Tretz-Discontinuous Galerkin Methods for Time-Harmonic Maxwells

ESCO 2012, PLZEŇ — 25–29TH JUNE 2012
Trefftz-DG methods
for time-harmonic
Maxwell’s equations
Andrea Moiola
DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF READING
Joint work with R. Hiptmair (ETH Zürich) & I. Perugia (Università di Pavia)

Time-harmonic PDEs
Propagation, scattering and interaction of time-harmonic
acoustic and electromagnetic waves are governed by:
the Helmholtz equation
−∆u −ω 2 u = 0
and the Maxwell equations
in Ω ⊂ R N , N = 2, 3,
∇×(∇×E)−ω 2 E = 0 in Ω ⊂ R 3 .
ω > 0 is the wavenumber.
We supplement them with impedance (Robin) boundary
conditions in bounded polygonal/polyhedral domains.

Oscillations
Time-harmonic solutions are inherently oscillatory: a lot of DOFs
needed for any polynomial discretization!
[Helmholtz BVP, picture by T. Betcke]
Wavenumber ω = 2π/λ is the crucial parameter.

Pollution effect
Big issue in FEM solution for high wavenumbers: pollution effect
��
��
��
��
��Galerkin
error��
��
��
��
��
≥ C ω
��best
approximation error��
a
a > 0, ω → ∞.
It affects every (low order) method in h: [BABUŠKA, SAUTER 2000].
⇓
Oscillating solutions + pollution effect
= standard FEM are too expensive at high frequencies!
Special schemes required, p-version preferred (hp even better!).

Trefftz methods
How to deal with these phenomena?
Trefftz methods incorporate the wavelength in the FEM space:
trial and test functions are solutions of Helmholtz/Maxwell’s
equations in each mesh element, e.g.:
Vp ⊂ T(Th) = � w ∈ L 2 (Ω) 3 : ∃s > 0 s.t. w |K ∈ H 1/2+s (K),
∇×(∇×w |K)−ω 2 �
w |K = 0 in each K ∈ Th .
Hope: reduce number of DOFs needed.
For instance:
vector plane waves x ↦→ ae iωx·d , d ∈ S 2 , a·d = 0;
spherical waves (vector Fourier–Bessel functions);
singular solutions;
. . .

Trefftz methods
How to deal with these phenomena?
Trefftz methods incorporate the wavelength in the FEM space:
trial and test functions are solutions of Helmholtz/Maxwell’s
equations in each mesh element, e.g.:
Vp ⊂ T(Th) = � w ∈ L 2 (Ω) 3 : ∃s > 0 s.t. w |K ∈ H 1/2+s (K),
∇×(∇×w |K)−ω 2 �
w |K = 0 in each K ∈ Th .
Hope: reduce number of DOFs needed.
For instance:
vector plane waves x ↦→ ae iωx·d , d ∈ S 2 , a·d = 0;
spherical waves (vector Fourier–Bessel functions);
singular solutions;
. . .

Wave-based methods
How to “match” traces across interelement boundaries?
Many techniques are available for Helmholtz:
Least squares schemes: Method of fundamental solutions
(MFS);
Lagrange multipliers: Discontinuous enrichment method
(DEM);
Partition of unity method (PUM/PUFEM), non-Trefftz;
Discontinuous Galerkin (DG): Ultraweak variational
formulation (UWVF).
For Maxwell: UWVF by Cessenat and Després 1996-2003,
For Maxwell: UWVF by Huttunen, Malinen and Monk 2007, . . .
Here: extension to Maxwell of a family of Trefftz-discontinuous
Galerkin (TDG) methods that includes the UWVF.
Main focus: p-version. Ideas from simpler model: Helmholtz.

Model problem
Homogeneous time-harmonic Maxwell equations with
impedance boundary conditions:
�
∇×(µ −1 ∇×E)−ω 2 ǫ E = 0 in Ω,
µ −1 (∇×E)×n−iωϑ(n×E)×n = g on ∂Ω,
Ω ⊂ R 3 , Lipschitz (star-shaped) bounded polyhedral
domain,
g ∈ L 2 T (∂Ω),
ω > 0 wavenumber,
ǫ,µ > 0 (piecewise) constant (here: ǫ ≡ µ ≡ 1),
0 �= ϑ ∈ R.

