Trefftz-Discontinuous Galerkin Methods for Maxwell s Equations

Transcription

1 Trefftz-Discontinuous Galerkin Methods for Maxwell s Equations Ilaria Perugia Faculty of Mathematics, University of Vienna Vienna P D E Ralf Hiptmair (ETH Zürich), Andrea Moiola (University of Reading) Building Bridges: Connections and Challenges in Modern Approaches to Numerical PDEs LMS-EPSRC Symposium, Durham, July 8-16, 2014

2 Outline The time-harmonic Maxwell equations Trefftz-discontinuous Galerkin methods Error analysis Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 1 / 19

3 Time-harmonic Maxwell s equations Electric-field based formulation with impedance boundary conditions { (µ 1 E) ω 2 ε E = 0 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 (University of Vienna) Trefftz-DG methods for Maxwell s equation 2 / 19

4 Time-harmonic Maxwell s equations 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 µ 1/2 E 0,Ω + ω ε 1/2 E 0,Ω C stab g 0, Ω (see P. Monk s book) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 3 / 19

5 Time-harmonic Maxwell s equations Numerical issues Oscillating solutions the number of d.o.f. to obtain a given accuracy increases with the frequency ω Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 4 / 19

6 Time-harmonic Maxwell s equations Numerical issues Oscillating solutions the number of d.o.f. to obtain a given accuracy increases with the frequency ω h-version FEM is affected by pollution effect: discretisation error }{{} u u hp where C(ω) is an increasing function of ω C(ω) best approximation error }{{} inf v hp V p(t h ) u v hp [Babuška & Sauter, 2000] Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 4 / 19

7 Trefftz-FEM Trefftz FEM: basis functions are, element by element, solutions to the PDE (for time-harmonic wave problems: oscillating functions with the same frequency as the problem) improved the accuracy vs. number of d.o.f. as compared to standard (polynomial-based) finite element methods Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 5 / 19

8 Trefftz-FEM emptyline { Lu = 0 in Ω (L elliptic operator) + b.c. emptyline Trefftz spaces for L mesh T h of Ω local Trefftz spaces T (E) = {v : Lv = 0} Trefftz spaces T (T h ): discontinuous functions whose restrictions to each E T h belong to T (E) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 6 / 19

9 Trefftz-FEM emptyline { Lu = 0 in Ω (L elliptic operator) + b.c. emptyline Trefftz spaces for L mesh T h of Ω local Trefftz spaces T (E) = {v : Lv = 0} Trefftz spaces T (T h ): discontinuous functions whose restrictions to each E T h belong to T (E) Trefftz finite element spaces let V p (E) T (E) be finite dimensional local spaces Trefftz finite element spaces V p (T h ): discontinuous functions whose restrictions to each E T h belong to V p (E) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 6 / 19

10 Trefftz-FEM for Maxwell s equations UWVF (Trefftz-DG) [Cessenat, 1996], [Cessenat & Després, 2002], [Huttunen, Malinen & Monk, 2007], [Darrigrand & Monk, 2007, 2012] Error analysis: [Hiptmair, Moiola & Perugia, 2013] Other Trefftz approaches [Copeland, 2009], [Copeland, Langer & Pusch, 2009] [Kretzschmar, Schnepp, Tsukerman & Weiland, 2014] Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 7 / 19

11 Trefftz-DG methods for time-harmonic Maxwell s equations Maxwell s equation: (µ 1 E) ω 2 ε E = 0 in Ω Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 8 / 19

12 Trefftz-DG methods for time-harmonic Maxwell s equations Maxwell s equation: (µ 1 E) ω 2 ε E = 0 in Ω Introduce a mesh T h ; multiply by test functions and integrate by parts twice on each element K (ultra weak formulation) ( ) E ( (µ 1 v) ω 2 ε v)+ n E µ 1 v + n (µ 1 E) v = 0 K K K Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 8 / 19

