Corner Singularities

Transcription

1 Corner Singularities Martin Costabel IRMAR, Université de Rennes 1 Analysis and Numerics of Acoustic and Electromagnetic Problems Linz, October 2016 Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

2 PART I : Motivation Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

3 Dirichlet problem in 2D: Some general observations Dirichlet problem: u = f in Ω, u = g on Ω Known (for g = 0) (i) If Ω is any bounded open set in R d, then f L 2 (Ω) 1 u H0 1 (Ω) : u = f in Ω (ii) If Ω is any open set in R d, then f L 2 (Ω) 1 u H0 1 (Ω) : u + u = f in Ω H0 1(Ω): closure of C 0 (Ω) in H1 (Ω). Norm: u 2 1 = Ω ( u 2 + u 2 )dx. Variational formulations: (i) u H0 1 (Ω) : (ii) u H0 1 (Ω) : u v dx = ( u v + uv)dx Ω = Ω v f dx Ω v f dx Ω v H 1 0 (Ω) v H 1 0 (Ω) (Counter-)Example in R 2 for (i): Ω = {x x > R}, f = 0, g = 1 No solution in H 1 (Ω), but two reasonable solutions in H 1 loc (Ω): u 1 : harmonic at infinity and u(x) = log x / log R (except for R = 1) : electrostatic potential. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

4 Harmonic functions near infinity Tool: Polar coordinates and separation of variables (Fourier series). Polar coordinates: r 0, θ R/(2πZ), (x 1, x 2 ) = (r cos θ, r sin θ) Laplace operator in polar coordinates: = r 2( ) (r r ) 2 + θ 2 Fourier series in θ: u continuous in neighborhood of = u(x) = û k (r)e ikθ k Z { u = 0 r(rû k ) = k 2 α k r k + β k r k if k 0 û k û k (r) = α 0 + β 0 log r if k = 0 Dirichlet condition u = 1 for r = R: α 0 + β 0 log R = 1 and α k R k + β k R k = 0. Two particular solutions: u 1 and u = log r/ log R, but also Infinitely many solutions of the homogeneous problem: u k (r) = ( (r/r) k (r/r) k ) e ikθ Observations for the exterior Dirichlet problem 1 Existence and uniqueness require a-priori assumptions on the asymptotic behavior of the solution at infinity. 2 There may be a choice between different reasonable a-priori assumptions. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

5 Dirichlet problem on bounded smooth domains What everybody knows: Existence, uniqueness, regularity Assume that Ω is a bounded C domain and f C (Ω). Consider the Dirichlet problem u = f in Ω, u = 0 on Ω. Then 1 The solution exists and is unique. 2 The solution is regular: u C (Ω). Right or wrong? Right for the solution u in H 1 0 (Ω), for u C(Ω), or even for u L2 (Ω). But wrong, in general! Example: Let Ω be the half-disk {0 < r < 1, 0 < θ < π} = {z C z < 1, Im z > 0}. Define Then u = ( r k r k ) sin kθ = Im ( z k + z k ) u = 0 in Ω and u = 0 on Ω. Observations for the Dirichlet problem in a smooth domain if z = x 1 + ix 2 = re iθ 1 Existence, uniqueness and regularity require a-priori assumptions on the behavior of the solution near the boundary. 2 There seems to be a unique choice for reasonable a-priori assumptions (equivalent: H 1, L 2, bounded etc.) Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

6 Dirichlet problem on a plane sector Definition: For 0 < ω < 2π, define Γ ω = {0 < r <, 0 < θ < ω} = {z 0 < arg z < ω} Assume f = 0 and g = 0 in a neighborhood of the corner, i.e. u = 0 in Γ ω B(0, R) and u = 0 on Γ ω B(0, R). Fourier series: u(x) = û k (r) sin kπ ω θ Radial differential equation: r(rû k ) = ( kπ ω )2 û k û k (r) = α k r kπ ω + β k r kπ ω Solutions of the totally homogeneous problem: k 1 s k = r kπ ω sin kπ ω θ s k = r kπ ω sin kπ ω θ Regular or singular at the corner? 1 { For ω = π, we saw (k N): s k = Im z k : regular (polynomial, Taylor expansion) s k = Im z k : singular (not even in L 2 ) 2 For general ω, derivatives of s k are singular: s k C if kπ ω N. More precisely: s k H s (Γ ω B(0, R)) s < kπ ω + 1 Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

