Static Maxwell Type Problems: Functional A Posteriori Error Estimates and Estimates for the Maxwell Constant in 3D. Dirk Pauly

Transcription

1 Static Maxwell Type Problems: Functional A Posteriori Error Estimates and Estimates for the Maxwell Constant in 3D Dirk Pauly Fakultät für Mathematik Universität Duisburg-Essen, Campus Essen, Germany partially joint work with Sergey Repin Steklov Institute, St. Petersburg, Russia & MIT, University of Jyväskylä, Finland LSSC 13 Sozopol, Bulgaria June 6 013

2 Functional A Posteriori Error Estimates Part 1 Functional A Posteriori Error Estimates for Static Maxwell Type Problems

3 Introduction: (simple) Electro Static Maxwell Problem simple: ε, µ = id, only hom. bc, only tang. (electric) bc, no mixed bc Ω R 3 bounded domain with Lipschitz boundary Γ = Ω E : Ω R 3 electric vector field F : Ω R 3, G : Ω R given right hand side data τ tangential trace, i.e., τe = n E Γ : Γ R 3 (for smooth E) orthogonality w.r.t. L (Ω)-scalar product E, H L (Ω) := E H Ω E H D (Ω) Dirichlet field E L (Ω) & rot E = 0, div E = 0 and τe = 0 rot E = F div E = G τe = 0 E H D (Ω) in Ω in Ω on Γ GOAL: non-conforming estimates for error e := E Ẽ, Ẽ approximation of E JUST: Ẽ L (Ω)

4 Introduction: Dirichlet Laplace Problem u H 1 (Ω) with u = div u = G, u Γ = 0 set E := u note: rot E = 0, n E Γ = Γ u Γ = 0 (since d ι = ι d) and E H D (Ω) rot E = 0 in Ω div E = G in Ω τe = 0 on Γ E H D (Ω) non-conforming estimates for error e := u Ẽ JUST: Ẽ L (Ω) approximation of E = u non-conforming estimates of energy norm

5 Introduction: rot rot-problem U R(Ω) with rot rot U = F, n rot U Γ = 0 set E := rot U note: div E = 0, n E Γ = 0 and E H D (Ω) rot E = F in Ω div E = 0 in Ω τe = 0 on Γ E H D (Ω) non-conforming estimates for error e := rot U Ẽ JUST: Ẽ L (Ω) approximation of E = rot U non-conforming estimates of energy norm

6 Solution Theory rot E = F div E = G τe = 0 E H D (Ω) in Ω in Ω on Γ split rot E rot = F, rot E = 0 in Ω div E rot = 0, div E = G in Ω ( electro static τe rot = 0, τe = 0 on Γ Maxwell problems) E rot H D (Ω), E H D (Ω) introducing scalar and vector potentials u and U solving U = rot rot U = F, u = div u = G in Ω div U = 0 in Ω τ rot U = 0, u = 0 on Γ variational formulations for u and U (right Hilbert spaces) E rot := rot U and E := u as well as E := E rot + E

7 Method for Error Estimates rot E = F in Ω div E = G in Ω (electro static Maxwell problem) τe = 0 on Γ E H D (Ω) method: funct. a post. error est. for linear second order elliptic problems pioneering work of Sergey Repin starting 1990 s later extended to all linear and non-linear second order elliptic problems (Laplace, elastic, parabolic, hyperbolic, even order problems,...) Maxwell system is first order! What to do? solution: Helmholtz decomposition scalar and vector potentials second order methods for the potentials

8 Sobolev Spaces spaces R(Ω) := {E L (Ω) : rot E L (Ω)} R 0 (Ω) := {E R(Ω) : rot E = 0} R(Ω) := {E R(Ω) : τe = 0} = C R(Ω) (Ω) R 0 (Ω) := R(Ω) R 0 (Ω) (Gauß theorem) analogously D(Ω), D 0 (Ω), D(Ω), D 0 (Ω) and H D (Ω) := R 0 (Ω) D 0 (Ω) (finite dimensionl since R(Ω) D(Ω) L (Ω) compact) = {E L (Ω) : rot E = 0, div E = 0, τe = 0}