13 Trefftz-DG methods for time-harmonic Maxwell s equations Maxwell s equation: (µ 1 E) ω 2 ε E = 0 in Ω Introduce a mesh T h ; multiply by test functions and integrate by parts twice on each element K (ultra weak formulation) ( ) E ( (µ 1 v) ω 2 ε v)+ n E µ 1 v + n (µ 1 E) v = 0 K K Replace traces by numerical fluxes on K : E Ê, µ 1 E iωĥ K Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 8 / 19

14 Trefftz-DG methods for time-harmonic Maxwell s equations Maxwell s equation: (µ 1 E) ω 2 ε E = 0 in Ω Introduce a mesh T h ; multiply by test functions and integrate by parts twice on each element K (ultra weak formulation) ( ) E ( (µ 1 v) ω 2 ε v)+ n E µ 1 v + n (µ 1 E) v = 0 K K Replace traces by numerical fluxes on K : E Ê, µ 1 E iωĥ Trefftz discrete spaces K (µ 1 v) ω 2 ε v = 0 in K Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 8 / 19

15 Trefftz-DG methods for time-harmonic Maxwell s equations Maxwell s equation: (µ 1 E) ω 2 ε E = 0 in Ω Introduce a mesh T h ; multiply by test functions and integrate by parts twice on each element K (ultra weak formulation) ( ) E ( (µ 1 v) ω 2 ε v)+ n E µ 1 v + n (µ 1 E) v = 0 K K Replace traces by numerical fluxes on K : E Ê, µ 1 E iωĥ Trefftz discrete spaces K (µ 1 v) ω 2 ε v = 0 in K Trefftz-DG formulation For every K T h, K n Ê (µ 1 v ) + n (iωĥ) v = 0 K Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 8 / 19

16 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 (University of Vienna) Trefftz-DG methods for Maxwell s equation 9 / 19

17 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 (3D): p = (q + 1) 2 directions {d l } p l=1 Basis (3D): {a l } p l=1 unit vectors s.t. a l d l Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 9 / 19

18 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 (3D): p = (q + 1) 2 directions {d l } p l=1 Basis (3D): {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, d l e iκd l x } p l=1 Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 9 / 19

19 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 (3D): p = (q + 1) 2 directions {d l } p l=1 Basis (3D): {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, d l e iκd l x } p l=1 (dimension 2(q + 1) 2 ) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 9 / 19

20 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 (3D): p = (q + 1) 2 directions {d l } p l=1 Basis (3D): {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, d l e iκd l x } p l=1 (dimension 2(q + 1) 2 ) Spherical wave Maxwell-Trefftz spaces [A. Moiola, PhD Thesis] Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 9 / 19

21 Error analysis Theoretical analysis [Hiptmair, Moiola & Perugia, MCOM (2013)] Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 10 / 19

22 Error analysis Theoretical analysis [Hiptmair, Moiola & Perugia, MCOM (2013)] Ingredients Results 1) suitable choice of numerical fluxes (stabilisation) unconditional well-posedness Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 10 / 19

23 Error analysis Theoretical analysis [Hiptmair, Moiola & Perugia, MCOM (2013)] Ingredients Results 1) suitable choice of numerical fluxes (stabilisation) unconditional well-posedness 2) regularity and stability estimates (duality argument) abstract error estimates for (inhomog.) adjoint problem in mesh-independent norm Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 10 / 19

24 Error analysis Theoretical analysis [Hiptmair, Moiola & Perugia, MCOM (2013)] Ingredients Results 1) suitable choice of numerical fluxes (stabilisation) unconditional well-posedness 2) regularity and stability estimates (duality argument) abstract error estimates for (inhomog.) adjoint problem in mesh-independent norm 3) best approximation error estimates of Maxwell solutions in Trefftz-finite element spaces convergence rates Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 10 / 19

25 Error analysis Theoretical analysis [Hiptmair, Moiola & Perugia, MCOM (2013)] Ingredients Results 1) suitable choice of numerical fluxes (stabilisation) unconditional well-posedness 2) regularity and stability estimates (duality argument) abstract error estimates for (inhomog.) adjoint problem in mesh-independent norm 3) best approximation error estimates of Maxwell solutions in Trefftz-finite element spaces convergence rates 1) = Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 10 / 19