7 Singular functions, kπ ω N s k = Im z kπ ω H s (Γ ω B(0, R)) s < kπ ω + 1 Observations 1 The condition u H0 1 ( finite potential energy ) excludes all k < 0. Same as H 1/2 ; same as boundedness. 2 The condition u L 2 does not exclude all s k : If ω > π (non-convex corner), then s 1 L 2 near the corner. ( Non-uniqueness, u = s 1 s 1 solves the homogeneous problem in Γ ω B(0, 1) ) 3 If ω > π, then s 1 has unbounded derivatives, s 1 H 1, hence s 1 H 2 near the corner. Questions 1 Are the singular functions we have seen sufficient to describe the asymptotic behavior near a corner also for the inhomogeneous Dirichlet problem? 2 Are they always there? 3 What is the best way to describe the a-priori regularity assumptions? 4 How to treat more general elliptic boundary value problems? And finally: 5 Why should we care?? Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

8 Exercise: Singular functions for the 2D Neumann problem and mixed D/N Neumann problem: u = 0 in Γ ω, nu = 0 on Γ ω n = + 1 r θ on Γ ω We look for homogeneous (of degree λ > 0) solutions of u = 0. We find z λ and z λ. Neumann condition in θ = 0 Re z λ = 1 2 (zλ + z λ ) = r λ cos λθ. Neumann condition in θ = ω: sin λω = 0 λ = kπ ω. Solution: s k = r kπ ω cos kπ θ (k Z). ω Mixed Dirichlet/Neumann problem: u = 0 in Γ ω, u = 0 for θ = 0, nu = 0 for θ = ω Homogeneous solution with Dirichlet condition in θ = 0 r λ sin λθ. Neumann condition in θ = ω: cos λω = 0 λ = (k ) π ω. Solution: s k = r (k+ 1 2 ) π ω cos(k ) π θ (k Z). ω Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

9 Exercise: Singular functions for a 2D transmission problem Transmission or interface problem: div a u = 0 in R 2, a(x) = µ for θ < ω, a(x) = 1 for ω < θ π Let Ω 1 = { ω < θ < ω}, Ω 2 = {ω < θ π} and u 1,2 = u Ω1,2. Then this problem, understood in the distributional sense, is equivalent to u j = 0 in Ω j, j = 1, 2 u 1 = u 2 for θ = + ω µ nu 1 = nu 2 for θ = + ω Angle 2ω Algorithm We write u 1,2 = α 1,2 a 1,2 (x) + β 1,2 b 1.2 (x), where a j, b j are a basis of harmonic functions in Ω j, homogeneous of degree λ. Then the 4 interface conditions give a 4 4 linear system for the 4 coefficients α 1,..., β 2 : ) M ω,µ(λ) The characteristic equation det M ω,µ(λ) = 0 gives the possible singularity exponents λ, depending on ω and µ. ( α1 α 2 β 1 β 2 = 0 Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

10 Exercise: Singular functions for a 2D transmission problem Transmission or interface problem: div a u = 0 in R 2, a(x) = µ for θ < ω, a(x) = 1 for ω < θ π Let Ω 1 = { ω < θ < ω}, Ω 2 = {ω < θ π} and u 1,2 = u Ω1,2. Then this problem, understood in the distributional sense, is equivalent to u j = 0 in Ω j, j = 1, 2 u 1 = u 2 for θ = + ω µ nu 1 = nu 2 for θ = + ω Angle 2ω Computation: We can split the problem into odd and even problems with respect to θ. For the odd problem, we have a Dirichlet condition on the line θ {0, π}, and we can choose u 1 = r λ sin θλ, u 2 = α sin(π { θ)λ. sin ωλ = α sin(π ω)λ Two interface conditions at θ = ω: µ cos ωλ = α cos(π ω)λ Elimination of α gives one characteristic equation tan ωλ + µ tan(π ω)λ = 0. For the even problem, we have a Neumann condition on the line θ {0, π}, and we can choose u 1 = r λ cos θλ, u 2 = β cos(π θ)λ. We end up with the second characteristic equation tan ωλ + 1 tan(π ω)λ = 0. µ Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

11 Re: Questions Questions 1 Are the singular functions we have seen sufficient to describe the asymptotic behavior near a corner also for the inhomogeneous Dirichlet problem? 2 Are they always there? 3 What is the best way to describe the a-priori regularity assumptions? 4 How to treat more general elliptic boundary value problems? And finally: 5 Why should we care?? Questions 1 to 4 will find systematic answers in the lectures by Monique Dauge. Here follow some observations to questions 2 and 4, followed by some stories concerning the last question 5. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