9 Results: Upper Bounds for Non-Conforming Approximations Ẽ L (Ω) approximation of E Theorem ( 11 DP, S.I. Repin) For all Ẽ L (Ω) and all D H D (Ω) E Ẽ D L (Ω) inf X R(Ω) holds. Here, π : H D (Ω) R d (e.g. isomorphic) and Mrot (Ẽ; X ) + inf Mdiv (Ẽ; Y ) + π(ẽ D) R Y D(Ω) d M rot(ẽ; X ) := cm F rot X L (Ω) + Ẽ X L (Ω), M div (Ẽ; Y ) := c p, G div Y L (Ω) + Ẽ Y L (Ω). only natural and well known continuity constants involved: c p, Poincaré constant u H 1 (Ω) u L (Ω) cp, u L (Ω) c m Maxwell constant E H E L (Ω) cm rot E L (Ω) here H := R(Ω) rot R(Ω) = R(Ω) D 0 (Ω) H N (Ω)

10 Proof: Tools 1 Rellich s selection theorems, i.e., H 1 (Ω), H 1 (Ω) L (Ω) compact Poincaré estimates, i.e., u H 1 (Ω) u H 1 (Ω) R u L (Ω) cp, u L (Ω) u L (Ω) cp u L (Ω) Maxwell selection theorems, i.e., R(Ω) D(Ω), R(Ω) D(Ω) L (Ω) compact Maxwell estimates, i.e., E H = R(Ω) D 0 (Ω) H D (Ω) E H = R(Ω) D 0 (Ω) H N (Ω) E L (Ω) cm rot E L (Ω) E L (Ω) cm rot E L (Ω) 3 Maxwell selection theorems dim H D (Ω), dim H N (Ω) < (Betti numbers) 4 Helmholtz decompositions (all 6 images are closed in L (Ω)) =D 0 (Ω) L (Ω) = H {}}{ 1 (Ω) H D (Ω) rot R(Ω), rot R(Ω) = rot H L (Ω) = H 1 (Ω) H N (Ω) rot R(Ω), rot R(Ω) = rot H }{{} = D 0 (Ω)

11 Proof e = E Ẽ L (Ω) Helmholtz decomposition of error e = e + e H + e rot H 1 (Ω) H D (Ω) rot H e = u with scalar potential u H 1 (Ω) e rot = rot U with vector potential U H e L (Ω) = e L (Ω) + e H L (Ω) + erot L (Ω)

12 Proof... 1 e = u, u H 1 (Ω): ϕ H 1 (Ω) Y D(Ω) e, ϕ L (Ω) = e, ϕ L (Ω) = E, ϕ L (Ω) Ẽ, ϕ L (Ω) = div Y G, ϕ L (Ω) + Y Ẽ, ϕ L (Ω) div Y G L (Ω) ϕ L (Ω) }{{} c p, ϕ L (Ω) + Y Ẽ L (Ω) ϕ L (Ω) ϕ := u e L (Ω) cp, div Y G L (Ω) + Y Ẽ L (Ω) e rot = rot U, U H: Φ H X R(Ω) e rot, rot Φ L (Ω) = e, rot Φ L (Ω) = E, rot Φ L (Ω) Ẽ, rot Φ L (Ω) = F rot X, Φ L (Ω) + X Ẽ, rot Φ L (Ω) F rot X L (Ω) Φ L (Ω) }{{} c m rot Φ L (Ω) + X Ẽ L (Ω) rot Φ L (Ω) Φ := U e rot L (Ω) cm F rot X L (Ω) + X Ẽ L (Ω) 3 e H : simple algebraic manipulation e H L (Ω) π(ẽ D) R d

13 Maxwell Constants Part The Maxwell Constants in 3D

14 Estimates for the Maxwell Constants open problem: estimates for the Maxwell constants c m in 3D or nd? E R(Ω) D(Ω) H D (Ω) E L (Ω) ( cm,t rot E L (Ω) + div ) 1/ E L (Ω) E R(Ω) D(Ω) H N (Ω) E L (Ω) ( cm,n rot E L (Ω) + div ) 1/ E L (Ω) question:? c m,t, c m,n? in D well known with Poincaré constants c p, c m,t, c m,n c p u H 1 (Ω) u H 1 (Ω) R u L (Ω) cp, u L (Ω) u L (Ω) cp u L (Ω) note always c p, = 1 λ1 < 1 µ = c p

