Edge Elements and Coercivity Ralf Hiptmair Seminar f ur Angewandte - - PowerPoint PPT Presentation

Sep 25-27, 2006 19th Chemnitz FEM Symposium

Edge Elements and Coercivity

Ralf Hiptmair Seminar f¨ ur Angewandte Mathematik ETH Z¨ urich

(e–mail: hiptmair@sam.math.ethz.ch) (Homepage: http://www.sam.math.ethz.ch/˜hiptmair)

Variational Problems

Maxwell Boundary Value Problem 1

Maxwell Boundary Value Problem

Bounded Lipschitz cavity ⊂ R3 with PMC walls Electric wave equation curl µ−1

r

curl E − κ2ǫrE = iκj0 in , µ−1

r

curl E × n =

• n ∂ .

j0 : exciting current κ : wavenumber, κ := ω√ǫ0µ0L ǫr : rel. dielectric constant µr : relative permeability Assumption: ǫr, µr uniformly positive, piecewise smooth Variational formulation Seek E ∈ H(curl; ) such that

• µ−1

r

curl E, curl v

• 0 − κ2 (ǫrE, v)0
• =:a(E,v)

= iκ (j0, v)0 ∀v ∈ H(curl; ) .

Coercivity 2

Coercivity

V Banach space, sesqui-linear form a(·, ·) : V × V → C satisfies generalized G˚ arding inequality, if ∃c > 0 : |a(u, Xu) + Ku, uV ′×V | ≥ c u2

V

∀u ∈ V . for some isomorphism X : V → V , compact K : V → V ′.

✬ ✫ ✩ ✪

plus a injective: a(u, v) = 0 ∀v ∈ V ⇒ u = 0

Fredholm alternative →

⇓ ∀ f ∈ V ′ : ∃1u ∈ V : a(u, v) = f, vV ′×V ∀v ∈ V . Example: Helmholtz equation −u − κ2u = f with V = H1(): a(u, v) := (grad u, grad v)0 − κ2 (u, v)0 , u, v ∈ H1() . principal part compact perturbation ⇒ K

• ⇒

X = Id

Splitting Idea

Maxwell Challenge (I) 3

Maxwell Challenge (I)

Acoustic waves Helmholtz equation −ρ − κ2ρ = 0 Potential “energy” :

• |1

κ grad ρ|2

Kinetic “energy” :

• |ρ|2

Electromagnetic waves Electric wave equation curl curl E − κ2E = 0 Magnetic “energy” :

• |1

κ curl E|2

Electric “energy” :

• |E|2

[Perfect symmetry of E and H!] Kinetic energy is compact perturbation

• f potential energy
• Strong ellipticity

Electric energy is no compact perturbation of magnetic energy

• Lack of strong ellipticity

Splitting Idea 4

Splitting Idea

Idea: Split E into predominantly electric and predominantly magnetic components. Example: L2-orthogonal Helmholtz decomposition of electric field: E = grad Electric component (curl-free) ⇓ No magnetic energy + curl A Magnetic component (divergence-free) ⇓ Magnetic energy dominates Recover (strong) ellipticity by restricting electric wave equation to components

• f Helmholtz decomposition

Generalization: Stability sufficient, orthogonality not required

Regular Decomposition 5

Regular Decomposition

Lemma (Girault, Raviart): (β2() = 0)

✬ ✫ ✩ ✪

There is a continuous operator L : H(div 0; ) → H1() with curl Lu = u , div Lu = 0 , LuH1() ≤ C uL2() . Projections: R := L ◦ curl , Z := Id − R [R2 = R, Z2 = Z, R ◦ Z = Z ◦ R = 0] Stable direct splitting: H(curl; ) = X() ⊕ N () X() := R(H(curl; )) ⊂ H1() , N () := Z(H(curl; )) = Ker(curl) . Compact embedding: X() ֒ → L2() Stability: uH1() ≤ C curl uL2(), ∀u ∈ X()

Coercivity 6

Coercivity

Split variational problem: Use E = E⊥ + E0, v = v⊥−v0, E⊥, v⊥ ∈ X(), E0, v0 ∈ N ()

• 1

µr curl E⊥, curl v⊥ 0 − κ2

ǫrE⊥, v⊥ − κ2 ǫrE0, v⊥ = iκ

• j0, v⊥

∀v⊥ , κ2 ǫrE⊥, v0 + κ2 ǫrE0, v0 = iκ

• j0, v0

∀v0 .

Note: red terms are compact ! Introduce “sign flipping isomorphism”: X := R − Z : H(curl; ) → H(curl; ) Generalized G˚ arding inequality:

✬ ✫ ✩ ✪

∃ compact sesqui-linear form k on H(curl; ) and c > 0 |a(u, Xu) + k(u, u)| ≥ c u2

H(curl;)

∀u ∈ H(curl; ) . Existence & uniqueness provided that κ = resonant frequency

Galerkin Discretization

Abstract Theory (I) 7

Abstract Theory (I)

Setting: • V Hilbert space, a : V × V → C continuous sesqui-linear form

• a satisfies generalized G˚

arding inequality & is injective

• sequence Vh ⊂ V , h ∈ H, of finite-dimensional subspaces,

asymptotically dense: ∀u ∈ V : lim

h→0 inf vh∈Vh

u − vhV = 0 . Goal: Asymptotic discrete inf-sup condition ∃˜ c > 0 : sup

vh∈Vh

|a(uh, vh)| vhV ≥ ˜ c uhV ∀uh ∈ Vh, ∀h < h0 . Existence & quasi-optimality of Galerkin solution uh

✤ ✣ ✜ ✢

u − uhV ≤ a ˜ c inf

vh∈Vh

u − vhV .

