Trefftz-discontinuous Galerkin methods for time-harmonic wave problems

Transcription

1 Trefftz-discontinuous Galerkin methods for time-harmonic wave problems Ilaria Perugia Dipartimento di Matematica - Università di Pavia (Italy) Joint work with Ralf Hiptmair, Andrea Moiola (SAM - ETH Zürich, CH) IMA Workshop on Numerical Solutions of Partial Differential Equations: Novel Discretization Techniques Minneapolis, November 1-5, 2010 Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 1 / 30

2 Model Problem: The Helmholtz Equation Given: Time-harmonic wave propagation in homogeneous media bounded polygonal/polyhedral domain Ω R d, d = 2, 3 angular frequency (wave number) ω ω 0 > 0 (wave length λ = 2π/ω) boundary datum g L 2 ( Ω) Model problem u ω 2 u = f u n + iω u = g in Ω on Ω Homogeneous case: f = 0 Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 2 / 30

3 Numerical Dispersion 1D numerical experiment u + ω 2 u = 0 in (0, 1), u(0) = 1, u (1) iωu(1) = 0 Analytical solution: u(x) = exp(iωx), ω = 40 (λ 0.16) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 3 / 30

4 Numerical Dispersion 1D numerical experiment u + ω 2 u = 0 in (0, 1), u(0) = 1, u (1) iωu(1) = 0 Analytical solution: u(x) = exp(iωx), ω = 40 (λ 0.16) Phase error in the Galerkin solution for piecewise linear finite elements on a uniform mesh ( 8 el. per wavelength) Galerkin solution Exact solution Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 3 / 30

5 Pollution Effect Numerical evidence with piecewise linear finite elements: 0.07 ωh = 0.2 ωh = ωh = 0.05 u uh L 2 (Ω) Galerkin error u u h L 2 (Ω) O(ω(ωh) 2 ) ω Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 4 / 30

6 Pollution Effect Numerical evidence with piecewise linear finite elements: 0.07 ωh = 0.2 ωh = ωh = 0.05 u uh L 2 (Ω) Galerkin error u u h L 2 (Ω) O(ω(ωh) 2 ) Best approximation error u u opt h L 2 (Ω) O((ωh) 2 ) ω Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 4 / 30

7 Pollution Effect Numerical evidence with piecewise linear finite elements: 0.07 ωh = 0.2 ωh = ωh = 0.05 u uh L 2 (Ω) Galerkin error u u h L 2 (Ω) O(ω(ωh) 2 ) Best approximation error u u opt h L 2 (Ω) O((ωh) 2 ) ω (see [Babuška & Sauter, 2000]) u u h L2 (Ω) u u opt h O(ω) for ω L 2 (Ω) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 4 / 30

8 Trefftz Finite Element Methods How to cope with these issues? Trefftz FEM Trial and test functions are solutions to the homogeneous equation within each mesh element: V h T(T h ) = {v H 2 (T h ) : v + ω 2 v = 0 in each K T h } Trefftz methods incorporate information on the problem within FEM spaces Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 5 / 30

9 Trefftz Finite Element Methods How to cope with these issues? Trefftz FEM Trial and test functions are solutions to the homogeneous equation within each mesh element: V h T(T h ) = {v H 2 (T h ) : v + ω 2 v = 0 in each K T h } Trefftz methods incorporate information on the problem within FEM spaces Strategy p version of Trefftz-FEM (immune to pollution effect) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 5 / 30

10 Trefftz Finite Element Methods Examples of Trefftz basis functions for Helmholtz: plane waves x e iωx d, d S N+1 Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 6 / 30

11 Trefftz Finite Element Methods Examples of Trefftz basis functions for Helmholtz: plane waves x e iωx d, d S N+1 Plane Wave Partition of Unit Method [Babuška & Melenk, 1997] (modulated plane waves conforming method) Ultra Weak Variational Formulation [Cessenat & Despres, 1998] (plane waves only; exotic formulation of the method) Plane Wave Least Square Methods [Monk & Wang, 1999] (LS functional containing transmission and boundary conditions) Discontinuous Enrichment Method [Farhat, Harari & Franca, 2001] (plane waves + Lagrange multipliers) Variational Theory of Complex Rays [Riou,Ladevèze&Sourcis, 2006] (plane waves with amplitude depending on the wave vector) Plane Wave Discontinuous Galerkin Method [Gittelson,Hiptmair,Moiola,Perugia, 2009-] (generalization of UWVF) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 6 / 30

12 Trefftz Finite Element Methods Examples of Trefftz basis functions for Helmholtz: plane waves x e iωx d, d S N+1 Fourier-Bessel functions (generalized harmonic polynomials) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 6 / 30