26 Error analysis Theoretical analysis [Hiptmair, Moiola & Perugia, MCOM (2013)] Ingredients Results 1) suitable choice of numerical fluxes (stabilisation) unconditional well-posedness 2) regularity and stability estimates (duality argument) abstract error estimates for (inhomog.) adjoint problem in mesh-independent norm 3) best approximation error estimates of Maxwell solutions in Trefftz-finite element spaces convergence rates 1) = 2) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 10 / 19

27 Error analysis Theoretical analysis [Hiptmair, Moiola & Perugia, MCOM (2013)] Ingredients Results 1) suitable choice of numerical fluxes (stabilisation) unconditional well-posedness 2) regularity and stability estimates (duality argument) abstract error estimates for (inhomog.) adjoint problem in mesh-independent norm 3) best approximation error estimates of Maxwell solutions in Trefftz-finite element spaces convergence rates 1) = 2) 3) but... Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 10 / 19

28 Error analysis: unconditional well-posedness Trefftz-DG elemental formulation For every K T h, n Ê (µ 1 v ) + n (iωĥ) v = 0 K K Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 11 / 19

29 Error analysis: unconditional well-posedness Trefftz-DG elemental formulation For every K T h, n Ê (µ 1 v ) + K Numerical fluxes on interior faces: K n (iωĥ) v = 0 Ê = {E } β iω [[µ 1 h E]] T iωĥ = {µ 1 h E } + α iω [[E]] T with α, β > 0 Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 11 / 19

30 Error analysis: unconditional well-posedness Trefftz-DG elemental formulation For every K T h, n Ê (µ 1 v ) + K Numerical fluxes on interior faces: K n (iωĥ) v = 0 Ê = {E } β iω [[µ 1 h E]] T iωĥ = {µ 1 h E } + α iω [[E]] T with α, β > 0 ( 1 On boundary faces: Ê = E δϑ 1 iω n (µ 1 h E) + ϑ(n E) n + 1 ) iω g iωĥ = 1 iωµ h E (1 δ) ( 1 iωµ h E ϑ(n E) 1 iω n g ) δ [0, 1] Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 11 / 19

31 Error analysis: unconditional well-posedness Trefftz-DG elemental formulation For every K T h, n Ê (µ 1 v ) + K Numerical fluxes on interior faces: K n (iωĥ) v = 0 Ê = {E } β iω [[µ 1 h E]] T iωĥ = {µ 1 h E } + α iω [[E]] T with α, β > 0 ( 1 On boundary faces: Ê = E δϑ 1 iω n (µ 1 h E) + ϑ(n E) n + 1 ) iω g iωĥ = 1 iωµ h E (1 δ) ( 1 iωµ h E ϑ(n E) 1 iω n g ) δ [0, 1] The choice α = β = δ = 1/2 gives the UWVF by Cessenat & Després [Gabard, 2007], [Buffa & Monk, 2008], [Gittelson, Hiptmair & Perugia, 2009] Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 11 / 19

32 Error analysis: unconditional well-posedness Trefftz-DG method Find E hp V p (T h ) such that A hp (E hp, v hp ) = l hp (v hp ) v hp V p (T h ) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 12 / 19

33 Error analysis: unconditional well-posedness Trefftz-DG method Find E hp V p (T h ) such that A hp (E hp, v hp ) = l hp (v hp ) v hp V p (T h ) In order to have coercivity, simplest possible choice of norm: v 2 DG := Im [A hp (v, v)] Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 12 / 19

34 Error analysis: unconditional well-posedness Trefftz-DG method Find E hp V p (T h ) such that A hp (E hp, v hp ) = l hp (v hp ) v hp V p (T h ) In order to have coercivity, simplest possible choice of norm: Explicitly, v 2 DG := Im [A hp (v, v)] 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 which is actually a norm on T (T h ) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 12 / 19

35 Error analysis: unconditional well-posedness Continuity + coercivity of A hp (, ) Well-posedness and quasi-optimality in DG norm (for any value of k and h) E E hp DG 3 inf E v hp DG + v hp V p(t h ) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 13 / 19

