Trefftz-Discontinuous Galerkin Methods for the Time-Harmonic Maxwell Equations

Transcription

1 Trefftz-Discontinuous Galerkin Methods for the Time-Harmonic Maxwell Equations Ilaria Perugia Dipartimento di Matematica - Università di Pavia (Italy) Joint work with Ralf Hiptmair, Andrea Moiola (SAM - ETH Zürich, CH) Discontinuous Galerkin Methods for Partial Differential Equations Heraklion, September 26-28, 2011 Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 1 / 26

2 Time-Harmonic Maxwell s Equations emptyline emptyline Electric-field based formulation with impedance boundary conditions: { (µ 1 E) ω 2 ε E = 0 James Clerk Maxwell ( ) in Ω (µ 1 E) n iωϑ(n E) n = g on Ω bounded polyhedral domain Ω R 3 angular frequency (wave number) ω ω 0 > 0 (wave length λ = 2π/ω) assume ε, µ, ϑ R to be constant, ε, µ > 0, ϑ 0 g L 2 T ( Ω) emptyline Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 2 / 26

3 Time-Harmonic Maxwell s Equations emptyline emptyline Electric-field based formulation with impedance boundary conditions: { (µ 1 E) ω 2 ε E = 0 James Clerk Maxwell ( ) in Ω (µ 1 E) n iωϑ(n E) n = g on Ω Fredholm alternative well-posedness of weak formulation in H imp (curl; Ω) = {v H(curl; Ω) : (n v) n L 2 T ( Ω)} with (εe) = 0 and stability bound (see P. Monk s book) µ 1/2 E 0,Ω + ω ε 1/2 E 0,Ω C stab g 0, Ω Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 2 / 26

4 Trefftz-Discontinuous Galerkin Methods emptyline { Lu = 0 + b.c. in Ω 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 Maxwell s Equations 3 / 26

5 Trefftz-Discontinuous Galerkin Methods emptyline { Lu = 0 + b.c. in Ω 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 Maxwell s Equations 3 / 26

6 Trefftz-Discontinuous Galerkin Methods Examples of finite element Trefftz functions: Laplace operator: harmonic polynomials Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 4 / 26

7 Trefftz-Discontinuous Galerkin Methods Examples of finite element Trefftz functions: Laplace operator: harmonic polynomials Helmholtz operator: plane waves, circular/spherical waves emptyspace emptyspace Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 4 / 26

8 Trefftz-Discontinuous Galerkin Methods Examples of finite element Trefftz functions: Laplace operator: harmonic polynomials Helmholtz operator: plane waves, circular/spherical waves emptyspace emptyspace Maxwell operator:... Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 4 / 26

9 Trefftz-Discontinuous Galerkin Methods How to match traces across interelement boundaries? Partition of unit method [Babuška & Melenk, ] Least square methods [Stojek, 1998], [Monk & Wang, 1999], [Barnett & Betcke, 2010] Lagrange multipliers [Farhat, Harari & Franca, 2001, Farhat, Tezaur,..., ] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 5 / 26

10 Trefftz-Discontinuous Galerkin Methods How to match traces across interelement boundaries? Partition of unit method [Babuška & Melenk, ] Least square methods [Stojek, 1998], [Monk & Wang, 1999], [Barnett & Betcke, 2010] Lagrange multipliers [Farhat, Harari & Franca, 2001, Farhat, Tezaur,..., ] Discontinuous Galerkin (DG) methods [Cessenat & Despres, ], [Gabard, 2007], [Buffa & Monk, 2008], [Gittelson, Hiptmair & Perugia, 2009] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 5 / 26

11 Trefftz-Discontinuous Galerkin Methods How to match traces across interelement boundaries? Partition of unit method [Babuška & Melenk, ] Least square methods [Stojek, 1998], [Monk & Wang, 1999], [Barnett & Betcke, 2010] Lagrange multipliers [Farhat, Harari & Franca, 2001, Farhat, Tezaur,..., ] Discontinuous Galerkin (DG) methods [Cessenat & Despres, ], [Gabard, 2007], [Buffa & Monk, 2008], [Gittelson, Hiptmair & Perugia, 2009] Boris Grigoryevich Galerkin ( ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 5 / 26