12 Dual singular function Ω R 2 bounded domain, coincides with Γ ω in a neighborhood of the origin, otherwise smooth. Dirichlet problem u H 1 0 (Ω), u = f in Ω f L2 (Ω). Define s 1 = 1 π s 1 + s 1, where s 1 (x) = r π ω sin π ω θ { s 1 and s 1 H0 1(Ω) satisfies = 0 in Ω s 1 = s 1 on Ω. (Note: s 1 is smooth on Ω) Example: If Ω = Γ ω B(0, 1), we can take s 1 = 1 π (s 1 s 1 ) = 1 π (r π ω r π ω ) sin π ω θ. We consider ω > π (non-convex corner). Then s 1 L2 (Ω) and s 1 solves the totally homogeneous Dirichlet problem. Assumption: u(x) = c s 1 (x) + o( x π ω ) as x 0. Let us compute the scalar product (s 1, f ) = Ω s 1 (x)f (x)dx. δ > 0, Green s formula in Ω δ = Ω \ B(0, δ): s 1 (x)f (x)dx = s 1 (x) u(x)dx = ( s 1 nu ns 1 u) ds Ω δ Ω δ Ω δ \ Ω ω ( = 1 r π ω c π π ω r π ω 1 ( π ω )r π ω 1 c r π ) ω + o(r 1 ) sin 2 π ω θ r dθ 0,r=δ = c + o(1) as δ 0. Hence (s 1, f ) = c ( dual singular function ). Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

13 Why care? Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

14 The failure of the Sleipner A offshore platform 6m $700,000,000 Richter magnitude 3 23 August 1991 Source: D. N. Arnold Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62 3 / 44

15 Standard finite elements can give bad results Official cause: Computations with NASTRAN... Shear stresses underestimated by 47% Loss: USD 700 million Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

16 Fractures emerge near corners Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

17 PART II : Maxwell singularities Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

18 Some simple numerical tests Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

19 Laplace operator: Dirichlet problem in a square ( no corners ) u = f in Ω = (0, 1) 2 ; u = g on Ω Meshes: Solution (f = 1, g = 0): Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

20 Dirichlet problem in a square ( no corners ) Error asymptotics (Rel. L 2 error vs. d.o.f.) 10 1 P1 P2 N Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

21 Dirichlet problem in an L ( one corner ) u = f in Ω = ( 1, 1) 2 \ (0, 1) ( 1, 0) ; u = g on Ω Meshes: Solution: !0.2!0.4!0.6! !1!1! Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

22 Dirichlet problem in an L, uniform meshes Error asymptotics (Rel. L 2 error vs. d.o.f.) 10 1 P1 P2 N Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

23 Dirichlet problem in an L, refined meshes Meshes: u = f in Ω = ( 1, 1) 2 \ (0, 1) ( 1, 0) ; u = g auf Ω Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

24 Dirichlet problem in an L, refined meshes Error asymptotics (Rel. L 2 error vs. d.o.f.) 10 1 P1 P2 N Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

25 Dirichlet problem in an L, hp version u = f in Ω = ( 1, 1) 2 \ (0, 1) ( 1, 0) ; u = g auf Ω Meshes: Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

26 Dirichlet problem in an L, hp version P1 raf Martin Costabel (Rennes) Corner Singularities P2 raf Linz, 11 14/10/ / 62 Error asymptotics (Rel. L 2 error vs. d.o.f.) 10 3 hp N

27 Worse than slow approximation: Good approximation of the wrong object Time-harmonic Maxwell equations curl E = iωµh curl H = iωɛe + J In this section: Domain Ω R 3, ɛ = µ = 1, J = 0. The condition div E = div H = 0 follows if ω 0. E n = 0 & H n = 0 on Ω Eigenfrequencies of a cavity with perfectly conducting walls. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

28 Worse than slow approximation: Good approximation of the wrong object Time-harmonic Maxwell equations curl E = iωµh curl H = iωɛe + J In this section: Domain Ω R 3, ɛ = µ = 1, J = 0. The condition div E = div H = 0 follows if ω 0. E n = 0 & H n = 0 on Ω Eigenfrequencies of a cavity with perfectly conducting walls. Second order system for E: curl curl E ω 2 E = 0 Simplest variational formulation Find ω 0, E H 0 (curl, Ω) \ {0} such that F H 0 (curl, Ω) : curl E curl F = ω 2 E F Ω Ω Energy space: H 0 (curl, Ω) = {u L 2 (Ω) curl u L 2 (Ω); u n = 0} = closure in H(curl, Ω) of C 0 (Ω)3 Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