13 Trefftz Finite Element Methods Examples of Trefftz basis functions for Helmholtz: plane waves x e iωx d, d S N+1 Fourier-Bessel functions (generalized harmonic polynomials) PUM [Babuška & Melenk, 1997] LSM [Stojek, 1998], [Monk & Wang, 1999], [Barnett & Betcke, 2010] UWVF [Luostari, Huttunen & Monk, 2010] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 6 / 30

14 Trefftz Finite Element Methods Examples of Trefftz basis functions for Helmholtz: plane waves x e iωx d, d S N+1 Fourier-Bessel functions (generalized harmonic polynomials) Matching traces across interelement boundaries: Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 6 / 30

15 Trefftz Finite Element Methods Examples of Trefftz basis functions for Helmholtz: plane waves x e iωx d, d S N+1 Fourier-Bessel functions (generalized harmonic polynomials) Matching traces across interelement boundaries: Partition of Unit Method Least Square Methods Lagrange multipliers Discontinuous Galerkin Methods Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 6 / 30

16 Trefftz Finite Element Methods Examples of Trefftz basis functions for Helmholtz: plane waves x e iωx d, d S N+1 Fourier-Bessel functions (generalized harmonic polynomials) Matching traces across interelement boundaries: Partition of Unit Method Least Square Methods Lagrange multipliers Discontinuous Galerkin Methods Trefftz vs Polynomial Methods: same accuracy with less d.o.f. (and for plane waves: possibility to choose directions) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 6 / 30

17 Trefftz Discontinuous Galerkin Methods Trefftz Discontinuous Galerkin Methods General DG finite element methods with Trefftz basis functions Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 7 / 30

18 Trefftz Discontinuous Galerkin Methods Trefftz Discontinuous Galerkin Methods General DG finite element methods with Trefftz basis functions In the following: derivation of Trefftz-DGFEM for the Helmholtz equation Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 7 / 30

19 Trefftz Discontinuous Galerkin Methods Trefftz Discontinuous Galerkin Methods General DG finite element methods with Trefftz basis functions In the following: derivation of Trefftz-DGFEM for the Helmholtz equation p version analysis of Trefftz-DGFEM for the Helmholtz equation [Hiptmair, Moiola & Perugia, Tech. Rep. 2009] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 7 / 30

20 Trefftz Discontinuous Galerkin Methods Trefftz Discontinuous Galerkin Methods General DG finite element methods with Trefftz basis functions In the following: derivation of Trefftz-DGFEM for the Helmholtz equation p version analysis of Trefftz-DGFEM for the Helmholtz equation [Hiptmair, Moiola & Perugia, Tech. Rep. 2009] best approximation estimates [Hiptmair, Moiola & Perugia, Tech. Rep. 2009] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 7 / 30

21 Trefftz Discontinuous Galerkin Methods Trefftz Discontinuous Galerkin Methods General DG finite element methods with Trefftz basis functions In the following: derivation of Trefftz-DGFEM for the Helmholtz equation p version analysis of Trefftz-DGFEM for the Helmholtz equation [Hiptmair, Moiola & Perugia, Tech. Rep. 2009] best approximation estimates [Hiptmair, Moiola & Perugia, Tech. Rep. 2009] p version analysis of Trefftz-DGFEM for the Maxwell equations [Hiptmair, Moiola & Perugia, in prep.] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 7 / 30

22 Trefftz-DG Discretization of the Helmholtz Equation New variable σ: iω σ = u in Ω u ω 2 u = 0 ω 2 u iω σ = 0 in Ω iωσ n + iω u = g on Ω Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 8 / 30

23 Trefftz-DG Discretization of the Helmholtz Equation New variable σ: iω σ = u in Ω u ω 2 u = 0 ω 2 u iω σ = 0 in Ω iωσ n + iω u = g on Ω T h = {K} mesh (with possible hanging nodes) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 8 / 30

24 Trefftz-DG Discretization of the Helmholtz Equation New variable σ: iω σ = u in Ω u ω 2 u = 0 ω 2 u iω σ = 0 in Ω iωσ n + iω u = g on Ω T h = {K} mesh (with possible hanging nodes) Multiply by test functions and integrate by parts element by element: τ H(div; K), v H 1 (K), iω σ τ + u τ u τ n = 0 K K ω 2 u v + iωσ v iωσ nv = 0 K K K K Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 8 / 30