Abstract Theory (II) 8

Abstract Theory (II)

Define S : V → V compact : a(v, Su) = Kv, uV ′×V ∀u, v ∈ V , Ph : V → Vh V -orthogonal projections .

✤ ✣ ✜ ✢

Ph

n→∞

− − − − → Id pointwise

S compact

• ⇒

(Id − Ph)S → 0 uniformly Yields discrete inf-sup condition for X = Id !

!

If X = R − Z , X(Vh) ⊂ Vh ➣ Need projector P X

h : X() → Vh satisfying

∃{ǫh} ⊂ RN

+, lim h→0 eh = 0: (Id − P Z h )RuhV ≤ ǫh uhV

∀uh ∈ Vh, ∀h ∈ H |a(uh, (P X

h X + PhS)uh)| =

|a(uh, Xuh) + Kuh, uhV ′×V − a((Id − P X

h )2Ruh, uh) − a((Id − Ph)Suh, uh)|

≥ (c − a(2ǫh + (Id − Ph)S)) uh2

V

Discrete inf-sup condition for sufficiently small h

Edge Elements 9

Edge Elements

Local space on tetrahedron: E(T) := {x → a + b × x, a, b ∈ R3} 6 local degrees of freedom: “edge voltages” Local shape functions (λi = barycentric coord.) bi j = λi grad λ j − λ j grad λi Edge FE space on tetrahedral mesh Mh: Eh h = FE interpolation onto Eh Ih = Nodal interpolation onto Sh

(Sh = p.w. linear continuous FE on Mh)

Jh = Face flux interpolation onto Fh

(Fh = H(div; )-conforming face elements)

➤ Commuting diagram properties h ◦ grad = grad ◦Ih Jh ◦ curl = curl ◦ h Note: h unbounded even on H1() ! But u − huH(curl;) < ∼ h(uH1() + curl uH1())

Z-Projection (I) 10

Z-Projection (I)

Goal: Find projection P X

h : R(Eh) → Eh such that

• (Id − P X

h )Ruh

• H(curl;)

< ∼ h curl uh0 ∀uh ∈ Eh . Surprise: P Z

h := h is eligible !

① curl Ru = curl u ➣ If uh ∈ Eh, then curl Ruh ⊂ Fh p.w. constant ② Poincar´ e mapping (Lw)(x) = 1 t (w(tx)) × x dt satisfies

• curl ◦L ◦ curl = curl,
• LwL2() ≤ diam() wL2(),
• if w ≡ const on tetrahedron T , then Lw ∈ E(T).

③ Pick tetrahedron T ∈ Mh If u ∈ (REh)|T ➢ curl(u − L(curl u)) = 0 ➢ u − L(curl u) = grad ϕ for some ϕ ∈ H2(T )

Z-Projection (II) 11

Z-Projection (II)

④ Use local inverse inequality & LwL2(T) ≤ hT wL2(T) |ϕ|H2(T ) < ∼ |u|H1(T ) + h−1

T L curl uL2(T)

< ∼ |u|H1(T ) + curl uL2(T) ⑤ Use commuting diagram property h ◦ grad = grad ◦Ih u − hu = L(curl u) − hL(curl u)

• =0

+ grad(ϕ − Ihϕ) u − huL2(T ) = |(Id − Ih)ϕ|H1(T ) < ∼ hT |ϕ|H2(T ) < ∼ hT |u|H1(T ) . ⑤ By commuting diagram property Jh ◦ curl = curl ◦ h curl(Ruh − hRuh) = (Id − Jh) curl Ruh = 0 .

Another Application: EFIE 12

Another Application: EFIE

EFIE = simplest boundary integral equation for electromagnetic scattering aŴ(η, µ) = 1 κ2 Vκ(divŴη), divŴµτ − Vκη, µτ , η, µ ∈ H−1

2(divŴ, Ŵ) .

Term of order 1 Term of order -1 Single layer BI-Op: Vκ :      H−1

2(Ŵ)

→ H

1 2(Ŵ)

ψ →

• Ŵ

eiκ|V x−y| 4π|x−y| ψ(y) dS(y)

coercive Tangential trace space of H(curl; ) Again:

• Use trace induced “Hodge-type” regular deomposition of H−1

2(divŴ, Ŵ)

➤ Generalized G˚ arding inequality for aŴ : H−1

2(divŴ, Ŵ) × H−1 2(divŴ, Ŵ) → C

• P Z

h -projection = FE interpolation for H−1

2(divŴ, Ŵ)-conforming BEM

• R. HIPTMAIR AND C. SCHWAB, Natural boundary element methods for the electric field integral equation on polyhedra,

SIAM J. Numer. Anal., 40 (2002), pp. 66–86.

References 13

References

☞ Survey of numerical analysis results for edge elements:

• R. HIPTMAIR, Finite elements in computational electromagnetism, Acta Nu-

merica, (2002), pp. 237–339. ☞ Coecivity arguments for edge boundary elements:

• A. BUFFA AND R. HIPTMAIR, Galerkin boundary element methods for electro-

magnetic scattering, in Topics in Computational Wave Propagation. Direct and inverse Problems, M. Ainsworth, P . Davis, D. Duncan, P . Martin, and B. Rynne, eds., vol. 31 of Lecture Notes in Computational Science and Engineering, Springer, Berlin, 2003, pp. 83–124. ☞ Presentation of abstract framework:

• A. BUFFA, Remarks on the discretization of some non-positive operators with

application to heterogeneous Maxwell problems, SIAM J. Numer. Anal., 43 (2005), pp. 1–18.