12 Trefftz-Discontinuous Galerkin Methods How to match traces across interelement boundaries? Partition of unit method [Babuška & Melenk, ] Least square methods [Stojek, 1998], [Monk & Wang, 1999], [Barnett & Betcke, 2010] Lagrange multipliers [Farhat, Harari & Franca, 2001, Farhat, Tezaur,..., ] Discontinuous Galerkin (DG) methods [Cessenat & Despres, ], [Gabard, 2007], [Buffa & Monk, 2008], [Gittelson, Hiptmair & Perugia, 2009] Boris Grigoryevich Galerkin ( )...Discontinuous Galerkin Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 5 / 26

13 FEM for Time-Harmonic Wave Problems FEM for mid/high-frequency problems: the number of d.o.f. to obtain a given accuracy increases with the wave number ω h version FEM is affected by pollution effect ([Babuška & Sauter, 2000]): Galerkin error C(ω) best approximation error where C(ω) is an increasing function of ω for hp version FEM, the conditions ωh p small and p C log(ω) ensure that quasi-optimality holds with constant independent of ω ([Melenk & Sauter, 2011]) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 6 / 26

14 FEM for Time-Harmonic Wave Problems high-order methods have less dispersion and show less pollution effect than lower-order methods ([Ihlenburg & Babuška, 1997]) analysis of the dispersion error for hp FEM on translation-invariant meshes ([Ainsworth, 2004]): 2p + 1 > ωh + C(ωh) 1/3 2p + 1 < ωh o(ωh) 1/3 superexponential decay for increasing p no decay spectral element methods (SEM): full resolution for 2p + 1 ωh and better accuracy than for FEM; optimally blended FEM-SEM to improve accuracy over FEM and SEM ([Ainsworth & Vajid, 2009, 2010]) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 7 / 26

15 FEM for Time-Harmonic Wave Problems The use of special basis functions can reduce the computational cost Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 8 / 26

16 FEM for Time-Harmonic Wave Problems The use of special basis functions can reduce the computational cost Trefftz FEM: they incorporate information on the problem within FEM spaces Trefftz Fe same accuracy with less d.o.f. than with high order polynomials Trefftz FEM: (and for plane waves: possibility to choose directions) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 8 / 26

17 FEM for Time-Harmonic Wave Problems The use of special basis functions can reduce the computational cost Trefftz FEM: they incorporate information on the problem within FEM spaces Trefftz Fe same accuracy with less d.o.f. than with high order polynomials Trefftz FEM: (and for plane waves: possibility to choose directions) Focus of this talk: p version of Trefftz-FEM Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 8 / 26

18 FEM for Time-Harmonic Wave Problems The use of special basis functions can reduce the computational cost Trefftz FEM: they incorporate information on the problem within FEM spaces Trefftz Fe same accuracy with less d.o.f. than with high order polynomials Trefftz FEM: (and for plane waves: possibility to choose directions) Focus of this talk: p version of Trefftz-FEM Outline: derivation of Trefftz-DG methods for the Maxwell problem p version error analysis [Hiptmair, Moiola & Perugia, Tech. Rep. 2011] [Hiptmair, Moiola & Perugia, M 3 AS online] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 8 / 26

19 Trefftz-DG Methods for the Maxwell Problem Maxwell s problem: { (µ 1 E) ω 2 ε E = 0 in Ω (µ 1 E) n iωϑ(n E) n = g on Ω James Clerk Maxwell ( ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 9 / 26

20 Trefftz-DG Methods for the Maxwell Problem Maxwell s problem: { (µ 1 E) ω 2 ε E = 0 in Ω (µ 1 E) n iωϑ(n E) n = g on Ω James Clerk Maxwell ( ) Introduce a mesh T h = {K} (with possible hanging nodes) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 9 / 26

21 Trefftz-DG Methods for the Maxwell Problem Maxwell s problem: { (µ 1 E) ω 2 ε E = 0 in Ω (µ 1 E) n iωϑ(n E) n = g on Ω James Clerk Maxwell ( ) Introduce a mesh T h = {K} (with possible hanging nodes) Multiply by test functions and integrate by parts element by element (terms with appear in the formulation) K Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 9 / 26