TDG: derivation — I
1 Introduce a mesh Th on Ω;
2 multiply the Maxwell equations with a test field ξ and
integrate by parts on a single element K ∈ Th:
�
�
� � 2
(∇×E)·(∇×ξ)−ω E·ξ dV + n×(∇×E)·ξ dS = 0;
3
�
K
K
∂K
integrate by parts again: ultraweak step
E· � ∇×∇×ξ−ω 2 ξ � �
� �
dV+ (n×E)·(∇×ξ)+n×(∇×E)·ξ dS = 0;
∂K

TDG: derivation — I
1 Introduce a mesh Th on Ω;
2 multiply the Maxwell equations with a test field ξ and
integrate by parts on a single element K ∈ Th:
�
�
� � 2
(∇×E)·(∇×ξ)−ω E·ξ dV + n×(∇×E)·ξ dS = 0;
3
�
K
K
∂K
integrate by parts again: ultraweak step
E· � ∇×∇×ξ−ω 2 ξ � �
� �
dV+ (n×E)·(∇×ξ)+n×(∇×E)·ξ dS = 0;
∂K

TDG: derivation — I
1 Introduce a mesh Th on Ω;
2 multiply the Maxwell equations with a test field ξ and
integrate by parts on a single element K ∈ Th:
�
�
� � 2
(∇×E)·(∇×ξ)−ω E·ξ dV + n×(∇×E)·ξ dS = 0;
3
�
K
K
∂K
integrate by parts again: ultraweak step
E· � ∇×∇×ξ−ω 2 ξ � �
� �
dV+ (n×E)·(∇×ξ)+n×(∇×E)·ξ dS = 0;
∂K

TDG: derivation — II
4 choose a discrete Trefftz space Vp(K) and replace traces
on ∂K with numerical fluxes �Ep and �Hp:
E → Ep (discrete solution) in K ,
E → �Ep ,
∇×E
iω → �Hp
on ∂K ;
5 use the Trefftz property: ∀ ξ p ∈ Vp(K)
�
K
Ep· � ∇×∇×ξ p −ω 2 �
ξp dV
� �� �
=0
�
� �
+ (n× �Ep)·(∇×ξ p)+iω(n× �Hp)·ξ p dS = 0.
∂K
� �� �
TDG eq. on 1 element
Two things to set:
discrete space Vp and numerical fluxes � Ep, � Hp.

TDG: derivation — II
4 choose a discrete Trefftz space Vp(K) and replace traces
on ∂K with numerical fluxes �Ep and �Hp:
E → Ep (discrete solution) in K ,
E → �Ep ,
∇×E
iω → �Hp
on ∂K ;
5 use the Trefftz property: ∀ ξ p ∈ Vp(K)
�
K
Ep· � ∇×∇×ξ p −ω 2 �
ξp dV
� �� �
=0
�
� �
+ (n× �Ep)·(∇×ξ p)+iω(n× �Hp)·ξ p dS = 0.
∂K
� �� �
TDG eq. on 1 element
Two things to set:
discrete space Vp and numerical fluxes � Ep, � Hp.

TDG: the space Vp
The abstract error analysis works for every discrete Trefftz space!
Possible choice: plane waves space (aℓ, dℓ ∈ S 2 , aℓ · dℓ = 0)
Vp(Th) =
�
v ∈ L 2 (Ω) 3 : v|K(x) =
p�
ℓ=1
α K 1,ℓaℓe iωd ℓ·x +α K 2,ℓ(aℓ × dℓ)e iωd ℓ·x ,
α K ν,ℓ ∈ C, ∀K ∈ Th
2p = number of basis plane waves (DOFs) in each element.
�
.

Numerical fluxes
We choose the numerical fluxes, on interior faces:
�Ep : = {Ep}− β
iω [∇×Ep]T ,
�Hp : = 1
iω {∇×Ep}+α [Ep]T ,
and on ∂Ω:
�Ep : = Ep −δϑ −1
�
n×(∇×Ep)
iω
�Hp : = 1
iω ∇×Ep
�
∇×Ep
−(1−δ)
iω
+ϑ(n×Ep)×n+ g
�
iω
−ϑ(n×Ep)− n×g
iω
{·} = averages, [·]T = tangential jumps on the interfaces,
α, β > 0, δ ∈ (0, 1
2 ] parameters at our disposal (in L∞ (Fh)),
independent of h, p, ω (other choices possible).
,
�
,

