Encounters with Maxwell Equations Martin Costabel IRMAR, Université de Rennes 1 Analysis and Numerics of Acoustic and Electromagnetic Problems Linz, 17–22 October 2016 Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 1 / 38
Stories of Encounters with Maxwell equations Four stories were planned: Strong Ellipticity: Beginning 1982 1 Maxwell Corner Singularities: Beginning 1997 2 Approximation of Eigenvalue Problems: Beginning 2002 3 Volume Integral Equations: Beginning 2005 4 1 and Today: Only Stories 3 Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 2 / 38
Story 1: Strong ellipticity Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 3 / 38
The Beginning R. C. MacCamy and E. Stephan. A boundary element method for an exterior problem for three-dimensional Maxwell’s equations. Applicable Anal. , 16(2):141–163, 1983. R. C. MacCamy and E. Stephan. Solution procedures for three-dimensional eddy current problems. J. Math. Anal. Appl. , 101(2):348–379, 1984. M. Costabel and E. P . Stephan. Strongly elliptic boundary integral equations for electromagnetic transmission problems. Proc. Roy. Soc. Edinburgh Sect. A , 109(3-4):271–296, 1988. M. Costabel. A coercive bilinear form for Maxwell’s equations. J. Math. Anal. Appl. , 157(2):527–541, 1991. Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 4 / 38
Oberwolfach 1988 Richard C. MacCamy 26/09/1925 – 6/07/2011 Suri, Hsiao, Stephan, Costabel, Wendland, MacCamy Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 5 / 38
Context: Numerical approximation of pseudodifferential equations Galerkin approximation u ∈ X : a ( u , v ) = � f , v � ∀ v ∈ X (Var) (Var h ) u h ∈ X h : a ( u h , v ) = � f , v � ∀ v ∈ X h Crucial Property: Strong ellipticity Strong ellipticity = ⇒ Gårding inequality, “ a pos. def. + compact” Result: Every Galerkin method is stable and convergent. Class of interest: Pseudodifferential operators Pseudodifferential operator A of order α with positive principal symbol ⇒ Strong ellipticity in X = H α/ 2 (Ω) , = a ( u , v ) = � Au , v � . Activity: Find relevant examples. Found: Acoustics, elasticity, fluid dynamics,... What about electromagnetics? Need to generalize strong ellipticity. Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 6 / 38
BIE for time-harmonic Maxwell equations The EFIE � e i κ | x − y | 1 V τ j + κ 2 ∇ τ V div τ j = E 0 V ψ ( x ) = 4 π | x − y | ψ ( y ) ds ( y ) , κ > 0 ∂ Ω Bilinear form ( u , v tangential vector fields) 1 a ( u , v ) = � Vu , v � − κ 2 � V div τ u , div τ v � • V is strongly elliptic (of order α = − 1), but a is indefinite! • Principal part (order α = + 1) not elliptic! Idea (“T-coercivity”): Replace a ( u , v ) by a ( u , T v ) with some T : X → X and prove that a ( · , T · ) is pos. def + compact. Two concretizations Keep a ( · , · ) in the computations, use T only in the theory. This works if one can find 1 T h : X h → X h that is “close” to T . Works for EFIE if T is defined by Hodge decomposition and X h are specially crafted finite elements (edge elements, discrete differential forms). Also: X = H − 1 / 2 ( div τ , ∂ Ω) . ( ∗ ) Change the equation: Replace A by T ∗ A . Then any Galerkin method will still do. 2 ( ∗ ) This would be another story, in which I only participated at the beginning. Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 7 / 38
The beginning of another story, later A. Buffa, M. Costabel, and D. Sheen. On traces for H ( curl , Ω) in Lipschitz domains. J. Math. Anal. Appl. , 276(2):845–867, 2002. A. Buffa, M. Costabel, and C. Schwab. Boundary element methods for Maxwell’s equations on non-smooth domains. Numer. Math. , 92(4):679–710, 2002. M. Costabel and C. Safa. A boundary integral formulation of antenna problems suitable for nodal-based wavelet approximations. In Numerical mathematics and advanced applications , pages 265–271. Springer Italia, Milan, 2003. Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 8 / 38
Back to the MacCamy-Stephan boundary integral equation In EFIE, introduce a scalar potential m as additional unknown, and add a second equation stating that div E = 0 on the boundary. Use (∆ + κ 2 ) Vm = 0 in Ω . � V τ j + ∇ τ Vm = E 0 V div τ j − κ 2 Vm = 0 Problem: This is an elliptic (in the sense of ADN) system of pseudodifferential operators on H 1 / 2 × H − 1 / 2 , but it is not strongly elliptic. 1 0 i ξ 1 1 0 1 i ξ 2 Principal symbol | ξ | − i ξ 1 − i ξ 2 0 Idea: Modify the system by “Gauss elimination”: Subtract div τ times the first equation from the second. If ∂ Ω is smooth, the commutator J = [ div τ , V τ ] is of lower order − 1. The system becomes � V τ j + ∇ τ Vm = E 0 J j + (∆ τ + κ 2 ) Vm = − div τ E 0 1 0 i ξ 1 1 triangular and pos. def. of orders ( − 1 , − 1 , + 1 ) . Principal symbol 0 1 i ξ 2 | ξ | | ξ | 2 0 0 This works (for smooth boundaries): Any Galerkin method is stable. No special FEM needed. Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 9 / 38