22 Trefftz-DG Methods for the Maxwell Problem Maxwell s problem: { (µ 1 E) ω 2 ε E = 0 in Ω (µ 1 E) n iωϑ(n E) n = g on Ω James Clerk Maxwell ( ) Introduce a mesh T h = {K} (with possible hanging nodes) Multiply by test functions and integrate by parts element by element (terms with appear in the formulation) K Integrate by parts a second time (ultra weak formulation) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 9 / 26

23 Trefftz-DG Methods for the Maxwell Problem Maxwell s problem: { (µ 1 E) ω 2 ε E = 0 in Ω (µ 1 E) n iωϑ(n E) n = g on Ω James Clerk Maxwell ( ) Introduce a mesh T h = {K} (with possible hanging nodes) Multiply by test functions and integrate by parts element by element (terms with appear in the formulation) K Integrate by parts a second time (ultra weak formulation) Replace continuous spaces by discrete spaces and traces at K s by numerical fluxes: E Êhp µ 1 E iωĥhp Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 9 / 26

24 Trefftz-DG Methods for the Maxwell Problem Z K Z E hp ( (µ 1 ξ hp ) ω 2 ε ξ hp ) + K n Eb µ hp 1 ξ Z hp + K iω n b H hp ξ hp = 0 Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 10 / 26

25 Trefftz-DG Methods for the Maxwell Problem Z K Z E hp ( (µ 1 ξ hp ) ω 2 ε ξ hp ) + K Using Trefftz discrete spaces V p (T h ) T (T h ), n Eb µ hp 1 ξ Z hp + K iω n b H hp ξ hp = 0 (µ 1 ξ hp ) ω 2 ε ξ hp = 0 in every K T h Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 10 / 26

26 Trefftz-DG Methods for the Maxwell Problem Z K Z E hp ( (µ 1 ξ hp ) ω 2 ε ξ hp ) + K Using Trefftz discrete spaces V p (T h ) T (T h ), n Eb µ hp 1 ξ Z hp + K iω n b H hp ξ hp = 0 (µ 1 ξ hp ) ω 2 ε ξ hp = 0 in every K T h Trefftz-DG formulation for the Maxwell problem ) For every K T h, n Êhp (µ 1 ξ hp + K K iω n Ĥhp ξ hp = 0 Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 10 / 26

27 Trefftz-DG Methods for the Maxwell Problem Z K Z E hp ( (µ 1 ξ hp ) ω 2 ε ξ hp ) + K Using Trefftz discrete spaces V p (T h ) T (T h ), n Eb µ hp 1 ξ Z hp + K iω n b H hp ξ hp = 0 (µ 1 ξ hp ) ω 2 ε ξ hp = 0 in every K T h Trefftz-DG formulation for the Maxwell problem ) For every K T h, n Êhp (µ 1 ξ hp + K K iω n Ĥhp ξ hp = 0 Numerical fluxes: Ê hp = {E hp } β iω [[µ 1 h E hp ]] T iω Ĥhp = {µ 1 h E hp } + α iω [[E hp ]] T Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 10 / 26

28 Trefftz-DG Methods for the Maxwell Problem The Ultra Weak Variational Formulation (UWVF) for the Maxwell problem ([Cessenat & Despres, ]) can be seen as a DG method with plane wave basis functions and flux parameters α = β = 1/2 For a derivation of the UWVF for the Maxwell problem as a DG method with upwind fluxes, see [Huttunen, Malinen & Monk, 2007]. Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 11 / 26

29 Trefftz-DG Methods for the Maxwell Problem The Ultra Weak Variational Formulation (UWVF) for the Maxwell problem ([Cessenat & Despres, ]) can be seen as a DG method with plane wave basis functions and flux parameters α = β = 1/2 For a derivation of the UWVF for the Maxwell problem as a DG method with upwind fluxes, see [Huttunen, Malinen & Monk, 2007]. For the Helmholtz problem, similar considerations were established in [Gabard, 2007], [Buffa & Monk, 2008], [Gittelson, Hiptmair & Perugia, 2009]. Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 11 / 26

30 Example of Maxwell-Trefftz Spaces Plane wave (PW) Maxwell-Trefftz spaces scalar PW: x e iκd x Helmholtz solutions vector PW: x a e iκd x, a d = 0 Maxwell solutions vector PW: x a e iκd x, (div = 0) κ = ω εµ Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 12 / 26