15 Step 1: Problem Reduction by Helmholtz Decomposition E R(Ω) D(Ω) H D (Ω) E L (Ω) ( cm,t rot E L (Ω) + div ) 1/ E L (Ω) E R(Ω) D(Ω) H N (Ω) E L (Ω) ( cm,n rot E L (Ω) + div ) 1/ E L (Ω) Helmholtz decomposition splits problems into 4 nicer problems E D(Ω) R 0 (Ω) H D (Ω) }{{} E L (Ω) c m,t,div div E L (Ω) = H 1 (Ω) E R(Ω) D 0 (Ω) H D (Ω) }{{} =rot R(Ω) E D(Ω) R 0 (Ω) H N (Ω) }{{} = H 1 (Ω) E R(Ω) D 0 (Ω) H N (Ω) }{{} E L (Ω) cm,t,rot rot E L (Ω) E L (Ω) c m,n,div div E L (Ω) E L (Ω) cm,n,rot rot E L (Ω) =rot R(Ω)

16 Step : First Results reminder E R(Ω) D(Ω) H D (Ω) E L (Ω) ( cm,t rot E L (Ω) + div ) 1/ E L (Ω) E R(Ω) D(Ω) H N (Ω) E L (Ω) ( cm,n rot E L (Ω) + div ) 1/ E L (Ω) E D(Ω) H 1 (Ω) E R(Ω) rot R(Ω) E D(Ω) H 1 (Ω) E R(Ω) rot R(Ω) E L (Ω) c m,t,div div E L (Ω) E L (Ω) cm,t,rot rot E L (Ω) E L (Ω) c m,n,div div E L (Ω) E L (Ω) cm,n,rot rot E L (Ω) trivially: c m,t,rot, c m,t,div c m,t and c m,n,rot, c m,n,div c m,n trivially: c m,t max{c m,t,rot, c m,t,div } and c m,n max{c m,n,rot, c m,n,div } (Helmholtz) trivially: c m,t = max{c m,t,rot, c m,t,div } and c m,n = max{c m,n,rot, c m,n,div } Theorem ( 13 DP) c m,t,div = c p, c m,n,div = c p c m,t,rot = c m,n,rot remains to estimate c m,rot := c m,t,rot = c m,n,rot

17 Step 3: Main Results E R(Ω) D(Ω) H D (Ω) E L (Ω) ( cm,t rot E L (Ω) + div ) 1/ E L (Ω) E R(Ω) D(Ω) H N (Ω) E L (Ω) ( cm,n rot E L (Ω) + div ) 1/ E L (Ω) E D(Ω) H 1 (Ω) E R(Ω) rot R(Ω) E D(Ω) H 1 (Ω) E R(Ω) rot R(Ω) E L (Ω) cp, div E L (Ω) E L (Ω) cm,rot rot E L (Ω) E L (Ω) cp div E L (Ω) E L (Ω) cm,rot rot E L (Ω) trivially: c m,t = max{c m,rot, c p, } and c m,n = max{c m,rot, c p} remains to estimate c m,rot Theorem ( 13 DP) Let Ω be bounded and convex. Then c m,rot c p. Moreover, c p, c m,t c m,n = c p. equivalent formulation for eigenvalues

18 Proof of First Theorem Proof... by some functional analysis... A : D(A) H 1 H lin., dens. def., closed with adjoint A : D(A ) H H 1 assume D(A) [ R(A ] ) H 1 compact! [ 0 A A define M := and note M A 0 := ] A 0 AA M, M, A A, AA self-adjoint with pure point spectra and σ p(m) = ± σ p(a A) = ± σ p(aa ) = ±{κ 1, κ,...}, 0 κ n looking at first resp. second eigenvalues Au A v H inf H 0 u D(A) R(A ) u = inf 1 0 v D(A ) R(A) v H 1 H