Variational formulation of the TDG
With this fluxes, summing over the elements K ∈ Th, the TDG
method reads: find Ep ∈ Vp(Th) s.t.
�
Ah(Ep,ξ p ) =
δ
iωϑ (n×g)·(∇×ξ �
p ) dS+ (1−δ)(n×g)·(n ×ξ p ) dS
∂Ω
∀ ξp ∈ Vp(Th), where (FI �
h = interior skeleton)
Ah(Ep,ξ p ) = −{Ep}· [∇×ξ p ]T − {∇×Ep}· [ξp ]T dS
F I h
�
+ (1−δ)(n×Ep)·(∇×ξ)−δ(∇×Ep)·(n×ξ) dS
∂Ω
�
− i
F I h
ω −1 β [∇×Ep]T · [∇×ξ p ]T +ωα [Ep]T · [ξ p ]T dS
�
δ
− i
∂Ω ωϑ n×(∇×Ep)·n×(∇×ξ p)+ω(1−δ)ϑ(n×Ep)·(n×ξ p) dS.
Ep ↦→ (−Im Ah(Ep,Ep)) 1
2 is a norm on the Trefftz space ⇒ ∃!Ep.
∂Ω

Variational formulation of the TDG
With this fluxes, summing over the elements K ∈ Th, the TDG
method reads: find Ep ∈ Vp(Th) s.t.
�
Ah(Ep,ξ p ) =
δ
iωϑ (n×g)·(∇×ξ �
p ) dS+ (1−δ)(n×g)·(n ×ξ p ) dS
∂Ω
∀ ξp ∈ Vp(Th), where (FI �
h = interior skeleton)
Ah(Ep,ξ p ) = −{Ep}· [∇×ξ p ]T − {∇×Ep}· [ξp ]T dS
F I h
�
+ (1−δ)(n×Ep)·(∇×ξ)−δ(∇×Ep)·(n×ξ) dS
∂Ω
�
− i
F I h
ω −1 β [∇×Ep]T · [∇×ξ p ]T +ωα [Ep]T · [ξ p ]T dS
�
δ
− i
∂Ω ωϑ n×(∇×Ep)·n×(∇×ξ p)+ω(1−δ)ϑ(n×Ep)·(n×ξ p) dS.
Ep ↦→ (−Im Ah(Ep,Ep)) 1
2 is a norm on the Trefftz space ⇒ ∃!Ep.
∂Ω

TDG norms
We define two DG norms on the Trefftz space T(Th):
|||w||| 2 F h := ω −1
�
�
�β 1/2 �
�
[∇×w]T
+ω −1
�
�
�(δ/ϑ) 1/2 �
�
n×(∇×w)
|||w||| 2
F +
h
+ω −1
∀ w, v ∈ T(Th):
:= |||w||| 2 F h +ω
�
�
�α −1/2 {(∇ × w)T }
− Im Ah(w,w) = |||w||| 2 Fh
|Ah(v,w)| ≤ 2|||v||| F +
h
� 2
0,FI h
� 2
0,∂Ω
�
�
�β −1/2 �
�
{wT }
�
�
� 2
0,FI h
|||w|||Fh
�
�
+ω �α 1/2 �
�
[w]T
� 2
0,F I h
�
�
+ω �(1−δ) 1/2 ϑ 1/2 �
�
(n×w)
� 2
0,FI h
�
�
+ω
�
�
�
�δ −1/2 ϑ 1/2 (n×w) � 2
0,∂Ω
⇒
.
� 2
0,∂Ω
Existence & uniqueness Ep,
quasi-optimality
in Fh −F +
h norms.
,

Quasi-optimality in mesh-independent norms
We have quasi-optimality / error estimates in DG norm:
|||E−Ep|||Fh
≤ 3 inf
ξp∈Vp(Th) |||E−ξ + p||| F ,
h
we want them in a more explicit mesh-independent norm
(e.g., L 2 (Ω)).
Tools:
Helmholtz decomposition;
special duality argument from MONK, WANG 1999;
new wavenumeber-explicit stability bounds for the
BVP (using Rellich identities for Maxwell);
new regularity estimates (in H 1/2+ǫ (curl;Ω)).
We obtain (for some ǫ > 0):
�
(E−Ep)·v dV Ω �E−Ep� H(div,Ω) ′ := sup �v�H(div;Ω) v∈H(div;Ω)
≤ C � h −1/2 1 3 −
(ω 2 −
+ω 2)+h ǫ (ω 1 3
2 −
+ω 2) � |||E−Ep|||Fh .