25 Trefftz-DG Discretization of the Helmholtz Equation New variable σ: iω σ = u in Ω u ω 2 u = 0 ω 2 u iω σ = 0 in Ω iωσ n + iω u = g on Ω T h = {K} mesh (with possible hanging nodes) Replace continuous spaces with discrete spaces Σ hp (K) and V hp (K), and traces by numerical fluxes: τ hp Σ hp (K), v hp V hp (K), iω σ hp τ hp + u hp τ hp û hp τ hp n = 0 K K ω 2 u hp v hp + iωσ hp v hp iω σ hp n v hp = 0 K K K K Discrete spaces and numerical fluxes to be defined; h V hp Σ hp required Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 8 / 30

26 Trefftz-DG Discretization of the Helmholtz Equation New variable σ: iω σ = u in Ω u ω 2 u = 0 ω 2 u iω σ = 0 in Ω iωσ n + iω u = g on Ω T h = {K} mesh (with possible hanging nodes) Replace continuous spaces with discrete spaces Σ hp (K) and V hp (K), and traces by numerical fluxes: τ hp Σ hp (K), v hp V hp (K), iω σ hp τ hp + u hp τ hp û hp τ hp n = 0 K K ω 2 u hp v hp + iωσ hp v hp iω σ hp n v hp = 0 K K K K Discrete spaces and numerical fluxes to be defined; h V hp Σ hp required Choose τ hp = v hp in the 1st equation and replace in the 2nd one Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 8 / 30

27 Trefftz-DG Discretization of the Helmholtz Equation Formulation in the u variable only: ( u hp v hp ω 2 u hp v hp ) (u hp û hp ) v hp n iω σ hp n v hp = 0 K K K Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 9 / 30

28 Trefftz-DG Discretization of the Helmholtz Equation Formulation in the u variable only: ( u hp v hp ω 2 u hp v hp ) (u hp û hp ) v hp n iω σ hp n v hp = 0 K K K A further integration by parts (ultra weak formulation): ( v hp ω 2 v hp )u hp + û hp v hp n iω σ hp n v hp = 0 K K K Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 9 / 30

29 Trefftz-DG Discretization of the Helmholtz Equation Formulation in the u variable only: ( u hp v hp ω 2 u hp v hp ) (u hp û hp ) v hp n iω σ hp n v hp = 0 K K K A further integration by parts (ultra weak formulation): ( v hp ω 2 v hp )u hp + û hp v hp n iω σ hp n v hp = 0 K K K Take V hp T(T h ) := {v H 2 (T h ) : v + ω 2 v = 0 in each K T h }; v hp + ω 2 v hp = 0 Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 9 / 30

30 Trefftz-DG Discretization of the Helmholtz Equation Formulation in the u variable only: ( u hp v hp ω 2 u hp v hp ) (u hp û hp ) v hp n iω σ hp n v hp = 0 K K K A further integration by parts (ultra weak formulation): ( v hp ω 2 v hp )u hp + û hp v hp n iω σ hp n v hp = 0 K K K Take V hp T(T h ) := {v H 2 (T h ) : v + ω 2 v = 0 in each K T h }; v hp + ω 2 v hp = 0 DG Formulation with Trefftz basis functions û hp v hp n iω σ hp n v hp = 0 K K Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 9 / 30

31 Trefftz-DG Discretization of the Helmholtz Equation Numerical fluxes on interior faces σ hp = 1 iω { h u hp } α [[u hp ]] N û hp = {u hp } β iω [[ hu hp ]] N Numerical fluxes on boundary faces σ hp = hu hp iω 1 δ iω ( hu hp iωu hp n gn) û hp = u hp δ iω ( hu hp n + iωu hp g) δ (0, 1/2] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 10 / 30

32 Trefftz-DG Discretization of the Helmholtz Equation The UWVF ([Cessenat & Despres, 1998]) can be seen as a DG method with plane wave basis functions and flux parameters α = β = δ = 1/2 (see [Buffa & Monk, 2008], [Gittelson, Hiptmair & Perugia, 2009]; for a derivation of the UWVF as a DG method with upwind fluxes, see [Huttunen, Malinen & Monk, 2007] and [Gabard, 2007]). Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 11 / 30

33 Trefftz-DG Discretization of the Helmholtz Equation The UWVF ([Cessenat & Despres, 1998]) can be seen as a DG method with plane wave basis functions and flux parameters α = β = δ = 1/2 (see [Buffa & Monk, 2008], [Gittelson, Hiptmair & Perugia, 2009]; for a derivation of the UWVF as a DG method with upwind fluxes, see [Huttunen, Malinen & Monk, 2007] and [Gabard, 2007]). General hp flux parameters: α = a ωh p log(p), log(p) β = bωh, δ = dωh log(p) p p with a a min and b > 0. Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 11 / 30