19 Proof of First Theorem... A : D(A) H 1 H lin., dens. def., cl., adjoint A : D(A ) H H 1 D(A) R(A ) H 1 cpt (note R(A ) = N(A) and R(A ) cl.) especially Au A v H inf H 0 u D(A) R(A ) u = inf 1 0 v D(A ) R(A) v H 1 H A := : H 1 (Ω) L (Ω) L (Ω), A := div : D(Ω) L (Ω) L (Ω) R(A ) = N(A) = {0} = L (Ω) H 1 (Ω) L (Ω) = H 1 (Ω) L (Ω) cpt by Rellich s selection theorem 1 c p, = λ 1 = inf u L (Ω) 0 u H u 1 (Ω) L (Ω) = inf 0 E D(Ω) H 1 (Ω) div E L (Ω) E L (Ω) = 1 c m,t,div

20 Proof of First Theorem... A : D(A) H 1 H lin., dens. def., cl., adjoint A : D(A ) H H 1 D(A) R(A ) H 1 cpt (note R(A ) = N(A) and R(A ) cl.) especially Au A v H inf H 0 u D(A) R(A ) u = inf 1 0 v D(A ) R(A) v H 1 H A := : H 1 (Ω) L (Ω) L (Ω), A := div : D(Ω) L (Ω) L (Ω) R(A ) = N(A) = R H 1 (Ω) R H 1 (Ω) L (Ω) cpt by Rellich s selection theorem 1 c p u L = µ = inf div E (Ω) L 0 u H 1 (Ω) R u = inf (Ω) L (Ω) 0 E D(Ω) H E = 1 1 (Ω) L c (Ω) m,n,div

21 Proof of First Theorem... A : D(A) H 1 H lin., dens. def., cl., adjoint A : D(A ) H H 1 D(A) R(A ) H 1 cpt (note R(A ) = N(A) and R(A ) cl.) Au A v H inf H 0 u D(A) R(A ) u = inf 1 0 v D(A ) R(A) v H 1 H especially A := rot : R(Ω) L (Ω) L (Ω), A := rot : R(Ω) L (Ω) L (Ω) R(A ) = rot R(Ω) R(Ω) rot R(Ω) R(Ω) D(Ω) L (Ω) cpt by MCP 1 c m,t,rot = κ = inf 0 E R(Ω) rot R(Ω) rot E L (Ω) E L (Ω) rot E L = inf (Ω) 0 E R(Ω) rot R(Ω) E L (Ω) = 1 c m,n,rot

22 Proof of Second (Main) Theorem Proof crucial estimate for convex domains Lemma ( 98 C. Amrouche, C. Bernardi, M. Dauge, V. Girault) Ω R bd and convex. Then E R(Ω) D(Ω), R(Ω) D(Ω) H 1 (Ω) continuous and E L (Ω) 1 ( rot E L (Ω) + div E L (Ω)). related, earlier, partial results by M. Costabel ( 91), J. Saranen ( 8), P. Grisvard ( 7, 85), R. Leis ( 68), J. Kadlec ( 64) pick E R(Ω) rot R(Ω) = R(Ω) D 0 (Ω) H N (Ω) = R(Ω) D 0 (Ω) (Ω convex) E, a L (Ω) = rot H, a L (Ω) = 0 for all a R3 since H R(Ω) E H 1 (Ω) (R 3 ) E L (Ω) c m,rot c p very simple! Poincare c p E L (Ω) Lemma 1 c p rot E L (Ω)

23 Last Slide! more results: Blagodarya / Thank You Functional A Posteriori Error Estimates: lower bounds usual features of Sergey s estimates: sharpness, only natural constants, simple implementation,... Ω exterior domain, polynomially weighted estimates differential forms, Ω R N, Ω Riemannian manifold hyperbolic problems, full time-dependent Maxwell system, eddy current,... with ε, µ and inhomogeneous bc diffusion problem, elasticity,... magnetic problem!!!!! mixed boundary conditions Maxwell Constants: also with ε, µ also Ω R N with differential forms also on non-convex polygons (not too pointy)