Trefftz-Discontinuous Galerkin Methods for Acoustic Scattering - Recent Advances

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1 Trefftz-Discontinuous Galerkin Methods for Acoustic Scattering - Recent Advances Ilaria Perugia Dipartimento di Matematica Università di Pavia (Italy) In collaboration with Ralf Hiptmair, Christoph Schwab, ETH Zürich Andrea Moiola, University of Reading EFEF May 31-June 1st, 2013, Heraklion, Crete, Greece Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 1 / 16

2 Outline Model problem Trefftz-discontinuous Galerkin methods Approximation properties of Trefftz FE spaces (Vekua s theory) Exponential convergence Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 2 / 16

3 Acoustic scattering problem Exterior boundary value problem sound-soft scatterer: Ω D R 3 Lipschitz polygonal star-shaped w.r.t. the origin time-harmonic incident field with complex amplitude u i and wave number k = ω/c u = u i +u s Γ D 0 Ω D Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 3 / 16

4 Acoustic scattering problem Exterior boundary value problem sound-soft scatterer: Ω D R 3 Lipschitz polygonal star-shaped w.r.t. the origin time-harmonic incident field with complex amplitude u i and wave number k = ω/c u = u i +u s u k 2 u = 0 in R 3 \Ω D u = 0 on Γ D := Ω D Γ D 0 Ω D (Helmholtz equation) (sound-soft obstacle) ( ) u lim x x s (x) x iku s (x) = 0 (radiation condition) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 3 / 16

5 Acoustic scattering problem Artificial bounded domain Ω R Ω D with boundary Γ R : dist(γ D,Γ R ) > 0 either Γ R smooth or Ω R Lipschitz polyhedron Ω R star-shaped w.r.t. a ball centered at the origin Ω R Γ D Γ R 0 Ω D Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 4 / 16

6 Acoustic scattering problem Model problem u k 2 u = 0 u = 0 u n ikϑu = g R in Ω := Ω R \Ω D on Γ D in Γ R (g R L 2 (Γ R ) impedance trace of u i ) Γ D Ω R Γ R 0 Ω D Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 4 / 16

7 Acoustic scattering problem Model problem u k 2 u = 0 u = 0 u n ikϑu = g R in Ω := Ω R \Ω D on Γ D in Γ R (g R L 2 (Γ R ) impedance trace of u i ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 5 / 16

8 Trefftz-finite element methods emptyline { Lu = 0 in Ω (L elliptic operator) + b.c. emptyline Trefftz spaces for L Erich Immanuel Trefftz ( ) Given a mesh T h of Ω, for all K T h we define the local Trefftz spaces and set T(K) = {v K : Lv = 0} T(T h ) = {v L 2 (Ω) : v K T(K) K T h } Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 6 / 16

9 Trefftz-finite element methods emptyline { Lu = 0 in Ω (L elliptic operator) + b.c. emptyline Trefftz spaces for L Erich Immanuel Trefftz ( ) Given a mesh T h of Ω, for all K T h we define the local Trefftz spaces and set T(K) = {v K : Lv = 0} T(T h ) = {v L 2 (Ω) : v K T(K) K T h } Trefftz finite element spaces Let V p (K) T(K) be finite dimensional local spaces; we set V p (T h ) = {v L 2 (Ω) : v K V p (K) K T h } Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 6 / 16

10 Trefftz-finite element methods Examples of finite element Trefftz functions Laplace operator: harmonic polynomials Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 7 / 16

11 Trefftz-finite element methods Examples of finite element Trefftz functions Laplace operator: harmonic polynomials Helmholtz operator: plane waves, circular/spherical waves emptyspace emptyspace x e ikd x Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 7 / 16

12 Trefftz-finite element methods How to match traces across interelement boundaries? Partition of unity [Babuška & Melenk, ] Least squares [Stojek, 1998], [Monk & Wang, 1999], [Barnett & Betcke, 2010] Lagrange multipliers [Farhat, Harari & Franca, 2001], [Farhat, Tezaur, Hetmaniuk, Harari,..., ] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 8 / 16

13 Trefftz-finite element methods How to match traces across interelement boundaries? Partition of unity [Babuška & Melenk, ] Least squares [Stojek, 1998], [Monk & Wang, 1999], [Barnett & Betcke, 2010] Lagrange multipliers [Farhat, Harari & Franca, 2001], [Farhat, Tezaur, Hetmaniuk, Harari,..., ] Discontinuous Galerkin methods (UWVF) [Cessenat & Després, ], [Huttunen, Kaipio, Luostari, Malinen & Monk, ], [Gabard, 2007], [Buffa & Monk, 2008], [Gittelson, Hiptmair, Moiola & Perugia, ] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 8 / 16