31 Example of Maxwell-Trefftz Spaces Plane wave (PW) Maxwell-Trefftz spaces scalar PW: x e iκd x Helmholtz solutions vector PW: x a e iκd x, a d = 0 Maxwell solutions vector PW: x a e iκd x, (div = 0) κ = ω εµ Basis: p = (q + 1) 2 directions {d l } p l=1 Basis: {a l } p l=1 unit vectors s.t. a l d l {a l e iκd l x, (d l a l )e iκd l x } p l=1 (dimension 2p) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 12 / 26

32 p Version Error Analysis Step Ingredients Results 1 choice of a suitable mesh-skeleton norm well-posedness and in T (T h ) s.t. the formulation is coercive quasi-optimality 2 elliptic regularity and stability estimates L 2 norm estimates by a for (inhomogeneous) adjoint problem modified duality argument 3 best approximation estimates of convergence rates Maxwell solutions in V p(t h ) Same analysis framework as in [Hiptmair, Moiola & Perugia, 2011] for Helmholtz Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 13 / 26

33 p Version Error Analysis Step Ingredients Results 1 choice of a suitable mesh-skeleton norm well-posedness and in T (T h ) s.t. the formulation is coercive quasi-optimality 2 elliptic regularity and stability estimates L 2 norm estimates by a for (inhomogeneous) adjoint problem modified duality argument 3 best approximation estimates of convergence rates Maxwell solutions in V p(t h ) Same analysis as in [Hiptmair, Moiola & Perugia, 2011] for Helmholtz Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 13 / 26

34 Step 1: Well-Posedness and Quasi-Optimality T (T h ) = {v H 1/2+s (curl; T h ), s > 0 : (µ 1 v) ω 2 ε v = 0 in every K T h } Mesh-dependent norms on T (T h ): v 2 DG = ω 1 β 1/2 [[µ 1 h v]] T 2 L 2 (F I h )3 + ω α 1/2 [[v]] T 2 L 2 (F I h )3 + ω 1 δ 1/2 ϑ 1/2 n (µ 1 h v) 2 L 2 (F B h )3 + ω (1 δ) 1/2 ϑ 1/2 (n v) 2 L 2 (F B h )3 v 2 DG + = v 2 DG + ω β 1/2 {v T } 2 L 2 (F I h )3 + ω 1 α 1/2 {(µ 1 h v) T } 2 L 2 (F I h )3 + ω δ 1/2 ϑ 1/2 (n v) 2 L 2 (F B h )3. Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 14 / 26

35 Step 1: Well-Posedness and Quasi-Optimality For all E, v T (T h ): Im [A h (v, v)] = v 2 DG (coercivity) A h (E, v) 2 E DG + v DG (continuity) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 15 / 26

36 Step 1: Well-Posedness and Quasi-Optimality For all E, v T (T h ): Im [A h (v, v)] = v 2 DG (coercivity) A h (E, v) 2 E DG + v DG (continuity) Well-posedness and quasi-optimal error estimates in DG norm E E hp DG 3 inf hp DG + v hp V p(t h ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 15 / 26

37 Step 1: Well-Posedness and Quasi-Optimality For all E, v T (T h ): Im [A h (v, v)] = v 2 DG (coercivity) A h (E, v) 2 E DG + v DG (continuity) Well-posedness and quasi-optimal error estimates in DG norm E E hp DG 3 inf hp DG + v hp V p(t h ) Remark: Local Trefftz property implies local divergence-free property: (µ 1 E hp ) ω 2 ε E hp = 0 (ε E hp ) = 0 in all K T h On the other hand, DG does not provide control on the normal jumps and traces no divergence-free property for E hp can be expected Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 15 / 26

38 p Version Error Analysis Step Ingredients Results 1 choice of a suitable mesh-skeleton norm well-posedness and in T (T h ) s.t. the formulation is coercive quasi-optimality 2 elliptic regularity and stability estimates L 2 norm estimates by a for (inhomogeneous) adjoint problem modified duality argument 3 best approximation estimates of convergence rates Maxwell solutions in V p(t h ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 16 / 26

39 Step 2: Error Estimates in a Mesh-Independent Norm We have an error bound in the DG norm. Now, we want to derive an error bound in a mesh-independent norm duality technique Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 17 / 26

