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

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) Sep 25-27, 2006 19th Chemnitz FEM Symposium

Variational Problems

Maxwell Boundary Value Problem Bounded Lipschitz cavity � ⊂ R 3 with PMC walls Electric wave equation curl µ − 1 curl E − κ 2 ǫ r E in � , = i κ j 0 r µ − 1 on ∂� . curl E × n = 0 r j 0 : exciting current ǫ r : rel. dielectric constant κ : wavenumber, κ := ω √ ǫ 0 µ 0 L µ r : relative permeability Assumption: ǫ r , µ r uniformly positive, piecewise smooth Variational formulation Seek E ∈ H ( curl ; �) such that � � 0 − κ 2 (ǫ r E , v ) 0 µ − 1 curl E , curl v = i κ ( j 0 , v ) 0 ∀ v ∈ H ( curl ; �) . r � �� � =: a ( E , v ) Maxwell Boundary Value Problem 1

Coercivity V Banach space, sesqui-linear form a ( · , · ) : V × V �→ C satisfies generalized G˚ arding inequality, if | a ( u , Xu ) + � Ku , u � V ′ × V | ≥ c � u � 2 ∃ c > 0 : ∀ u ∈ V . 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 ′ : a ( u , v) = � f , v � V ′ × V ∃ 1 u ∈ V : ∀ v ∈ V . ✫ ✪ Example: Helmholtz equation − � u − κ 2 u = f with V = H 1 (�) : a ( u , v) := ( grad u , grad v) 0 − κ 2 ( u , v) 0 , u , v ∈ H 1 (�) . � ⇒ X = Id principal part compact perturbation ⇒ K Coercivity 2

Splitting Idea

Maxwell Challenge (I) Acoustic waves Electromagnetic waves Helmholtz equation Electric wave equation − �ρ − κ 2 ρ = 0 curl curl E − κ 2 E = 0 � � | 1 | 1 κ grad ρ | 2 κ curl E | 2 Potential “energy” : Magnetic “energy” : � � | ρ | 2 | E | 2 Kinetic “energy” : Electric “energy” : [Perfect symmetry of E and H !] Kinetic energy is compact perturbation Electric energy is no compact of potential energy perturbation of magnetic energy � � Strong ellipticity Lack of strong ellipticity Maxwell Challenge (I) 3

Splitting Idea Idea: Split E into predominantly electric and predominantly magnetic components. Example: L 2 -orthogonal Helmholtz decomposition of electric field: E = grad � + curl A Electric component Magnetic component ( curl -free) (divergence-free) ⇓ ⇓ No magnetic energy Magnetic energy dominates Recover (strong) ellipticity by restricting electric wave equation to components of Helmholtz decomposition Generalization: Stability sufficient, orthogonality not required Splitting Idea 4

Regular Decomposition Lemma (Girault, Raviart): ( β 2 (�) = 0 ) ✬ ✩ There is a continuous operator L : H ( div 0 ; �) �→ H 1 (�) with curl Lu = u , div Lu = 0 , � Lu � H 1 (�) ≤ C � u � L 2 (�) . ✫ ✪ Projections: , R := L ◦ curl Z := Id − R [ R 2 = R , Z 2 = Z , R ◦ Z = Z ◦ R = 0 ] Stable direct splitting: H ( curl ; �) = X (�) ⊕ N (�) X (�) := R ( H ( curl ; �)) ⊂ H 1 (�) , N (�) := Z ( H ( curl ; �)) = Ker ( curl ) . → L 2 (�) Compact embedding: X (�) ֒ Stability: � u � H 1 (�) ≤ C � curl u � L 2 (�) , ∀ u ∈ X (�) Regular Decomposition 5

Coercivity Split variational problem: Use E = E ⊥ + E 0 , v = v ⊥ − v 0 , E ⊥ , v ⊥ ∈ X (�) , E 0 , v 0 ∈ N (�) � µ r curl E ⊥ , curl v ⊥ � ∀ v ⊥ , 1 0 − κ 2 � ǫ r E ⊥ , v ⊥ � κ 2 � ǫ r E 0 , v ⊥ � � j 0 , v ⊥ � − = i κ 0 0 0 ∀ v 0 . κ 2 � ǫ r E ⊥ , v 0 � κ 2 � ǫ r E 0 , v 0 � � j 0 , v 0 � + = i κ 0 0 0 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 � u � 2 ∀ u ∈ H ( curl ; �) . H ( curl ; �) ✫ ✪ Existence & uniqueness provided that κ �= resonant frequency Coercivity 6

Galerkin Discretization

Abstract Theory (I) Setting: • V Hilbert space, a : V × V �→ C continuous sesqui-linear form • a satisfies generalized G˚ arding inequality & is injective • sequence V h ⊂ V , h ∈ H , of finite-dimensional subspaces, asymptotically dense: � u − v h � V = 0 . ∀ u ∈ V : h → 0 inf lim v h ∈ V h Goal: Asymptotic discrete inf-sup condition | a ( u h , v h ) | ∃˜ c > 0 : sup ≥ ˜ c � u h � V ∀ u h ∈ V h , ∀ h < h 0 . � v h � V v h ∈ V h Existence & quasi-optimality of Galerkin solution u h ✤ ✜ � u − u h � V ≤ � a � � u − v h � V . inf c ˜ v h ∈ V h ✣ ✢ Abstract Theory (I) 7