29 Regularized formulation Simple variational formulation E H 0 (curl, Ω) \ {0} : F H 0 (curl, Ω) : curl E curl F = ω 2 E F Ω Ω Galerkin discretization: Restriction to finite-dimensional subspace V h, h 0. Good: Eigenfrequencies are non-negative, discrete. Big Problem: ω = 0 has infinite multiplicity Kernel: Electrostatic fields: gradients of all φ H0 1 (Ω) (+ harmonic forms). Idea: div E = 0, so we can add a multiple of 0 = Ω div E div F Regularized formulation: E X N \ {0} : F X N : (Reg X ) curl E curl F + s div E div F = ω 2 E F Ω Ω Ω Energy space: X N = H 0 (curl, Ω) H(div, Ω) Second order system: curl curl E s div E = ω 2 E: Strongly elliptic. OK Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

30 Approximation on the square Ω = (0, π) (0, π), s = 0 Good approximation: Triangular edge elements (15 nodes per side, P 1 ) Eigenvalue ωk 2 vs. rank k Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

31 Approximation on the square Ω = (0, π) (0, π), s = 0 Bad approximation: Nodal triangular elements (15 nodes per side, P 1 ) Eigenvalue ωk 2 vs. rank k Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

32 Regularized formulation in the square ω[s] 2 vs. s Blue circles: computed ω[s] 2 with curl-dominant eigenfunctions. Red stars: computed ω[s] 2 with div-dominant eigenfunctions. div E satisfies s div E = ω 2 div E Extra eigenvalues: s times Dirichlet eigenvalues. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

33 Regularized formulation in the L ω[s] 2 vs. s Gray triangles: computed ω[s] 2 with indifferent eigenfunctions. Cyan-Lines: true Maxwell eigenvalues Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

34 Regularized formulation in the L Error of the first eigenvalue 10 1 Valeur propre Maxwell n 1 pénalisée au bord par λ = 10 1 ; Maillage 1 Interp 1 Interp 2 Interp 3 Interp Interp Error remains larger than 90% Error vs. number of d.o.f. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

35 Solution of the source problem, regularized formulation Exact solution (2nd component E 2 = r 1 3 cos θ 3 )). Computation with Q 3 elements. curl curl E div E = 0 in Ω; E n = E 0 on Ω Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

36 Analysis of Maxwell corner singularities is needed! Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

37 An integration by parts formula for Maxwell s equations Lemma 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) = (uτ n) v τ ) + div n u n v n Ω Corollary 1 If either u τ = v τ = 0 or u n = v n = 0, then ( ) curl u curl v + div u div v = u v + b(u, v) Ω Ω With the help of Corollary 1 one can prove that if Ω is smooth or convex (approximation by smooth domains with boundaries of positive curvature), then [Saranen 1982, Nedelec 1982] X N X T H 1. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

38 An integration by parts formula for Maxwell s equations Lemma 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) = (uτ n) v τ ) + div n u n v n 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 For the proof one has to show that smooth functions that are zero near the edges and corners are dense in H N and H T. From Corollary 2 follows that the subspaces H N of X N and H T of X T are closed. This implies that approximation of elements of X N \ H N or X T \ H T by conforming finite elements is impossible. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

39 Form of Maxwell singular functions, 2D On a sector Γ = Γ ω, 0 < ω 2π, we define spaces of homogeneous functions of degree λ N: S λ (Γ) = {u = r λ φ(θ) φ C inf ([0, ω]), S λ (Γ) = S λ (Γ) 2 S λ Dir (Γ) = {u Sλ (Γ) u Γ = 0}, S λ N (Γ) = {u Sλ (Γ) u n Γ = 0}. We consider homogeneous solutions of the principal part of the regularized Maxwell system curl curl u s div u = 0 in Γ u n = 0 and div u = 0 on Γ u S λ (Γ) We rewrite the system by introducing ψ = curl u, q = div u as a triangular system q= 0 in Γ, q S λ 1 Dir (Γ) (1) curl ψ= s q in Γ, ψ S λ 1 Neu (Γ) = Sλ 1 (Γ) (2) curl u = ψ, div u= q in Γ, u S λ N (Γ) (3) Solutions: Sums of the following Type 1: q = 0, ψ = 0, u general non-zero solution of (3) Type 2: q = 0, ψ general solution of (2), u particular solution of (3) Type 3: q general solution of (1), ψ and u particular solutions of (2) and (3). In 2D, Type 2 is easy: curl ψ = 0 means ψ = const. This doesn t exist in S λ 1 (Γ). Type 1: curl u = 0 u = φ, φ = 0 and φ S λ+1 Dir (Γ). That is, φ is a Laplace/Dirichlet singular function, λ + 1 = kπ ω and φ = c Im z λ+1 = u = c(λ + 1)r λ (sin λθ, cos λθ). Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