40 Step 2: Error Estimates in a Mesh-Independent Norm We have an error bound in the DG norm. Now, we want to derive an error bound in a mesh-independent norm duality technique We need stability/regularity estimates for the adjoint problem: { (µ 1 Φ) ω 2 ε Φ = w in Ω (µ 1 Φ) n + iωϑ(n Φ) n = 0 on Ω with w = E E hp T(T h ), thus w 0 no stability/regularity Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 17 / 26

41 Step 2: Error Estimates in a Mesh-Independent Norm We have an error bound in the DG norm. Now, we want to derive an error bound in a mesh-independent norm duality technique We need stability/regularity estimates for the adjoint problem: { (µ 1 Φ) ω 2 ε Φ = w in Ω (µ 1 Φ) n + iωϑ(n Φ) n = 0 on Ω with w = E E hp T(T h ), thus w 0 no stability/regularity Helmholtz decomposition (L 2 orth.): w = w 0 + p, w 0 H(div 0 ; Ω), p H 1 0 (Ω) then estimate w 0 and p separately Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 17 / 26

42 Step 2: Error Estimates in a Mesh-Independent Norm Estimate of w 0 By a modified duality argument [Monk & Wang, 1999], assuming shape-regular and quasi-uniform meshes, we prove w 0 L 2 (Ω) 3 C 0(ω, h) w DG w T (T h ) with explicit expression of C 0 (ω, h) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 18 / 26

43 Step 2: Error Estimates in a Mesh-Independent Norm Estimate of w 0 By a modified duality argument [Monk & Wang, 1999], assuming shape-regular and quasi-uniform meshes, we prove w 0 L 2 (Ω) 3 C 0(ω, h) w DG w T (T h ) with explicit expression of C 0 (ω, h) Essential tools: stability/regularity properties of the adjoint problem, with Essential tools: explicit dependence of the constants on ω (here we need Essential tools: ε, µ, ϑ to be constant) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 18 / 26

44 Step 2: Error Estimates in a Mesh-Independent Norm Extension to Maxwell of stability and elliptic regularity results proved in [Melenk, 1995], [Cumming & Feng, 2006], [Hetmaniuk, 2007] for Helmholtz: Φ L 2 (Ω) 3 + ω Φ L 2 (Ω) 3 C w 0 L 2 (Ω) 3 with C indep. of ω! Φ 1/2+s,Ω + ω Φ 1/2+s,Ω C (1 + ω) w 0 L 2 (Ω) 3 (0 < s < 1/2) using a Rellich-type identity for star-shaped polyhedra [Hiptmair, Moiola & Perugia, M3AS online] [Moiola, PhD Thesis, 2011] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 19 / 26

45 Step 2: Error Estimates in a Mesh-Independent Norm Estimate of p The poor regularity of p does not allow to obtain an L 2 norm estimate of p Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 20 / 26

46 Step 2: Error Estimates in a Mesh-Independent Norm Estimate of p The poor regularity of p does not allow to obtain an L 2 norm estimate of p We can estimate p only in a weaker norm: ( p, v) Ω p H(div;Ω) := sup C 1 (ω, h) w DG w T (T h ) v H(div;Ω) v H(div;Ω) with explicit expression of C 1 (ω, h) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 20 / 26

47 Step 2: Error Estimates in a Mesh-Independent Norm Final estimate w := E E hp = w 0 + p { w 0 L 2 (Ω) 3 p H(div;Ω) C 0(ω, h) E E hp DG C 1 (ω, h) E E hp DG + quasi-optimal estimate of E E hp DG Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 21 / 26

48 Step 2: Error Estimates in a Mesh-Independent Norm Final estimate w := E E hp = w 0 + p { w 0 L 2 (Ω) 3 p H(div;Ω) C 0(ω, h) E E hp DG C 1 (ω, h) E E hp DG + quasi-optimal estimate of E E hp DG For solutions E H 1/2+δ (curl; Ω), δ > 0 (minimal regularity), E E hp H(div;Ω) C(ω, h) inf v hp V p(t h ) E v hp DG + with explicit expression of C(ω, h) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 21 / 26