14 Trefftz-finite element methods How to match traces across interelement boundaries? Partition of unity [Babuška & Melenk, ] Least squares [Stojek, 1998], [Monk & Wang, 1999], [Barnett & Betcke, 2010] Lagrange multipliers [Farhat, Harari & Franca, 2001], [Farhat, Tezaur, Hetmaniuk, Harari,..., ] Discontinuous Galerkin methods (UWVF) [Cessenat & Després, ], [Huttunen, Kaipio, Luostari, Malinen & Monk, ], [Gabard, 2007], [Buffa & Monk, 2008], [Gittelson, Hiptmair, Moiola & Perugia, ] In this talk: Trefftz discontinuous Galerkin (TDG) methods Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 8 / 16

15 TDG methods Helmholtz equation: u k 2 u = 0 in Ω Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 9 / 16

16 TDG methods Helmholtz equation: u k 2 u = 0 in Ω Introduce a mesh T h = {K}; multiply by test functions and integrate by parts element by element twice (ultra weak formulation) u( v k 2 v)dv + u v n K ds u n K v ds = 0 K K K Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 9 / 16

17 TDG methods Helmholtz equation: u k 2 u = 0 in Ω Introduce a mesh T h = {K}; multiply by test functions and integrate by parts element by element twice (ultra weak formulation) u( v k 2 v)dv + u v n K ds u n K v ds = 0 K K K Replace continuous by discrete functions, traces by numerical fluxes u,v u hp,v hp, on K : u û hp, u ik σ hp Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 9 / 16

18 TDG methods Helmholtz equation: u k 2 u = 0 in Ω Introduce a mesh T h = {K}; multiply by test functions and integrate by parts element by element twice (ultra weak formulation) u( v k 2 v)dv + u v n K ds u n K v ds = 0 K K K Replace continuous by discrete functions, traces by numerical fluxes u,v u hp,v hp, on K : u û hp, u ik σ hp Trefftz discrete spaces V p (T h ) T(T h ) v hp k 2 v hp = 0 in K Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 9 / 16

19 TDG methods Helmholtz equation: u k 2 u = 0 in Ω Introduce a mesh T h = {K}; multiply by test functions and integrate by parts element by element twice (ultra weak formulation) u( v k 2 v)dv + u v n K ds u n K v ds = 0 K K K Replace continuous by discrete functions, traces by numerical fluxes u,v u hp,v hp, on K : u û hp, u ik σ hp Trefftz discrete spaces V p (T h ) T(T h ) v hp k 2 v hp = 0 in K TDG formulation For every K T h, û hp v hp n K ds ik σ hp n K v hp ds = 0 K K Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 9 / 16

20 TDG methods Numerical fluxes { h u hp } αik [[u hp ]] N on interior faces ik σ hp = h u hp αiku hp n on faces on Γ D h u hp (1 δ)( h u hp +ikϑu hp n g R n) on faces on Γ R {u hp } β(ik) 1 [[ h u hp ]] N on interior faces û hp = 0 on faces on Γ D u hp δ ( (ikϑ) 1 h u hp n+u hp (ikϑ) 1 ) g R on faces on Γ R with α,β > 0, 0 < δ 1/2 Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 10 / 16

21 TDG methods Numerical fluxes { h u hp } αik [[u hp ]] N on interior faces ik σ hp = h u hp αiku hp n on faces on Γ D h u hp (1 δ)( h u hp +ikϑu hp n g R n) on faces on Γ R {u hp } β(ik) 1 [[ h u hp ]] N on interior faces û hp = 0 on faces on Γ D u hp δ ( (ikϑ) 1 h u hp n+u hp (ikϑ) 1 ) g R on faces on Γ R with α,β > 0, 0 < δ 1/2 The choice α = β = δ = 1/2 gives the UWVF by Cessenat & Després (see [Gabard, 2007], [Buffa & Monk, 2008], [Gittelson, Hiptmair & Perugia, 2009]). Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 10 / 16

22 Trefftz-discontinuous Galerkin (TDG) methods A priori error estimates [Hiptmair, Moiola & Perugia, Appl. Num. Math. (2013)] regularity/stability of the dual solution (ϕ H 3 2 +s (Ω) with 0 < s 1/2) in non star-shaped domains, with explicit dependence of the stability constant on k (extending the results of [Melenk, 1995], [Cummings & Feng, 2006], [Hetmaniuk, 2007]) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 11 / 16

23 Trefftz-discontinuous Galerkin (TDG) methods A priori error estimates [Hiptmair, Moiola & Perugia, Appl. Num. Math. (2013)] regularity/stability of the dual solution (ϕ H 3 2 +s (Ω) with 0 < s 1/2) in non star-shaped domains, with explicit dependence of the stability constant on k (extending the results of [Melenk, 1995], [Cummings & Feng, 2006], [Hetmaniuk, 2007]) flux parameters depending on the local meshsize Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 11 / 16