36 Error analysis: unconditional well-posedness Continuity + coercivity of A hp (, ) Well-posedness and quasi-optimality in DG norm (for any value of k and h) E E hp DG 3 inf E v hp DG + v hp V p(t h ) = Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 13 / 19

37 Error analysis: unconditional well-posedness Continuity + coercivity of A hp (, ) Well-posedness and quasi-optimality in DG norm (for any value of k and h) E E hp DG 3 inf E v 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 (University of Vienna) Trefftz-DG methods for Maxwell s equation 13 / 19

38 Error analysis: error estimates in a mesh-independent norm Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 14 / 19

39 Error analysis: error estimates in a mesh-independent norm Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 14 / 19

40 Error analysis: error estimates in a mesh-independent norm Error bound in DG norm duality error bound in mesh-independent norm w := E E hp = w 0 + p with w 0 H(div 0 ; Ω), p H 1 0 (Ω) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 14 / 19

41 Error analysis: error estimates in a mesh-independent norm Error bound in DG norm duality error bound in mesh-independent norm w := E E hp = w 0 + p with w 0 H(div 0 ; Ω), p H 1 0 (Ω) Estimate of w 0 in L 2 (Ω) 3 : modified duality argument [Monk & Wang, 1999] Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 14 / 19

42 Error analysis: error estimates in a mesh-independent norm Error bound in DG norm duality error bound in mesh-independent norm w := E E hp = w 0 + p with w 0 H(div 0 ; Ω), p H 1 0 (Ω) Estimate of w 0 in L 2 (Ω) 3 : modified duality argument [Monk & Wang, 1999] w 0 L 2 (Ω) 3 C 0(Ω, ω, h) w DG w T (T h ) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 14 / 19

43 Error analysis: error estimates in a mesh-independent norm Error bound in DG norm duality error bound in mesh-independent norm w := E E hp = w 0 + p with w 0 H(div 0 ; Ω), p H 1 0 (Ω) Estimate of w 0 in L 2 (Ω) 3 : modified duality argument [Monk & Wang, 1999] w 0 L 2 (Ω) 3 C 0(Ω, ω, h) w DG w T (T h ) Estimate of p: the poor regularity of p does not allow to obtain Estimate of p: an L 2 -bound estimate in a weaker norm Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 14 / 19

44 Error analysis: error estimates in a mesh-independent norm Error bound in DG norm duality error bound in mesh-independent norm w := E E hp = w 0 + p with w 0 H(div 0 ; Ω), p H 1 0 (Ω) Estimate of w 0 in L 2 (Ω) 3 : modified duality argument [Monk & Wang, 1999] w 0 L 2 (Ω) 3 C 0(Ω, ω, h) w DG w T (T h ) Estimate of p: the poor regularity of p does not allow to obtain Estimate of p: an L 2 -bound estimate in a weaker norm ( p, v) Ω p H(div;Ω) := sup C 1 (Ω, ω, h) w DG w T (T h ) v H(div;Ω) v H(div;Ω) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 14 / 19

45 Error analysis: error estimates in a mesh-independent norm For the duality argument: regularity/stability of solutions to the adjoint problem with div-free rhs star-shaped polyhedral domains constant coefficients [Hiptmair, Moiola & Perugia, M 3 AS (2011)] Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 15 / 19

46 Error analysis: error estimates in a mesh-independent norm For the duality argument: regularity/stability of solutions to the adjoint problem with div-free rhs star-shaped polyhedral domains constant coefficients [Hiptmair, Moiola & Perugia, M 3 AS (2011)] shape regular meshes, and constant flux parameters α, β, δ quasi-uniform mesh assumption Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 15 / 19

47 Error analysis: error estimates in a mesh-independent norm For the duality argument: regularity/stability of solutions to the adjoint problem with div-free rhs star-shaped polyhedral domains constant coefficients [Hiptmair, Moiola & Perugia, M 3 AS (2011)] shape regular meshes, and constant flux parameters α, β, δ quasi-uniform mesh assumption in order to allow for locally refined meshes: flux parameters depending on the local mesh size (α h max/h loc ) [Hiptmair, Moiola & Perugia, APNUM 2014] Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 15 / 19