Quasi-optimality in mesh-independent norms
We have quasi-optimality / error estimates in DG norm:
|||E−Ep|||Fh
≤ 3 inf
ξp∈Vp(Th) |||E−ξ + p||| F ,
h
we want them in a more explicit mesh-independent norm
(e.g., L 2 (Ω)).
Tools:
Helmholtz decomposition;
special duality argument from MONK, WANG 1999;
new wavenumeber-explicit stability bounds for the
BVP (using Rellich identities for Maxwell);
new regularity estimates (in H 1/2+ǫ (curl;Ω)).
We obtain (for some ǫ > 0):
�
(E−Ep)·v dV Ω �E−Ep� H(div,Ω) ′ := sup �v�H(div;Ω) v∈H(div;Ω)
≤ C � h −1/2 1 3 −
(ω 2 −
+ω 2)+h ǫ (ω 1 3
2 −
+ω 2) � |||E−Ep|||Fh .

Quasi-optimality in mesh-independent norms
We have quasi-optimality / error estimates in DG norm:
|||E−Ep|||Fh
≤ 3 inf
ξp∈Vp(Th) |||E−ξ + p||| F ,
h
we want them in a more explicit mesh-independent norm
(e.g., L 2 (Ω)).
Tools:
Helmholtz decomposition;
special duality argument from MONK, WANG 1999;
new wavenumeber-explicit stability bounds for the
BVP (using Rellich identities for Maxwell);
new regularity estimates (in H 1/2+ǫ (curl;Ω)).
We obtain (for some ǫ > 0):
�
(E−Ep)·v dV Ω �E−Ep� H(div,Ω) ′ := sup �v�H(div;Ω) v∈H(div;Ω)
≤ C � h −1/2 1 3 −
(ω 2 −
+ω 2)+h ǫ (ω 1 3
2 −
+ω 2) � |||E−Ep|||Fh .

Orders of convergence
Last missing ingredient: best approximation estimates for vector
plane/spherical waves.
�
aℓeiωx·dℓ , aℓ × dℓeiωx·dℓ �
ℓ=1,...,(q+1) 2
|aℓ| = |dℓ| = 1, dℓ · aℓ = 0.
We show them by proving analogous results for the Helmholtz
equation; tools: Vekua’s theory, harmonic polynomial
approximation, spherical waves, “inversion” of Jacobi–Anger
expansion. . .
Ak x dk
Algebraic orders in h and q, using 2(q+1) 2 spherical/plane
waves per element:
|||E−Ep|||Fh ≤ C (ωh)ω −5/2
�
h
qλ �k−3/2 �∇×E� k+1,ω,Ω ,
�E−Ep� H(div;Ω) ′ ≤ C (ωh)(ω −5/2 +ω −4 ) hk−2
q λ(k−3/2) �∇×E� k+1,ω,Ω .
Ak
dk

Orders of convergence
Last missing ingredient: best approximation estimates for vector
plane/spherical waves.
�
aℓeiωx·dℓ , aℓ × dℓeiωx·dℓ �
ℓ=1,...,(q+1) 2
|aℓ| = |dℓ| = 1, dℓ · aℓ = 0.
We show them by proving analogous results for the Helmholtz
equation; tools: Vekua’s theory, harmonic polynomial
approximation, spherical waves, “inversion” of Jacobi–Anger
expansion. . .
Ak x dk
Algebraic orders in h and q, using 2(q+1) 2 spherical/plane
waves per element:
|||E−Ep|||Fh ≤ C (ωh)ω −5/2
�
h
qλ �k−3/2 �∇×E� k+1,ω,Ω ,
�E−Ep� H(div;Ω) ′ ≤ C (ωh)(ω −5/2 +ω −4 ) hk−2
q λ(k−3/2) �∇×E� k+1,ω,Ω .
Ak
dk

Summary and open problems
TDG method for Maxwell:
formulation, well-posedness,
h- and p-convergence in energy and H(div;Ω) ′ norms,
approximation estimates for plane and spherical waves.
Not discussed: strong ill-conditioning for (moderately) high p.
Open problems:
hp-version,
non-constant coefficients ǫ(x), µ(x)
and non-homogeneous eqs.,
adaptivity on PW directions, ongoing work
with T. Betcke and J. Phillips (UCL),
non star-shaped domains,
better approximation estimates,
other PDEs,
. . .

THANK YOU!