40 Form of Maxwell singular functions, 2D Type 1: q = 0, ψ = 0, u general non-zero solution of (3) Type 2: q = 0, ψ 0 general solution of (2), u particular solution of (3) Type 3: q 0 general solution of (1), ψ and u particular solutions of (2) and (3). In 2D, Type 2 is easy: curl ψ = 0 means ψ = const. This doesn t exist in S λ 1 (Γ). Type 1: curl u = 0 u = φ, φ = 0 and φ S λ+1 Dir (Γ). That is, φ is a Laplace/Dirichlet singular function, λ + 1 = kπ ω and φ = c Im z λ+1 = u = c(λ + 1)r λ (sin λθ, cos λθ). Type 3: (1) means q is a Laplace/Dirichlet singular function, λ 1 = kπ ω and q = c Im z λ 1. From (2) we see that ψ is conjugate harmonic to q, hence ψ = c Re z λ 1. A particular solution of (3) is then Theorem u = c 2λ r λ (sin λθ, cos λθ). At a polygonal corner of opening ω, the non-integer singular exponents of the principal part of the regularized Maxwell system are of the form λ = kπ ω + 1, k Z The divergence-free Maxwell singular functions are of the form u = r λ ((sin λθ, cos λθ), λ = kπ ω 1, k Z. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

41 Form of Maxwell singular functions, 3D conical point Cone in spherical coordinates Γ = {(ρ, ϑ) ρ > 0, ϑ G S 2 } Spaces of homogeneous functions (λ R \ N) S λ (Γ) = {u = ρ λ φ(ϑ) φ H 1 (G)}, S λ, S λ Dir, etc S λ N (Γ) = {u = ρλ φ(ϑ) L 2 loc (Γ \ {0}) curl u, div u L2 loc (Γ \ {0}), u n = 0 on Γ} S λ T (Γ) = {u = ρλ φ(ϑ) L 2 loc (Γ \ {0}) curl u, div u L2 loc (Γ \ {0}), u n = 0 on Γ} Homogeneous solutions of the principal part of the regularized Maxwell system (s > 0) curl curl u s div u = 0 u n = 0 and div u = 0 u S λ N (Γ), div u Sλ 1 Dir (Γ) in Γ on Γ We rewrite the system by introducing ψ = curl u, q = div u as a triangular system q = 0 in Γ, q S λ 1 Dir (Γ) (1) curl ψ = s q in Γ, ψ S λ 1 T (Γ) (2) curl u = ψ, div u = q in Γ, u S λ N (Γ) (3) Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

42 Form of Maxwell singular functions, 3D conical point Solutions of the triangular system: Type 1: q = 0, ψ = 0, u S λ N (Γ) general non-zero solution of (3): curl u = ψ, div u = q Type 2: q = 0, ψ S λ 1 T (Γ) general solution of (2) : curl ψ = s q, ( s = 1) u particular solution of (3) Type 3: q S λ 1 Dir (Γ) general solution of (1): q = 0, ψ and u particular solutions of (2) and (3). Lemma Explicit solutions, [Co-Dauge ARMA2000] λ 1, u S λ 1 N (Γ) = φ = λ+1 u x Sλ+1 Dir (Γ) ; curl u = 0 = φ = u λ 1, ψ S λ 1 T (Γ) = u = 1 λ+1 ψ x Sλ N (Γ) ; div ψ = 0 = curl u = ψ λ 1 2, q Sλ 1 Dir (Γ) u = 2qx+ρ2 q S λ 4λ+2 N (Γ) ; q = 0 curl u = 0, div u = q Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

43 Form of Maxwell singular functions, 3D conical point Solutions of the triangular system: Type 1: q = 0, ψ = 0, u S λ N (Γ) general non-zero solution of (3): curl u = ψ, div u = q Type 2: q = 0, ψ S λ 1 T (Γ) general solution of (2) : curl ψ = s q, ( s = 1) u particular solution of (3) Type 3: q S λ 1 Dir (Γ) general solution of (1): q = 0, ψ and u particular solutions of (2) and (3). Lemma Explicit solutions, [Co-Dauge ARMA2000] λ 1, u S λ 1 N (Γ) = φ = λ+1 u x Sλ+1 Dir (Γ) ; curl u = 0 = φ = u λ 1, ψ S λ 1 T (Γ) = u = 1 λ+1 ψ x Sλ N (Γ) ; div ψ = 0 = curl u = ψ λ 1 2, q Sλ 1 Dir (Γ) u = 2qx+ρ2 q S λ 4λ+2 N (Γ) ; q = 0 curl u = 0, div u = q Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