34 p Version Error Analysis Scheme of the analysis Step Ingredients Results 1 choice of a suitable mesh-skeleton well-posedness and norm in T(T h ) quasi-optimality 2 elliptic regularity and stability estimates L 2 norm estimates by for (inhomogeneous) adjoint problem modified duality argument 3 best approximation estimates convergence rates Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 12 / 30

35 p Version Error Analysis: Step 1 Trefftz-DG discretization of the Helmholtz equation Find u hp V hp such that, v hp V hp, A h (u hp, v hp ) = l h (v hp ) The Trefftz-DG method is consistent by construction: if u is the analytical solution then, v hp V hp, A h (u, v hp ) = l h (v hp ) Z Z Z A h (u, v) := {u }[[ h v]] N { h u } [[v]] N + F h I F h I Z Z +iω 1 β [[ h u]] N [[ h v]] N + iω +iω 1 Z l h (v) :=iω 1 Z F B h F I h F B h Z δ h u n h v n + iω Z δ g h v n + F B h F I h F B h (1 δ)g v F B h Z (1 δ) u h v n δ h u nv F h B α[[u]] N [[v]] N (1 δ)u v Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 13 / 30

36 p Version Error Analysis: Step 1 Trefftz-DG bilinear form: Z Z A h (u,v) = {u }[[ h v]] N F I h +iω 1 Z +iω 1 Z F I h F B h F I h Z { h u } [[v]] N + Z β [[ h u]] N [[ h v]] N + iω Z δ h u n h v n + iω F I h F B h F B h Z (1 δ)u h v n δ h u n v F h B α[[u]] N [[v]] N (1 δ)u v Provided that w T(T h ), local integration by parts gives Im [A h (w, w)] = ω 1 β 1/2 [[ h w]] N 2 0,F + ω α 1/2 [[w]] h I N 2 0,Fh I + ω 1 δ 1/2 h w n 2 0,F B h + ω (1 δ) 1/2 w 2 0,F B h Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 14 / 30

37 p Version Error Analysis: Step 1 Trefftz-DG bilinear form: Z Z A h (u,v) = {u }[[ h v]] N F I h +iω 1 Z +iω 1 Z F I h F B h F I h Z { h u } [[v]] N + Z β [[ h u]] N [[ h v]] N + iω Z δ h u n h v n + iω F I h F B h F B h Z (1 δ)u h v n δ h u n v F h B α[[u]] N [[v]] N (1 δ)u v Provided that w T(T h ), local integration by parts gives Im [A h (w, w)] = ω 1 β 1/2 [[ h w]] N 2 0,F + ω α 1/2 [[w]] h I N 2 0,Fh I + ω 1 δ 1/2 h w n 2 0,F B h + ω (1 δ) 1/2 w 2 0,F B h Define the seminorm w 2 F h := Im [A h (w, w)] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 14 / 30

38 p Version Error Analysis: Step 1 Trefftz-DG bilinear form: Z Z A h (u,v) = {u }[[ h v]] N F I h +iω 1 Z +iω 1 Z F I h F B h F I h Z { h u } [[v]] N + Z β [[ h u]] N [[ h v]] N + iω Z δ h u n h v n + iω F I h F B h F B h Z (1 δ)u h v n δ h u n v F h B α[[u]] N [[v]] N (1 δ)u v Provided that w T(T h ), local integration by parts gives Im [A h (w, w)] = ω 1 β 1/2 [[ h w]] N 2 0,F + ω α 1/2 [[w]] h I N 2 0,Fh I + ω 1 δ 1/2 h w n 2 0,F B h + ω (1 δ) 1/2 w 2 0,F B h Define the seminorm w 2 F h := Im [A h (w, w)] The seminorm Fh is actually a norm in T(T h ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 14 / 30

39 p Version Error Analysis: Step 1 DG norms: For every w T(T h ): w 2 F h = ω 1 β 1/2 [[ h w]] N 2 0,F h I + ω α 1/2 [[w]] N 2 0,F h I + ω 1 δ 1/2 h w n 2 0,F B h + ω (1 δ) 1/2 w 2 0,F B h Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 15 / 30

40 p Version Error Analysis: Step 1 DG norms: For every w T(T h ): w 2 F h = ω 1 β 1/2 [[ h w]] N 2 0,F h I + ω α 1/2 [[w]] N 2 0,F h I + ω 1 δ 1/2 h w n 2 0,F B h + ω (1 δ) 1/2 w 2 0,F B h w 2 F + h := w 2 F h + ω β 1/2 {w } 2 0,F I h + ω 1 α 1/2 { h w } 2 0,F h I + ω δ 1/2 w 2 0,F h B. Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 15 / 30