24 Trefftz-discontinuous Galerkin (TDG) methods A priori error estimates [Hiptmair, Moiola & Perugia, Appl. Num. Math. (2013)] regularity/stability of the dual solution (ϕ H 3 2 +s (Ω) with 0 < s 1/2) in non star-shaped domains, with explicit dependence of the stability constant on k (extending the results of [Melenk, 1995], [Cummings & Feng, 2006], [Hetmaniuk, 2007]) flux parameters depending on the local meshsize hp-error analysis allowing for meshes which are locally refined near the scatterer space Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 11 / 16

25 Trefftz-discontinuous Galerkin (TDG) methods L 2 -norm abstract error estimates u u hp 0,Ω C(Ω,k,h) inf v hp V p(t h ) u v hp DG + with C(Ω,k,h) = d Ω [(kh) 1 2 +(dω k) 1 2 (d 1 Ω h)s ] (best approximation) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 12 / 16

26 Trefftz-discontinuous Galerkin (TDG) methods L 2 -norm abstract error estimates u u hp 0,Ω C(Ω,k,h) inf v hp V p(t h ) u v hp DG + with C(Ω,k,h) = d Ω [(kh) 1 2 +(dω k) 1 2 (d 1 Ω h)s ] (best approximation) Approximation properties of Trefftz FE spaces to be investigated Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 12 / 16

27 Approximation properties of Trefftz FE spaces Vekua s theory for a 2nd order elliptic operator L harmonic functions V 2 V 1 {v : Lv = 0} harmonic polynomials V 2 V 1 generalized harmonic polynomials Ilya Nestorovich Vekua ( ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 13 / 16

28 Approximation properties of Trefftz FE spaces Vekua s theory for a 2nd order elliptic operator L harmonic functions V 2 V 1 {v : Lv = 0} harmonic polynomials V 2 V 1 generalized harmonic polynomials Ilya Nestorovich Vekua ( ) Then: approximation of generalized harmonic polynomials in V p (T h ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 13 / 16

29 Approximation properties of Trefftz FE spaces Vekua s theory for a 2nd order elliptic operator L harmonic functions V 2 V 1 {v : Lv = 0} harmonic polynomials V 2 V 1 generalized harmonic polynomials Ilya Nestorovich Vekua ( ) Then: approximation of generalized harmonic polynomials in V p (T h ) V p (T h ) plane wave or circular/spherical wave spaces: sharp best approximation estimates of Helmholtz solutions in weighted H r -norms [Moiola, Hiptmair & Perugia, ZAMP (2011)] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 13 / 16

30 Exponential convergence hp-tdg on geometric meshes We aim at proving that, if u is a solution of the acoustic scattering problem, then inf v hp V p(t h ) u v hp DG + C exp( b #DOFs) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 14 / 16

31 Exponential convergence hp-tdg on geometric meshes We aim at proving that, if u is a solution of the acoustic scattering problem, then inf v hp V p(t h ) u v hp DG + C exp( b #DOFs) In particular, we need exponential convergence of the approximation of Helmholtz solutions by generalized harmonic polynomials in W 1, -norm in the elements far from corners Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 14 / 16

32 Exponential convergence hp-tdg on geometric meshes We aim at proving that, if u is a solution of the acoustic scattering problem, then inf v hp V p(t h ) u v hp DG + C exp( b #DOFs) In particular, we need exponential convergence of the approximation of Helmholtz solutions by generalized harmonic polynomials in W 1, -norm in the elements far from corners Vekua s theory: transfer approximation properties of harmonic polynomials for harmonic functions to generalized harmonic polynomials for Helmholtz solutions Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 14 / 16

33 Exponential convergence Approximation of harmonic functions by harmonic polynomials [Hiptmair, Moiola, Perugia & Schwab, Tech. Rep. 2012] Assuming D R 2 0 < ρ 1/2 s.t. D contains B ρ 0 < ρ 0 < ρ s.t. D is star-shaped w.r.t. B ρ0 if u is harmonic and belongs to W 1, (D δ ), then there is a sequence of harmonic polynomials {Q p } p of degree at most p such that u Q p W j, (D) C exp( bp) u W 1, (D δ ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 15 / 16

34 Exponential convergence TDG approximations of the Laplace equation on geometric meshes Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 16 / 16

35 Exponential convergence TDG approximations of the Laplace equation on geometric meshes #DOFs piecewise polynomials of degree p on geometric meshes with l layers full polynomials: O(p 2 l) harmonic polynomials: O(p l) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 16 / 16

36 Exponential convergence TDG approximations of the Laplace equation on geometric meshes #DOFs piecewise polynomials of degree p on geometric meshes with l layers full polynomials: O(p 2 l) harmonic polynomials: O(p l) Exponential convergence for the Laplace equation TDG for the Laplace equation on geometric meshes: exponential convergence with rate exp( b #DOFs) (instead of exp( b 3 #DOFs)) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Acoustic Scattering 16 / 16