44 Form of Maxwell singular functions, 3D conical point Solutions of the triangular system: Type 1: q = 0, ψ = 0, u S λ N (Γ) general non-zero solution of (3): curl u = ψ, div u = q Type 2: q = 0, ψ S λ 1 T (Γ) general solution of (2) : curl ψ = s q, ( s = 1) u particular solution of (3) Type 3: q S λ 1 Dir (Γ) general solution of (1): q = 0, ψ and u particular solutions of (2) and (3). Lemma Explicit solutions, [Co-Dauge ARMA2000] λ 1, u S λ 1 N (Γ) = φ = λ+1 u x Sλ+1 Dir (Γ) ; curl u = 0 = φ = u λ 1, ψ S λ 1 T (Γ) = u = 1 λ+1 ψ x Sλ N (Γ) ; div ψ = 0 = curl u = ψ λ 1 2, q Sλ 1 Dir (Γ) u = 2qx+ρ2 q S λ 4λ+2 N (Γ) ; q = 0 curl u = 0, div u = q Corollary, Type 1 If λ 1, then u solution of Type 1 u = φ, φ S λ+1 Dir (Γ), φ = 0. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

45 Form of Maxwell singular functions, 3D conical point Solutions of the triangular system: Type 1: q = 0, ψ = 0, u S λ N (Γ) general non-zero solution of (3): curl u = ψ, div u = q Type 2: q = 0, ψ S λ 1 T (Γ) general solution of (2) : curl ψ = s q, ( s = 1) u particular solution of (3) Type 3: q S λ 1 Dir (Γ) general solution of (1): q = 0, ψ and u particular solutions of (2) and (3). Lemma Explicit solutions, [Co-Dauge ARMA2000] λ 1, u S λ 1 N (Γ) = φ = λ+1 u x Sλ+1 Dir (Γ) ; curl u = 0 = φ = u λ 1, ψ S λ 1 T (Γ) = u = 1 λ+1 ψ x Sλ N (Γ) ; div ψ = 0 = curl u = ψ λ 1 2, q Sλ 1 Dir (Γ) u = 2qx+ρ2 q S λ 4λ+2 N (Γ) ; q = 0 curl u = 0, div u = q Corollary, Type 2 If λ {0, 1}, then u solution of Type 2 curl u = φ, φ SNeu λ 1 (Γ), φ = 0 ; u = λ+1 φ x Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

46 Form of Maxwell singular functions, 3D conical point Solutions of the triangular system: Type 1: q = 0, ψ = 0, u S λ N (Γ) general non-zero solution of (3): curl u = ψ, div u = q Type 2: q = 0, ψ S λ 1 T (Γ) general solution of (2) : curl ψ = s q, ( s = 1) u particular solution of (3) Type 3: q S λ 1 Dir (Γ) general solution of (1): q = 0, ψ and u particular solutions of (2) and (3). Lemma Explicit solutions, [Co-Dauge ARMA2000] λ 1, u S λ 1 N (Γ) = φ = λ+1 u x Sλ+1 Dir (Γ) ; curl u = 0 = φ = u λ 1, ψ S λ 1 T (Γ) = u = 1 λ+1 ψ x Sλ N (Γ) ; div ψ = 0 = curl u = ψ λ 1 2, q Sλ 1 Dir (Γ) u = 2qx+ρ2 q S λ 4λ+2 N (Γ) ; q = 0 curl u = 0, div u = q Corollary, Type 3 If λ { 1, 0}, then 2 u solution of Type 3 div u = q, q S λ 1 ψ = 1 λ φ x, u = 1 λ(2λ+1) Dir (Γ), q = 0. ( (2λ 1)qx ρ 2 q ) Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

47 Some formulas from problems vector (6.3) and analysis (6.4). All these formulas are based on the scalar product or the vector product with the vector x, with x denoting the vector of Cartesian coordinates (x, y, z), and ρ = x. We begin with three From series [Co-Da of formulas. ARMA First2000]: we give product laws: a and b denoting vector fields and γ being a scalar function on R 3, we have grad(a b) = (a grad) b + (b grad) a + a curl b + b curl a, (6.5a) curl(a b) = (b grad) a (a grad) b + a div b b div a, div(a b) = b curl a a curl b, curl(γ a) = γ curl a + grad γ a, div(γ a) = γ div a + grad γ a. Now, using the above formulas for the field x which satisfies div x = 3, curl x = 0, x grad = ρ ρ and grad x = I, we obtain for any field a and scalar q grad(a x) = (ρ ρ + 1)a + x curl a, curl(a x) = (ρ ρ + 2)a x div a, div(a x) = x curl a, curl(qx) = grad q x, div(qx) = (ρ ρ + 3)q. Finally, with γ = ρ 2 and a = grad q, (6.5d) and (6.5e) yield curl(ρ 2 grad q) = 2 grad q x, div(ρ 2 grad q) = 2ρ ρq + ρ 2 q. (6.5b) (6.5c) (6.5d) (6.5e) (6.6a) (6.6b) (6.6c) (6.6d) (6.6e) (6.6f) (6.6g) Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