49 p Version Error Analysis Step Ingredients Results 1 choice of a suitable mesh-skeleton norm well-posedness and in T (T h ) s.t. the formulation is coercive quasi-optimality 2 elliptic regularity and stability estimates estimates in mesh-indep. norm by for (inhomogeneous) adjoint problem a modified duality argument 3 best approximation estimates of convergence rates Maxwell solutions in V p(t h ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 22 / 26

50 Step 3: Convergence Rates Abstract error estimates: E E hp DG 3 inf v hp V p(t h ) E v hp DG + E E hp H(div;Ω) C(ω, h) inf v hp V p(t h ) E v hp DG + Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 23 / 26

51 Step 3: Convergence Rates Abstract error estimates: E E hp DG 3 inf v hp V p(t h ) E v hp DG + E E hp H(div;Ω) C(ω, h) inf v hp V p(t h ) E v hp DG + choose the Maxwell-Trefftz finite element spaces V p (T h ) and derive best approximation estimates of the Maxwell solution E in V p (T h ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 23 / 26

52 Step 3: Convergence Rates Abstract error estimates: E E hp DG 3 inf v hp V p(t h ) E v hp DG + E E hp H(div;Ω) C(ω, h) inf v hp V p(t h ) E v hp DG + choose the Maxwell-Trefftz finite element spaces V p (T h ) and derive best approximation estimates of the Maxwell solution E in V p (T h ) Main tool: Vekua s theory Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 23 / 26

53 Step 3: Convergence Rates Vekua s theory for a 2nd order elliptic operator L Harmonic functions V 2 V 1 {v : Lv = 0} approximation by HP V 2 V 1 approximation by GHP Ilya Nestorovich Vekua ( ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 24 / 26

54 Step 3: Convergence Rates Vekua s theory for a 2nd order elliptic operator L Harmonic functions V 2 V 1 {v : Lv = 0} approximation by HP V 2 V 1 approximation by GHP Ilya Nestorovich Vekua ( ) Then: approximation of GHP in V p (T h ) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 24 / 26

55 Step 3: Convergence Rates Vekua s theory for a 2nd order elliptic operator L Harmonic functions V 2 V 1 {v : Lv = 0} approximation by HP V 2 V 1 approximation by GHP Ilya Nestorovich Vekua ( ) Then: approximation of GHP in V p (T h ) Helmholtz: for V p (T h ) plane wave spaces, sharp best approx. estimates in weighted Sobolev norms [Moiola, Hiptmair & Perugia, ZAMP online] Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 24 / 26

56 Step 3: Convergence Rates Vekua s theory for a 2nd order elliptic operator L Harmonic functions V 2 V 1 {v : Lv = 0} approximation by HP V 2 V 1 approximation by GHP Ilya Nestorovich Vekua ( ) Then: approximation of GHP in V p (T h ) Helmholtz: for V p (T h ) plane wave spaces, sharp best approx. estimates in weighted Sobolev norms [Moiola, Hiptmair & Perugia, ZAMP online] Maxwell: we used the estimates for Helmholtz: E Maxwell-Trefftz E = H, with H Helmholtz-Trefftz one order less than expected Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 24 / 26

57 Step 3: Convergence Rates Using trace inequalities, we derive estimates of inf v hp V p(t h ) E v p DG + from best approximation estimates in weighted Sobolev norms Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 25 / 26

58 Step 3: Convergence Rates Using trace inequalities, we derive estimates of inf v hp V p(t h ) E v p DG + from best approximation estimates in weighted Sobolev norms Error estimates: Provided that E H r+1 (curl; Ω), for large p = (q + 1) 2, E E h,p DG q (r 3 2 ) (not sharp because of best approx. estimates) E E h,p H(div;Ω) q (r 3 2 ) (same as in DG norm because of modified duality) Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 25 / 26

59 Concluding Remarks Summary: p version Trefftz-DG methods for the time-harmonic Maxwell problem Stability/regularity results to develop duality argument Vekua s theory to derive best approximation estimates in Trefftz finite element spaces Open issues: to remove the following restrictions: constant material parameters ε, µ, ϑ star-shaped domains (Rellich identity) quasi-uniform meshes (to extend to hp) to derive sharp orders of convergence (Maxwell-Vekua s theory?) to deal with ill-conditioning... Ilaria Perugia (Pavia, Italy) Trefftz-DGFEM for Maxwell s Equations 26 / 26