41 p Version Error Analysis: Step 1 DG norms: For every w T(T h ): w 2 F h = ω 1 β 1/2 [[ h w]] N 2 0,F h I + ω α 1/2 [[w]] N 2 0,F h I + ω 1 δ 1/2 h w n 2 0,F B h + ω (1 δ) 1/2 w 2 0,F B h w 2 F + h := w 2 F h + ω β 1/2 {w } 2 0,F I h + ω 1 α 1/2 { h w } 2 0,F h I + ω δ 1/2 w 2 0,F h B. Continuity and coercivity: For every v, w T(T h ): A h (v, w) C v F + h w F h Im [A h (w, w)] = w 2 F h Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 15 / 30

42 p Version Error Analysis: Step 1 DG norms: For every w T(T h ): w 2 F h = ω 1 β 1/2 [[ h w]] N 2 0,F h I + ω α 1/2 [[w]] N 2 0,F h I + ω 1 δ 1/2 h w n 2 0,F B h + ω (1 δ) 1/2 w 2 0,F B h w 2 F + h := w 2 F h + ω β 1/2 {w } 2 0,F I h + ω 1 α 1/2 { h w } 2 0,F h I + ω δ 1/2 w 2 0,F h B. Continuity and coercivity: For every v, w T(T h ): A h (v, w) C v F + h w F h Im [A h (w, w)] = w 2 F h Well-posedness and quasi-optimality of the Trefftz-DG method Existence and uniqueness of discrete solutions Quasi-optimal error estimate: u u hp Fh C inf u v hp F + v hp V h hp Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 15 / 30

43 p Version Error Analysis: Step 2 We have an error bound in Fh norm. Now, we want to derive an error bound in the L 2 norm duality technique Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 16 / 30

44 p Version Error Analysis: Step 2 We have an error bound in Fh norm. Now, we want to derive an error bound in the L 2 norm duality technique Domain assumption: Ω convex (elliptic regularity needed) Mesh assumptions: T h shape-regular and quasi-uniform Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 16 / 30

45 p Version Error Analysis: Step 2 We have an error bound in Fh norm. Now, we want to derive an error bound in the L 2 norm duality technique Domain assumption: Ω convex (elliptic regularity needed) Mesh assumptions: T h shape-regular and quasi-uniform Problem: standard duality does not work for a p version error analysis of Problem: Trefftz-DG! Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 16 / 30

46 p Version Error Analysis: Step 2 We have an error bound in Fh norm. Now, we want to derive an error bound in the L 2 norm duality technique Domain assumption: Ω convex (elliptic regularity needed) Mesh assumptions: T h shape-regular and quasi-uniform Problem: standard duality does not work for a p version error analysis of Problem: Trefftz-DG! Adjoint problem (0 ϕ L 2 (Ω)) v ω 2 v = ϕ in Ω v n iω v = 0 on Ω v T(T h ) we can not expect high order approximation of v in V hp : lim p inf v v hp 0 v hp V hp Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 16 / 30

47 p Version Error Analysis: Step 2 Main tool: duality technique from [Monk & Wang, 1999] For every w T(T h ): (w, ϕ) 0,Ω w L 2 (Ω) = sup ϕ L 2 (Ω) = sup ϕ L 2 (Ω) (Adj. pbl. v ω 2 v = ϕ) ϕ 0,Ω P K T h (w, v ω 2 v) 0,K w Fh sup ϕ L 2 (Ω) ϕ 0,Ω C diam(ω) ϕ 0,Ω X K T h «ω γ 1/2 v 2 0, K + ω 1 α 1/2 v , K ω 1/2 h 1/2 + ω 1/2 h 1/2 w Fh using integration by parts, Trefftz property of w, trace inequality (γ = β, δ) and stability estimates [Melenk 2D, Cummings & Feng 3D]: { v 1,Ω + ω v 0,Ω C 1 ϕ 0,Ω v 2,Ω C 2 (1 + ω) ϕ 0,Ω Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 17 / 30

48 p Version Error Analysis: Step 2 Bound of the L 2 norm For every w T(T h ): w L 2 (Ω) C diam(ω) [ (ωh) 1/2 + (ωh) 1/2] w Fh Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 18 / 30

49 p Version Error Analysis: Step 2 Bound of the L 2 norm For every w T(T h ): w L 2 (Ω) C diam(ω) [ (ωh) 1/2 + (ωh) 1/2] w Fh From (u u hp ) T(T h ) and quasi-optimality in Fh norm: L 2 norm error estimate [ u u hp L2 (Ω) C diam(ω) (ωh) 1/2 + (ωh) 1/2] inf u v hp F + v hp V h hp Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 18 / 30

50 p Version Error Analysis Scheme of the analysis Step Ingredients Results 1 choice of a suitable mesh-skeleton well-posedness and norm in T(T h ) quasi-optimality 2 elliptic regularity and stability estimates L 2 norm estimates by for (inhomogeneous) adjoint problem modified duality argument 3 best approximation estimates convergence rates Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 19 / 30