48 Form of Maxwell singular functions, 3D conical point, Electric b.c. Theorem Electric Maxwell Singularities At a 3D conical point, the singular exponents λ 0, λ > 1 of the principal part of the regularized Maxwell equations are of the form λ Dir 1, λ Neu or λ Dir + 1, where λ Dir and λ Neu are Dirichlet resp. Neumann singular exponents of the Laplacian. The last type does not appear for the divergence-free Maxwell equations. The exponents λ = 0 and λ = 1 appear for certain non-lipschitz topologies. Similar theorem for Magnetic Boundary Conditions... Rule of thumb: The main singularity is a gradient. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

49 Birman-Solomyak decomposition theorems ( Helmholtz-like ) Decomposition Theorem: Electric b. c. [Birman-Solomyak 1987] Let Ω be a bounded Lipschitz domain (+ cracks, in R 2 or R 3 ). Then K N : X N (Ω) H N (Ω) bounded such that curl K N = curl. That is, for u X N (Ω) we have u = φ + w with w = K N u H N (Ω), φ H0 1 (, Ω). Decomposition Theorem: Magnetic b. c. [Birman-Solomyak Filonov 1997] Let Ω be a bounded Lipschitz domain (+ cracks, in R 2 or R 3 ). Then K T : X T (Ω) H T (Ω) bounded such that curl K T = curl, that is, for u X T (Ω) we have if and only if u = φ + w with w = K T u H T (Ω), φ HNeu 1 (, Ω), v H 1 (Ω) ψ H 2 (Ω) : n v = nψ on Ω. This is satisfied if Ω is piecewise C 3/2+ε, ε > 0. There exists a domain Ω C 3/2 for which it is not satisfied. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

50 X N and H N We have seen: u X N \ H N φ, φ H 1 0 ( ) \ H2. Corollary (i) Let Ω be a Lipschitz polygon in R 2. Then H N (Ω) is a closed subspace of finite codimension of X N. The codimension is equal to the number of non-convex corners of Ω. (ii) Let Ω be a Lipschitz polyhedron in R 3. Then either X N (Ω) = H N (Ω) (when Ω is convex) or H N (Ω) is a closed subspace of infinite codimension of X N (due to non-convex edges). Consequences of X N H N? Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

51 Two variational formulations Regularized time-harmonic Maxwell boundary value problem (ε = µ = 1) curl curl u s div u ω 2 u = f in Ω (BVP) u n = 0 and div u = 0 on Ω u X N (Ω) Weak formulation: Integration by parts against v C 1 (Ω) with v n = 0 gives ( (P) curl u curl v + s div u div v ω 2 u v ) = f v Ω Ω We assume now that Ω is a Lipschitz polyhedron in R 3. Then C (Ω) X N (Ω) is dense in H N (Ω), and we see that The boundary value problem (BVP) is equivalent to Find u X N (Ω) such that (P) is satisfied v H N (Ω). Non-symmetric! We have two symmetric versions: (P X ) Find u X N (Ω) such that (P) is satisfied v X N (Ω) Maxwell (P H ) Find u H N (Ω) such that (P) is satisfied v H N (Ω) Lamé Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

52 Two variational formulations Two unique solutions For f L 2 (Ω) with div f = 0 and all ω > 0 except for a discrete set of frequences, both problems (P X ) and (P H ) have unique solutions. If Ω is non-convex, these solutions are, in general, different. The eigenfrequencies are different. The solution of (P X ) satisfies div u = 0 (Maxwell solution), the solution of (P H ) has, in general, div u H 1 (Ω) (Lamé or pseudo-maxwell solution). Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

53 Different versions of the decomposition The splitting into a singular and regular part u = u sing + u reg is not unique. Different motivations different splittings. Motivation 1: Numerical approximation by singular function method. The principle: Assume u V is solution of a variational problem v V : a(u, v) =< f, v >. Consider Galerkin approximation by subspaces V h V : u h V h such that v h V h : a(u h, v h ) =< f, v h >. We assume that there holds Céa s Lemma: u u h C inf{ u v h v h V h }. If there is a splitting u = γs + u reg with a known singular function s, one can define the approximation space as an augmented finite element space Céa s Lemma now gives V h := span{s} V 0 h. u u h C inf{ u reg v h v h V 0 h }. If Vh 0 is a regular finite element space, the convergence order is thus determined by the regularity of u reg H 1+ɛ, which should therefore be as high as possible. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