48 Error analysis: error estimates in a mesh-independent norm For the duality argument: regularity/stability of solutions to the adjoint problem with div-free rhs star-shaped polyhedral domains constant coefficients [Hiptmair, Moiola & Perugia, M 3 AS (2011)] shape regular meshes, and constant flux parameters α, β, δ quasi-uniform mesh assumption in order to allow for locally refined meshes: flux parameters depending on the local mesh size (α h max/h loc ) [Hiptmair, Moiola & Perugia, APNUM 2014] Scattering problems? Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 15 / 19

49 Error analysis: error estimates in a mesh-independent norm Regularity/stability results in star-shaped polyhedra [Hiptmair, Moiola & Perugia, M 3 AS (2011)] The solution Φ to the Maxwell adjoint problem with rhs w 0 H(div 0 ; Ω) satisfies: Φ, Φ H 1 2 +s (Ω) 3, 0 < s < 1/2, and 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 L2 (Ω) 3 Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 16 / 19

50 Error analysis: error estimates in a mesh-independent norm Regularity/stability results in star-shaped polyhedra [Hiptmair, Moiola & Perugia, M 3 AS (2011)] The solution Φ to the Maxwell adjoint problem with rhs w 0 H(div 0 ; Ω) satisfies: Φ, Φ H 1 2 +s (Ω) 3, 0 < s < 1/2, and 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 L2 (Ω) 3 Extension to time-harmonic Maxwell of previous results proved for Helmholtz ([Melenk, 1995], [Cummings & Feng, 2006], [Hetmaniuk, 2007]) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 16 / 19

51 Error analysis: error estimates in a mesh-independent norm Final estimate w := E E hp = w 0 + p with w 0 H(div 0 ; Ω), p H 1 0 (Ω) w 0 L 2 (Ω) 3 C 0(Ω, ω, h) w DG = C 0 (Ω, ω, h) E E hp DG p H(div;Ω) C 1 (Ω, ω, h) w DG = C 1 (Ω, ω, h) E E hp DG Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 16 / 19

52 Error analysis: error estimates in a mesh-independent norm Final estimate w := E E hp = w 0 + p with w 0 H(div 0 ; Ω), p H 1 0 (Ω) w 0 L 2 (Ω) 3 C 0(Ω, ω, h) w DG = C 0 (Ω, ω, h) E E hp DG p H(div;Ω) C 1 (Ω, ω, h) w DG = C 1 (Ω, ω, h) E E hp DG E E hp H(div;Ω) (C 0 (Ω, ω, h) + C 1 (Ω, ω, h)) E E hp DG Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 16 / 19

53 Error analysis: error estimates in a mesh-independent norm Final estimate w := E E hp = w 0 + p with w 0 H(div 0 ; Ω), p H 1 0 (Ω) w 0 L 2 (Ω) 3 C 0(Ω, ω, h) w DG = C 0 (Ω, ω, h) E E hp DG p H(div;Ω) C 1 (Ω, ω, h) w DG = C 1 (Ω, ω, h) E E hp DG E E hp H(div;Ω) (C 0 (Ω, ω, h) + C 1 (Ω, ω, h)) E E hp DG + quasi-optimal estimate of E E hp DG Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 16 / 19

54 Error analysis: error estimates in a mesh-independent norm Final estimate w := E E hp = w 0 + p with w 0 H(div 0 ; Ω), p H0 1 (Ω) w 0 L 2 (Ω) 3 C 0(Ω, ω, h) w DG = C 0 (Ω, ω, h) E E hp DG p H(div;Ω) C 1 (Ω, ω, h) w DG = C 1 (Ω, ω, h) E E hp DG E E hp H(div;Ω) (C 0 (Ω, ω, h) + C 1 (Ω, ω, h)) E E hp DG + quasi-optimal estimate of E E hp DG Error estimate in H(div; Ω) -norm For solutions E H 1/2+σ (curl; Ω), σ > 0 (datum g H τ ( Ω), τ > 0), E E hp H(div;Ω) C(Ω, ω, h) inf v hp V p(t h ) E v hp DG + }{{} best approximation error Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 16 / 19

