V h. Skeletal interpolant I Wh u Ω := γ S (V h u Ω ) W h. = γ i (u Ω I Vh u Ω ) = u i I Wh u Ω.

54 Estimating the Dirichlet approximation error Conforming global triangulation Ξ = i Ξ i. Introduce globally cont., pw. linear FE space V h over Ξ. Choose suitable FE interpolant I Vh u Ω V h. Skeletal interpolant I Wh u Ω := γ S (V h u Ω ) W h. = γ i (u Ω I Vh u Ω ) = u i I Wh u Ω.

55 Estimating the Dirichlet approximation error Conforming global triangulation Ξ = i Ξ i. Introduce globally cont., pw. linear FE space V h over Ξ. Choose suitable FE interpolant I Vh u Ω V h. Skeletal interpolant I Wh u Ω := γ S (V h u Ω ) W h. = γ i (u Ω I Vh u Ω ) = u i I Wh u Ω.

56 Estimating the Dirichlet approximation error Conforming global triangulation Ξ = i Ξ i. Introduce globally cont., pw. linear FE space V h over Ξ. Choose suitable FE interpolant I Vh u Ω V h. Skeletal interpolant I Wh u Ω := γ S (V h u Ω ) W h. = γ i (u Ω I Vh u Ω ) = u i I Wh u Ω.

57 Estimating the Dirichlet approximation error Conforming global triangulation Ξ = i Ξ i. Introduce globally cont., pw. linear FE space V h over Ξ. Choose suitable FE interpolant I Vh u Ω V h. Skeletal interpolant I Wh u Ω := γ S (V h u Ω ) W h. = γ i (u Ω I Vh u Ω ) = u i I Wh u Ω. inf v h W h u v h 2 S

58 Estimating the Dirichlet approximation error Conforming global triangulation Ξ = i Ξ i. Introduce globally cont., pw. linear FE space V h over Ξ. Choose suitable FE interpolant I Vh u Ω V h. Skeletal interpolant I Wh u Ω := γ S (V h u Ω ) W h. = γ i (u Ω I Vh u Ω ) = u i I Wh u Ω. inf v h W h u v h 2 S u I Wh u Ω 2 S

59 Estimating the Dirichlet approximation error Conforming global triangulation Ξ = i Ξ i. Introduce globally cont., pw. linear FE space V h over Ξ. Choose suitable FE interpolant I Vh u Ω V h. Skeletal interpolant I Wh u Ω := γ S (V h u Ω ) W h. = γ i (u Ω I Vh u Ω ) = u i I Wh u Ω. inf v h W h u v h 2 S u I Wh u Ω 2 S = N H i (u I Wh u Ω ) 2 H 1 (T i ) i=1

60 Estimating the Dirichlet approximation error Conforming global triangulation Ξ = i Ξ i. Introduce globally cont., pw. linear FE space V h over Ξ. Choose suitable FE interpolant I Vh u Ω V h. Skeletal interpolant I Wh u Ω := γ S (V h u Ω ) W h. = γ i (u Ω I Vh u Ω ) = u i I Wh u Ω. inf v h W h u v h 2 S u I Wh u Ω 2 S = N H i (u I Wh u Ω ) 2 H 1 (T i ) i=1 N u Ω I Vh u Ω 2 H 1 (T i ) i=1

61 Estimating the Dirichlet approximation error Conforming global triangulation Ξ = i Ξ i. Introduce globally cont., pw. linear FE space V h over Ξ. Choose suitable FE interpolant I Vh u Ω V h. Skeletal interpolant I Wh u Ω := γ S (V h u Ω ) W h. = γ i (u Ω I Vh u Ω ) = u i I Wh u Ω. inf v h W h u v h 2 S u I Wh u Ω 2 S = N H i (u I Wh u Ω ) 2 H 1 (T i ) i=1 N u Ω I Vh u Ω 2 H 1 (T i ) = u Ω I Vh u Ω 2 H 1 (Ω) i=1

62 Estimating the Neumann approximation error Steps in this estimation are quite standard, but: choice of norms is delicate, results need to be explicit in the regularity parameters. (C is a generic constant depending only on the reg. param.) inf z h,i Z h,i u Ω n i z h,i Vi