51 p Version Error Analysis Scheme of the analysis Step Ingredients Results 1 choice of a suitable mesh-skeleton well-posedness and norm in T(T h ) quasi-optimality 2 elliptic regularity and stability estimates L 2 norm estimates by for (inhomogeneous) adjoint problem modified duality argument 3 best approximation estimates convergence rates Remark [h version analysis]: For the h version analysis, a similar technique was used in [Buffa & Monk, 2008] for UWVF (constant flux parameters). Using hp flux parameters, one can do better (see [Gittelson, Hiptmair & Perugia, 2009] and [Moiola, Tech. Rep. 2009]) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 19 / 30

52 Best Approximation Estimates (Hints) Using trace inequalities, we can derive estimates of inf u v hp F + v hp V h hp from best approximation estimates in the weighted Sobolev norms v 2 s,ω,d = s ω 2(s j) v 2 j,d j=0 Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 20 / 30

53 Best Approximation Estimates (Hints) Using trace inequalities, we can derive estimates of inf u v hp F + v hp V h hp from best approximation estimates in the weighted Sobolev norms v 2 s,ω,d = s ω 2(s j) v 2 j,d j=0 Let V hp be your favorite Trefftz discrete space; one can derive best approximation estimates in s,ω,d norm using Vekua s theory. Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 20 / 30

54 Best Approximation Estimates (Hints) Vekua s operators: Harmonic functions V 2 V 1 Helmholtz solutions Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 21 / 30

55 Best Approximation Estimates (Hints) Vekua s operators: Harmonic functions V 2 V 1 Helmholtz solutions approximation estimates of harmonic functions by harmonic polynomials (cont. of V 1 and V 2 ) (cont. of V 1 and V 2 ) approximation estimates of homogeneous Helmholtz solutions by generalized harmonic polynomials := V 1 [harmonic polynomials] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 21 / 30

56 Best Approximation Estimates (Hints) Vekua s operators: Harmonic functions V 2 V 1 Helmholtz solutions approximation estimates of harmonic functions by harmonic polynomials (cont. of V 1 and V 2 ) (cont. of V 1 and V 2 ) approximation estimates of homogeneous Helmholtz solutions by generalized harmonic polynomials := V 1 [harmonic polynomials] + approximation estimates of generalized harmonic polynomials by functions in V hp Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 21 / 30

57 Best Approximation Estimates (Hints) Vekua s operators: Harmonic functions V 2 V 1 Helmholtz solutions approximation estimates of harmonic functions by harmonic polynomials (cont. of V 1 and V 2 ) (cont. of V 1 and V 2 ) approximation estimates of homogeneous Helmholtz solutions by generalized harmonic polynomials := V 1 [harmonic polynomials] + approximation estimates of generalized harmonic polynomials by functions in V hp (triangle inequality) (triangle inequality) approximation estimates of homogeneous Helmholtz solutions by functions in V hp Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 21 / 30

58 Best Approximation Estimates (Hints) Generalized harmonic polynomials: circular/spherical waves in 2D/3D Q(x) = e ilψ J l (ωr) (x = re iψ, l Z) Q(x) = Y l,m ( x x ) jl (ω x ) 0 m l N Y l,m = spherical harmonics, j l = spherical Bessel functions Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 22 / 30

59 Best Approximation Estimates (Hints) Generalized harmonic polynomials: circular/spherical waves in 2D/3D In 2D: hp convergence rates fully determined [Melenk, 1999] In 3D: h convergence rates fully determined; In 3D: p convergence rates determined up to a parameter depending on In 3D: the element shape [Moiola, Hiptmair & Perugia, Tech. Rep. 2009] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 22 / 30

60 Best Approximation Estimates (Hints) Generalized harmonic polynomials: circular/spherical waves in 2D/3D In 2D: hp convergence rates fully determined [Melenk, 1999] In 3D: h convergence rates fully determined; In 3D: p convergence rates determined up to a parameter depending on In 3D: the element shape [Moiola, Hiptmair & Perugia, Tech. Rep. 2009] Plane waves: Let V hp = PW p ω (T h), where PW p ω (T h) := {v L 2 (Ω) : v(x) K = p α K j exp(iωd j x) in each K T h }. hp approximation of generalized harmonic polynomials by functions in V hp derived in [Moiola, Hiptmair & Perugia, Tech. Rep. 2009] j=1 Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 22 / 30