55 Error analysis: best approximation error estimates Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 17 / 19

56 Error analysis: best approximation error estimates Helmholtz: plane wave or circular/spherical wave spaces sharp best approximation estimates in weighted Sobolev norms [Moiola, Hiptmair & Perugia, ZAMP (2011)] Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 17 / 19

57 Error analysis: best approximation error estimates Helmholtz: plane wave or circular/spherical wave spaces sharp best approximation estimates in weighted Sobolev norms [Moiola, Hiptmair & Perugia, ZAMP (2011)] Vekua s theory for a 2nd order elliptic operator L harmonic functions appr V 2 {v : Lv = 0} V 1 appr harmonic polynomials V 2 V 1 generalised harmonic polynomials Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 17 / 19

58 Error analysis: best approximation error estimates Helmholtz: plane wave or circular/spherical wave spaces sharp best approximation estimates in weighted Sobolev norms [Moiola, Hiptmair & Perugia, ZAMP (2011)] Vekua s theory for a 2nd order elliptic operator L harmonic functions appr V 2 {v : Lv = 0} V 1 appr harmonic polynomials V 2 V 1 generalised harmonic polynomials appr Trefftz space V p(t h ) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 17 / 19

59 Error analysis: best approximation error estimates but... Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 18 / 19

60 Error analysis: best approximation error estimates For Maxwell: E Maxwell-Trefftz, i.e. E = ω 2 E but... Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 18 / 19

61 Error analysis: best approximation error estimates For Maxwell: but... E Maxwell-Trefftz, i.e. E = ω 2 E set H := 1 iω E E = 1 iω H Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 18 / 19

62 Error analysis: best approximation error estimates For Maxwell: but... E Maxwell-Trefftz, i.e. E = ω 2 E set H := 1 iω E E = 1 iω H H Helmholtz-Trefftz ( H = ( H) ( H) = (iω E) = ω 2 H) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 18 / 19

63 Error analysis: best approximation error estimates For Maxwell: but... E Maxwell-Trefftz, i.e. E = ω 2 E set H := 1 iω E E = 1 iω H H Helmholtz-Trefftz ( H = ( H) ( H) = (iω E) = ω 2 H) E MaxPW H j (K) = 1 iω H ( 1 iω HelPW ) H j (K) = 1 iω (H HelPW ) H j (K) H HelPW H j+1 (K) Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 18 / 19

64 Error analysis: best approximation error estimates For Maxwell: but... E Maxwell-Trefftz, i.e. E = ω 2 E set H := 1 iω E E = 1 iω H H Helmholtz-Trefftz ( H = ( H) ( H) = (iω E) = ω 2 H) E MaxPW H j (K) = 1 iω H ( 1 iω HelPW ) H j (K) = 1 iω (H HelPW ) H j (K) H HelPW H j+1 (K) then use the estimates for Helmholtz one order less than expected Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 18 / 19

65 Error analysis: best approximation error estimates For Maxwell: but... E Maxwell-Trefftz, i.e. E = ω 2 E set H := 1 iω E E = 1 iω H H Helmholtz-Trefftz ( H = ( H) ( H) = (iω E) = ω 2 H) E MaxPW H j (K) = 1 iω H ( 1 iω HelPW ) H j (K) = 1 iω (H HelPW ) H j (K) H HelPW H j+1 (K) then use the estimates for Helmholtz one order less than expected Sharp best approximation estimates of Maxwell solutions in V p (T h )? Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 18 / 19

66 Summary Extension of the analysis framework for Trefftz-DG methods developed for the Helmholtz problem to the time-harmonic Maxwell s equations unconditional well-posedness and quasi-optimality in mesh-dependent norm error analysis in a mesh-independent norm hp-convergence rates derived from best approximation error estimates of Maxwell solutions in Maxwell-Trefftz spaces Open issues sharp best approximation error estimates extension to scattering problems: stability/regularity results in non star-shaped domains. Ilaria Perugia (University of Vienna) Trefftz-DG methods for Maxwell s equation 19 / 19