63 Estimating the Neumann approximation error Steps in this estimation are quite standard, but: choice of norms is delicate, results need to be explicit in the regularity parameters. (C is a generic constant depending only on the reg. param.) inf z h,i Z h,i u Ω n i z h,i Interpolation Vi u Ω n i ( ) uω Q h,i n i V i

64 Estimating the Neumann approximation error Steps in this estimation are quite standard, but: choice of norms is delicate, results need to be explicit in the regularity parameters. (C is a generic constant depending only on the reg. param.) inf z h,i Z h,i u Ω n i z h,i Interpolation Vi [Pechstein] CV ( ) u Ω uω Q h,i n i n i V ( ) i u Ω uω Q h,i n i n i H 1/2 (Γ i )

65 Estimating the Neumann approximation error Steps in this estimation are quite standard, but: choice of norms is delicate, results need to be explicit in the regularity parameters. (C is a generic constant depending only on the reg. param.) inf z h,i Z h,i u Ω n i z h,i Interpolation Vi [Pechstein] CV Approx. thm. ( ) u Ω uω Q h,i n i n i V ( ) i u Ω uω Q h,i n i n i u Ω C h i n i H 1/2 pw(γ i ) H 1/2 (Γ i )

66 Estimating the Neumann approximation error Steps in this estimation are quite standard, but: choice of norms is delicate, results need to be explicit in the regularity parameters. (C is a generic constant depending only on the reg. param.) inf z h,i Z h,i u Ω n i z h,i Interpolation Vi [Pechstein] CV Approx. thm. ( ) u Ω uω Q h,i n i n i V ( ) i u Ω uω Q h,i n i n i u Ω C h i n i H 1/2 pw(γ i ) N. trace thm. C h i u Ω H 2 (T i ). H 1/2 (Γ i )

67 On the Neumann approximation theorem Standard approximation on pw. constant boundary function space (e.g. Steinbach): v Q h,i v L2 (Γ i ) C (diam T i ) 1/2 v H 1/2 pw(γ i ). We use the norm relations (for v H 1/2 (Γ i )) to obtain v H 1/2 pw(γ i ) C v H 1/2 pw (Γ i ) C #F i v H 1/2 (Γ i ) v Q h v L2 (Γ i ) C #F i (diam T i ) 1/2 v H 1/2 (Γ i ). If #F i bounded, this gives us the desired approximation result for H 1/2 (Γ i ) using a duality argument.

68 On the Neumann approximation theorem Standard approximation on pw. constant boundary function space (e.g. Steinbach): v Q h,i v L2 (Γ i ) C (diam T i ) 1/2 v H 1/2 pw(γ i ). We use the norm relations (for v H 1/2 (Γ i )) to obtain v H 1/2 pw(γ i ) C v H 1/2 pw (Γ i ) C #F i v H 1/2 (Γ i ) v Q h v L2 (Γ i ) C #F i (diam T i ) 1/2 v H 1/2 (Γ i ). If #F i bounded, this gives us the desired approximation result for H 1/2 (Γ i ) using a duality argument.

69 On the Neumann approximation theorem Standard approximation on pw. constant boundary function space (e.g. Steinbach): v Q h,i v L2 (Γ i ) C (diam T i ) 1/2 v H 1/2 pw(γ i ). We use the norm relations (for v H 1/2 (Γ i )) to obtain v H 1/2 pw(γ i ) C v H 1/2 pw (Γ i ) C #F i v H 1/2 (Γ i ) v Q h v L2 (Γ i ) C #F i (diam T i ) 1/2 v H 1/2 (Γ i ). If #F i bounded, this gives us the desired approximation result for H 1/2 (Γ i ) using a duality argument.

70 On the Neumann approximation theorem Standard approximation on pw. constant boundary function space (e.g. Steinbach): v Q h,i v L2 (Γ i ) C (diam T i ) 1/2 v H 1/2 pw(γ i ). We use the norm relations (for v H 1/2 (Γ i )) to obtain v H 1/2 pw(γ i ) C v H 1/2 pw (Γ i ) C #F i v H 1/2 (Γ i ) v Q h v L2 (Γ i ) C #F i (diam T i ) 1/2 v H 1/2 (Γ i ). If #F i bounded, this gives us the desired approximation result for H 1/2 (Γ i ) using a duality argument.