61 Convergence Result for PWDG p error estimates in 2D Assume u H k+1, p = 2m + 1, constant flux parameters; u u hp Fh C(ωh)ω 1/2 h k 1/2( log(p) p ω u u hp 0,Ω C(ωh) diamωh k 1( log(p) p ) k 1/2 u k+1,ω,ω ) k 1/2 u k+1,ω,ω Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 23 / 30

62 Convergence Result for PWDG p error estimates in 2D Assume u H k+1, p = 2m + 1, constant flux parameters; u u hp Fh C(ωh)ω 1/2 h k 1/2( log(p) p ω u u hp 0,Ω C(ωh) diamωh k 1( log(p) p ) k 1/2 u k+1,ω,ω ) k 1/2 u k+1,ω,ω Quasi-optimal error estimates in energy -norm in p (exponential convergence for smooth analytical solutions) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 23 / 30

63 Convergence Result for PWDG p error estimates in 2D Assume u H k+1, p = 2m + 1, constant flux parameters; u u hp Fh C(ωh)ω 1/2 h k 1/2( log(p) p ω u u hp 0,Ω C(ωh) diamωh k 1( log(p) p ) k 1/2 u k+1,ω,ω ) k 1/2 u k+1,ω,ω Quasi-optimal error estimates in energy -norm in p (exponential convergence for smooth analytical solutions) No quasi-optimal estimate available in L 2 norm Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 23 / 30

64 Convergence Result for PWDG p error estimates in 2D Assume u H k+1, p = 2m + 1, constant flux parameters; u u hp Fh C(ωh)ω 1/2 h k 1/2( log(p) p ω u u hp 0,Ω C(ωh) diamωh k 1( log(p) p ) k 1/2 u k+1,ω,ω ) k 1/2 u k+1,ω,ω Quasi-optimal error estimates in energy -norm in p (exponential convergence for smooth analytical solutions) No quasi-optimal estimate available in L 2 norm No estimate available for the gradient of the error, but only for u P(u p ), where P(u p ) is a computable projection of u p onto the space of piecewise polynomials of degree p Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 23 / 30

65 Trefftz-DGFEM for Maxwell s Equations Electric-field based formulation: { (µ 1 E) ω 2 ε E = 0 in Ω (µ 1 E) n iωλ(n E) n = g L 2 T ( Ω) on Ω Assume constant µ, ε, λ globally constant. Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 24 / 30

66 Trefftz-DGFEM for Maxwell s Equations Electric-field based formulation: { (µ 1 E) ω 2 ε E = 0 in Ω (µ 1 E) n iωλ(n E) n = g L 2 T ( Ω) on Ω Assume constant µ, ε, λ globally constant. Derivation and formulation of Trefftz-DGFEM analogous to Helmholtz [Huttunen, Malinen & Monk, 2006] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 24 / 30

67 Trefftz-DGFEM for Maxwell s Equations Electric-field based formulation: { (µ 1 E) ω 2 ε E = 0 in Ω (µ 1 E) n iωλ(n E) n = g L 2 T ( Ω) on Ω Assume constant µ, ε, λ globally constant. Derivation and formulation of Trefftz-DGFEM analogous to Helmholtz [Huttunen, Malinen & Monk, 2006] Local Trefftz property implies local divergence-free property: 0 = ( (µ 1 v hp ) ω 2 ε v hp ) = (ε vhp ) divergence-free vector plane (or spherical) waves as basis functions: x a e iω εµd x, (a, d) = 0 Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 24 / 30

68 Trefftz-DGFEM for Maxwell s Equations Reminder: scheme of the analysis for Helmholtz Step Ingredients Results 1 choice of a suitable mesh-skeleton well-posedness and norm in T(T h ) quasi-optimality 2 elliptic regularity and stability estimates L 2 norm estimates by for (inhomogeneous) adjoint problem modified duality argument 3 best approximation estimates convergence rates Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 25 / 30

69 Trefftz-DGFEM for Maxwell s Equations Set w 2 F h := Im [A h (w,w)]; this is a norm in the Trefftz space T(T h ) := {w H 2 (T h ) 3 : (µ 1 w) ω 2 ε w = 0 in each K T h } Define the enriched F + norm such that h for every v,w T(T h ) A h (v,w) C v F + h w F h Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 26 / 30

70 Trefftz-DGFEM for Maxwell s Equations Set w 2 F h := Im [A h (w,w)]; this is a norm in the Trefftz space T(T h ) := {w H 2 (T h ) 3 : (µ 1 w) ω 2 ε w = 0 in each K T h } Define the enriched F + norm such that h for every v,w T(T h ) A h (v,w) C v F + h w F h Well-posedness and quasi-optimal error estimates in Fh norm Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 26 / 30