54 Different splittings in 2D Assumption: Ω is a bounded domain, coincides near the origin with Γ ω, ω > π, and is smooth elsewhere. u X N is solution of the regularized Maxwell system (P X ) with ω = 0 and smooth right hand side f, div f = 0. Decomposition 1. Natural decomposition. s = χ r λ 1 sin λ 1 θ with λ 1 = π ω. Limit 1 + ɛ of regularity determined by second singular function: u reg H s, s < 1 + ɛ for 1 + ɛ = λ 2 = 2π ω. Decomposition 2. Divergence-free singular function: s = χ curl r λ 1 cos λ 1 θ. This is the same as 1. Decomposition 3. Projection on H N. Let φ H0 1(Ω) solve φ = s 1 with the dual singular function s 1 of the Dirichlet problem, and define s = φ. Here u reg is computable: It is the pseudo-maxwell or Lamé solution of (P H ). Then s, and hence u reg, contains a term in S π ω +2 1, hence ɛ = min{ 2π ω 1, 1 π ω }. For π < ω < 3π/2, this is worse than Decomposition 1, worst for ω close to π. Decomposition 4. Orthogonal decomposition in X N, s H N. Here ɛ is the same as in Decomposition 3. Note: In Decomposition 3 & 4, additional (spurious) singular functions appear in both s and in u reg that are absent in u. Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

55 Making convergence possible again : The WRM Idee: The problem comes from the fact that the regularized Maxwell bilinear form ( ) a(u, v) = curl u curl v + s div u div v Ω defines a space X N = H 0 (curl) H(div) in which smooth functions (and piecewise polynomials) are not dense. The term Ω div u div v was rather arbitrary, because div u = 0 for Maxwell solutions. It can be replaced by a different bilinear form, that is, the norm div u L 2 (Ω) can be replaced by a norm div u Y with a space Y satisfying the two requirements: 1 The corresponding space is still compactly embedded in L 2 (Ω). 2 C (Ω) X Y N is dense in X Y N. X Y N = H 0(curl) {u L 2 div u Y } This was first proposed in [Co-Dauge NuMa 2002] Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation of nodal finite elements. There, Y is a weighted L 2 space: q 2 Y = ρ 2γ q 2, ρ distance to the non-convex edges and corners Ω Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

56 Weighted regularization Theorem [Co-Da 2002] Let Ω be a polygon in R 2 or a polyhedron in R 3. Define D Y ( ) = {q H0 1 (Ω) q Y }. Then (i) Decomposition X Y N = H N + D Y ( ). (ii) If H 2 H 1 0 (Ω) is dense in DY ( ), then condition 2 is satisfied. (iii) There exists 0 < γ < 1 such that for weight exponents γ < γ < 1, conditions 1 and 2 are satisfied. γ = max{1 π ω e, 1 2 λdir c }. As a consequence, for this choice of weights, any conforming finite element Galerkin method for the regularized Maxwell system (source problem or eigenvalue problem) converges in X Y N. More recently, results with other choices of Y have appeared: e.g. Y = H s for 0 < s < 1, or a discretized version thereof, Y,h = h s L 2 [Bonito-Guermond MathComp 2011] Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

57 Weighted Regularization in the L Computations with Q 10 elements on refined mesh. Legend: Blue circles: computed ω[s] 2 with curl-dominant eigenfunctions. Red stars: computed ω[s] 2 with div-dominant eigenfunctions. Gray triangles: computed ω[s] 2 with indifferent eigenfunctions. Cyan Lines: true Maxwell eigenvalues (calculated from scalar Neumann problem) Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

58 The ideal: Computations in the square ω[s] 2 vs. s Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

59 The unweighted reality: Computations in the L (γ = 0) ω[s] 2 vs. s This is Lamé, not Maxwell! Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

60 Towards the ideal: WRM in the L (γ = 0.35) ω[s] 2 vs. s Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

61 Towards the ideal: WRM in the L (γ = 0.5) ω[s] 2 vs. s Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

62 The ideal recovered: WRM in the L (γ = 1) ω[s] 2 vs. s Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

63 WRM on Fichera s corner Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

64 WRM on Fichera s corner 1st Maxwell eigenmode on Fichera s corner Q 4 elements Weighted Regularization Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

65 WRM on Fichera s corner Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62

66 Another singularity Martin Costabel (Rennes) Corner Singularities Linz, 11 14/10/ / 62