Interpretation, generalization to transmission problem Trying to understand: The “Dirichlet data” ( E τ , div E ) were changed into ( E τ , div E − div τ E τ ) . This mapping T Dir defines an isomorphism of H − 1 / 2 ( div ) × H − 1 / 2 to itself and also of H 1 / 2 × H − 1 / 2 to itself. τ Strong ellipticity was obtained in the latter space. There is a corresponding mapping for the “Neumann” data T Neu : ( E n , n × curl E ) �→ ( E n , n × curl E + ∇ τ E n ) Consider now the Maxwell transmission problem in regularized form: (∆ + k 2 ) u = 0 in Ω ∪ Ω c µ 2 n × u 0 , [ λ div u ] = λ 2 div u 0 [ u τ ] = u 0 τ , [ ε u n ] = ε 2 u 0 n , [ 1 1 µ n × curl u ] = on ∂ Ω Here κ, ε, µ, λ are piecewise constant, taking values κ 1 in Ω and κ 2 in Ω c . The constants κ, ε, µ are physical, λ is suitably chosen so as to enforce div u = 0 if the incident field u 0 is divergence free. We introduce Cauchy data corresponding to these jump conditions ( w , φ, ψ , v ) = ( u τ , λ div u , − 1 µ n × curl u , ε u n ) and finally the modification operator T Cauchy : ( w , φ, ψ , v ) �→ ( w , ηφ − div τ w , ψ + θ ∇ τ v , θ v ) where η, θ ∈ C are yet to be chosen. Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 10 / 38
The regularized Maxwell transmission problem Theorem [Costabel-Stephan 1988] Consider the “direct” boundary integral equation method for the Maxwell transmission problem, where the system of boundary integral operators A is the difference between the exterior and interior Calderón projectors for the vector Helmholtz equations, defined using the modified Cauchy data as above. If Ω is smooth, then under rather general natural conditions on κ, ε, µ , the parameters λ, η, θ can be chosen such that A is a strongly elliptic system of pseudodifferential operators defined on the boundary function space × H − 1 / 2 × H − 1 / 2 1 / 2 × H 1 / 2 . H τ τ Proof by explicit calculation of the principal symbol. Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 11 / 38
Replacing calculations of symbols by Green’s formula Lemma [Co 1991] Let Ω ⊂ R 3 be a bounded smooth domain and u , v ∈ C 2 (Ω) . Then � � � � curl u · curl v + div u div v + c ( u , v ) = ∇ u · ∇ v + b ( u , v ) Ω Ω � � � where c ( u , v ) = ∇ τ u n · v τ − div τ u τ v n ∂ Ω � � � b ( u , v ) = ( u τ · ∇ n ) · v τ ) + div n u n v n ∂ Ω On the left hand side, we recognize the modification of the Cauchy data: n × curl u �→ n × curl u + ∇ τ u n and div u �→ div u − div τ u τ On the right hand side, we find the curvature of the boundary, which for a C 2 boundary consists of bounded functions. Corollary For a bounded C 2 domain Ω , the bilinear form on the left hand side is strongly elliptic on H 1 (Ω) . This allows to prove the strong ellipticity of the boundary integral system studied in [Costabel-Stephan 1988] by integration by parts, without computing symbols of pseudodifferential operators. Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 12 / 38
What about non-smooth domains? In the paper [Co 1991], there was a remark in the last paragraph that for a polyhedron and functions that are in X = H ( curl ) ∩ H ( div ) and have either vanishing tangential component ( X N ) or vanishing normal component ( X T ) on the boundary, this integration-by-parts formula simplifies considerably, if one knows that the function is in H 1 (Ω) . Corollary 2 [Co 1991] If Ω is a polyhedron and u ∈ H N ∪ H T , then � � | curl u | 2 + | div u | 2 � | ∇ u | 2 � = Ω Ω As a consequence, H N = X N ∩ H 1 and H T = X T ∩ H 1 are closed subspaces of X N and X T , respectively, of infinite codimension if Ω is a non-convex polyhedron in R 3 . This remark turned out to be the only part of that paper that wasn’t forgotten... It implies that approximation of elements of X N \ H N or X T \ H T by conforming finite elements is impossible. And this had consequences... Martin Costabel (Rennes) Encounters with Maxwell Equations Linz, 17/10/2016 13 / 38
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