71 Trefftz-DGFEM for Maxwell s Equations Set w 2 F h := Im [A h (w,w)]; this is a norm in the Trefftz space T(T h ) := {w H 2 (T h ) 3 : (µ 1 w) ω 2 ε w = 0 in each K T h } Define the enriched F + norm such that h for every v,w T(T h ) A h (v,w) C v F + h w F h Well-posedness and quasi-optimal error estimates in Fh norm Remark: The Fh norm only contains tangential jumps and traces no weak divergence-free property can be expected Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 26 / 30

72 Duality for Maxwell s Equations We would like to prove w 0,Ω C(ω) w Fh for Trefftz functions Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 27 / 30

73 Duality for Maxwell s Equations We would like to prove w 0,Ω C(ω) w Fh for Trefftz functions Adjoint problem { (µ 1 Φ) ω 2 ε Φ = w in Ω (µ 1 Φ) n + iωλ(n Φ) n = 0 on Ω Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 27 / 30

74 Duality for Maxwell s Equations We would like to prove w 0,Ω C(ω) w Fh for Trefftz functions Adjoint problem { (µ 1 Φ) ω 2 ε Φ = w in Ω (µ 1 Φ) n + iωλ(n Φ) n = 0 on Ω No duality argument possible with w T(T h ), since T(T h ) H(div 0 ; Ω) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 27 / 30

75 Duality for Maxwell s Equations We would like to prove w 0,Ω C(ω) w Fh for Trefftz functions Adjoint problem { (µ 1 Φ) ω 2 ε Φ = w in Ω (µ 1 Φ) n + iωλ(n Φ) n = 0 on Ω No duality argument possible with w T(T h ), since T(T h ) H(div 0 ; Ω) Helmholtz decomposition (L 2 orth) w = w 0 + p with w 0 H(div 0 ; Ω) and p H 1 0 (Ω), then estimate w 0 and p separately Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 27 / 30

76 Duality for Maxwell s Equations Estimate of w 0 H(div 0 ; Ω) by duality Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 28 / 30

77 Duality for Maxwell s Equations Estimate of w 0 H(div 0 ; Ω) by duality { (µ 1 Φ) ω 2 ε Φ = w 0 in Ω (µ 1 Φ) n + iωλ(n Φ) n = 0 on Ω New stability and elliptic regularity results: Φ 0,Ω + ω Φ 0,Ω C w 0 0,Ω with C indep. of ω! Φ 1/2+s,Ω + ω Φ 1/2+s,Ω C (1 + ω) w 0 0,Ω (0 < s < 1/2) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 28 / 30

78 Duality for Maxwell s Equations Estimate of w 0 H(div 0 ; Ω) by duality w 0 0,Ω C [ (ωh) 1/2 + ω 1/2 h s] w Fh. Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 28 / 30

79 Duality for Maxwell s Equations Estimate of w 0 H(div 0 ; Ω) by duality w 0 0,Ω C [ (ωh) 1/2 + ω 1/2 h s] w Fh. We can estimate p only in a weaker norm ( p, v) Ω p H(div;Ω) := sup C ω 3/2 (1 + h 1/2 ) w Fh v H(div;Ω) v H(div;Ω) using Laplace elliptic regularity and Helmholtz decomposition Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 28 / 30

80 Duality for Maxwell s Equations Estimate of w 0 H(div 0 ; Ω) by duality w 0 0,Ω C [ (ωh) 1/2 + ω 1/2 h s] w Fh. We can estimate p only in a weaker norm ( p, v) Ω p H(div;Ω) := sup C ω 3/2 (1 + h 1/2 ) w Fh v H(div;Ω) v H(div;Ω) using Laplace elliptic regularity and Helmholtz decomposition Adding together and using estimates in Fh norm: Error estimates in H(div; Ω) norm. Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 28 / 30

81 Trefftz-DGFEM for Maxwell s Equations Scheme of the analysis for Maxwell Step Ingredients Results 1 choice of a suitable mesh-skeleton well-posedness and norm in T(T h ) quasi-optimality 2 elliptic regularity and stability H(div; Ω) norm estimates by estimates for (inhomogeneous) modified duality argument adjoint problem (new) using Helmholtz decomposition 3 best approximation estimates convergence rates Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 29 / 30

82 Conclusions Derivation of Trefftz-DG methods for Helmholtz and Maxwell Abstract (h and) p convergence analysis Convergence rates for generalized harmonic polynomials and plane waves: h version p version 2D 3D up to a parameter (dep. on element shape) Main drawback: ill-conditioning for (moderately) high p Open issues: explicit order of p convergence in 3D p-convergence in (broken) H 1 -norm non quasi-uniform meshes non constant coefficients plane waves: adaptive choice of directions Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for time-harmonic wave problems 30 / 30