Abstract Theory (II) Define S : V �→ V compact : a (v, Su ) = � K v, u � V ′ × V ∀ u , v ∈ V , V -orthogonal projections . P h : V �→ V h ✤ ✜ S compact n →∞ pointwise uniformly � ⇒ ( Id − P h ) S → 0 − − − − → Id P h ✣ ✢ Yields discrete inf-sup condition for X = Id ! ! Need projector P X ➣ If X = R − Z , X ( V h ) �⊂ V h h : X (�) �→ V h satisfying ∃{ ǫ h } ⊂ R N h → 0 e h = 0 : � ( Id − P Z ∀ u h ∈ V h , ∀ h ∈ H + , lim h ) R u h � V ≤ ǫ h � u h � V | a ( u h , ( P X h X + P h S ) u h ) | = | a ( u h , Xu h ) + � Ku h , u h � V ′ × V − a (( Id − P X h ) 2 R u h , u h ) − a (( Id − P h ) Su h , u h ) | ≥ ( c − � a � ( 2 ǫ h + � ( Id − P h ) S � )) � u h � 2 V Discrete inf-sup condition for sufficiently small h Abstract Theory (II) 8

Edge Elements Local space on tetrahedron: E ( T ) := { x �→ a + b × x , a , b ∈ R 3 } 6 local degrees of freedom: “edge voltages” Local shape functions ( λ i = barycentric coord.) b i j = λ i grad λ j − λ j grad λ i Edge FE space on tetrahedral mesh M h : E h FE interpolation onto E h � h = Commuting diagram properties Nodal interpolation onto S h I h = � h ◦ grad = grad ◦ I h ➤ ( S h = p.w. linear continuous FE on M h ) Face flux interpolation onto F h J h = J h ◦ curl = curl ◦ � h ( F h = H ( div ; �) -conforming face elements) Note: � h unbounded even on H 1 (�) ! But � u − � h u � H ( curl ; �) < ∼ h ( � u � H 1 (�) + � curl u � H 1 (�) ) Edge Elements 9

Z -Projection (I) Goal: Find projection P X h : R ( E h ) �→ E h such that � � � ( Id − P X � � < ∼ h � curl u h � 0 h ) Ru h ∀ u h ∈ E h . � H ( curl ; �) P Z Surprise: h := � h is eligible ! ① ➣ If u h ∈ E h , then curl Ru h ⊂ F h p.w. constant curl Ru = curl u � 1 ② Poincar´ e mapping ( L w )( x ) = satisfies t ( w ( t x )) × x d t 0 • curl ◦ L ◦ curl = curl , • � L w � L 2 (�) ≤ diam (�) � w � L 2 (�) , • if w ≡ const on tetrahedron T , then L w ∈ E ( T ) . ③ Pick tetrahedron T ∈ M h ➢ If u ∈ ( R E h ) | T curl ( u − L ( curl u )) = 0 for some ϕ ∈ H 2 ( T ) ➢ u − L ( curl u ) = grad ϕ Z -Projection (I) 10

Z -Projection (II) ④ Use local inverse inequality & � L w � L 2 ( T ) ≤ h T � w � L 2 ( T ) | u | H 1 ( T ) + h − 1 < | ϕ | H 2 ( T ) T � L curl u � L 2 ( T ) ∼ < | u | H 1 ( T ) + � curl u � L 2 ( T ) ∼ ⑤ Use commuting diagram property � h ◦ grad = grad ◦ I h u − � h u = L ( curl u ) − � h L ( curl u ) + grad (ϕ − I h ϕ) � �� � = 0 � u − � h u � L 2 ( T ) = | ( Id − I h )ϕ | H 1 ( T ) < ∼ h T | ϕ | H 2 ( T ) < ∼ h T | u | H 1 ( T ) . ⑤ By commuting diagram property J h ◦ curl = curl ◦ � h curl ( Ru h − � h Ru h ) = ( Id − J h ) curl Ru h = 0 . Z -Projection (II) 11

Another Application: EFIE EFIE = simplest boundary integral equation for electromagnetic scattering a Ŵ ( η , µ ) = 1 η , µ ∈ H − 1 κ 2 � V κ ( div Ŵ η ), div Ŵ µ � τ − � V κ η , µ � τ , 2 ( div Ŵ , Ŵ) . Term of order 1 Term of order -1  H − 1 1  2 (Ŵ) �→ H 2 (Ŵ)  Single layer BI-Op: coercive V κ : e i κ | V x − y | � ψ �→ 4 π | x − y | ψ( y ) dS ( y )   Ŵ Tangential trace space of H ( curl ; �) Again: • Use trace induced “Hodge-type” regular deomposition of H − 1 2 ( div Ŵ , Ŵ) arding inequality for a Ŵ : H − 1 2 ( div Ŵ , Ŵ) × H − 1 ➤ Generalized G˚ 2 ( div Ŵ , Ŵ) �→ C h -projection = FE interpolation for H − 1 • P Z 2 ( div Ŵ , Ŵ) -conforming BEM R. H IPTMAIR AND C. S CHWAB , Natural boundary element methods for the electric field integral equation on polyhedra , SIAM J. Numer. Anal., 40 (2002), pp. 66–86. Another Application: EFIE 12

References ☞ Survey of numerical analysis results for edge elements: R. H IPTMAIR , Finite elements in computational electromagnetism , Acta Nu- merica, (2002), pp. 237–339. ☞ Coecivity arguments for edge boundary elements: A. B UFFA AND R. H IPTMAIR , 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. B UFFA , Remarks on the discretization of some non-positive operators with application to heterogeneous Maxwell problems , SIAM J. Numer. Anal., 43 (2005), pp. 1–18. References 13