Analysis of Block-Jacobi Preconditioners for Local Multi-Trace Formulations A.Ayala ∗ , P.Marchand ∗ , X.Claeys ∗ , V.Dolean † , M.Gander § ∗ Labo. J.-L. Lions UPMC / INRIA Alpines, † Labo. J.A. Dieudonné, Univ. Nice, § Université de Genève.
Multi-domain scattering Notations n 1 R d = ∪ n Ω j ∩ Ω k = ∅ for j � = k j = 0 Ω j , Γ = ∪ n n 2 Ω 2 Ω 1 j = 0 Γ j , Γ j := ∂ Ω j κ j ∈ C effective wave number in Ω j , n 0 n 3 Ω 3 Transmission problem (well posed) : Ω 0 = H 1 Find u ∈ (∆ , Ω j ) exterieur su h that lo ∆ u + κ 2 j u = 0 Ω j , j = 0 , . . . n u in in u − u in Ω 0 outgoing , in � ∂ n j u | Γ j + ∂ n k u | Γ k = 0 , u | Γ j − u | Γ k = 0 , on Γ j ∩ Γ k ∀ j , k Concern : Solution by means of boundary integral equation methods?
Difficulties related to the conditioning Problem : BEM = ill conditioned Boundary element methods lead to dense linear systems that require, for large scale problems (industrial context), iterative solvers. But they induce ill conditioned matrices in general, which prevents the convergence of solvers. Two alternatives : i) equation of the 2nd kind : Id+compact ii) preconditioned equation Operator preconditionning In the case of isolated homogeneous scatterers, the Calderón preconditioner is now a very popular technique because it is rather easy to implement, and is adapted to a reasonnably large frequency bandwidth. It stabilizes the condition number with respect to the meshwidth. [Steinbach & Wendland, 1998], [Christiansen & Nédélec, 2000], [Antoine & Boubendir, 2008], [Cools, Andriulli & Olyslager, 2009],...
Objective and literature To obtain well conditioned (or easily preconditionable) integral formulations, a natural idea consists in adapting what already exists for isolated scatterers. But usually orientation of interfaces is required for applying those techniques. On the other hand, the skeleton Γ is not a priori orientable. Already existing : • Rumsey/PMCHWT/EFIE = "Single-trace formulation" • Boundary element tearing and interconnecting method (BETI) [Steinbach & Windisch, 2010], [Langer & Steinbach, 2003], [Hsiao, Steinbach & Wendland, 2000], ...
Objective and literature To obtain well conditioned (or easily preconditionable) integral formulations, a natural idea consists in adapting what already exists for isolated scatterers. But usually orientation of interfaces is required for applying those techniques. On the other hand, the skeleton Γ is not a priori orientable. Already existing : Calderón technique not possible • Rumsey/PMCHWT/EFIE = "Single-trace formulation" • Boundary element tearing and interconnecting method (BETI) [Steinbach & Windisch, 2010], [Langer & Steinbach, 2003], [Hsiao, Steinbach & Wendland, 2000], ...
Objective and literature To obtain well conditioned (or easily preconditionable) integral formulations, a natural idea consists in adapting what already exists for isolated scatterers. But usually orientation of interfaces is required for applying those techniques. On the other hand, the skeleton Γ is not a priori orientable. Already existing : Calderón technique not possible • Rumsey/PMCHWT/EFIE = "Single-trace formulation" • Boundary element tearing and interconnecting method (BETI) [Steinbach & Windisch, 2010], [Langer & Steinbach, 2003], [Hsiao, Steinbach & Wendland, 2000], ... Recently : • Global multi-trace [Claeys & Hiptmair, 2011] • Local multi-trace [Jerez & Hiptmair, 2011]
Objective and literature To obtain well conditioned (or easily preconditionable) integral formulations, a natural idea consists in adapting what already exists for isolated scatterers. But usually orientation of interfaces is required for applying those techniques. On the other hand, the skeleton Γ is not a priori orientable. Already existing : Calderón technique not possible • Rumsey/PMCHWT/EFIE = "Single-trace formulation" • Boundary element tearing and interconnecting method (BETI) [Steinbach & Windisch, 2010], [Langer & Steinbach, 2003], [Hsiao, Steinbach & Wendland, 2000], ... Calderón available Recently : • Global multi-trace [Claeys & Hiptmair, 2011] • Local multi-trace [Jerez & Hiptmair, 2011]
Objective and literature To obtain well conditioned (or easily preconditionable) integral formulations, a natural idea consists in adapting what already exists for isolated scatterers. But usually orientation of interfaces is required for applying those techniques. On the other hand, the skeleton Γ is not a priori orientable. Already existing : • Rumsey/PMCHWT/EFIE = "Single-trace formulation" • Boundary element tearing and interconnecting method (BETI) [Steinbach & Windisch, 2010], [Langer & Steinbach, 2003], [Hsiao, Steinbach & Wendland, 2000], ... Recently : • Global multi-trace [Claeys & Hiptmair, 2011] • Local multi-trace [Jerez & Hiptmair, 2011]
Objective and literature To obtain well conditioned (or easily preconditionable) integral formulations, a natural idea consists in adapting what already exists for isolated scatterers. But usually orientation of interfaces is required for applying those techniques. On the other hand, the skeleton Γ is not a priori orientable. Already existing : • Rumsey/PMCHWT/EFIE = "Single-trace formulation" • Boundary element tearing and interconnecting method (BETI) [Steinbach & Windisch, 2010], [Langer & Steinbach, 2003], [Hsiao, Steinbach & Wendland, 2000], ... Recently : • Global multi-trace [Claeys & Hiptmair, 2011] • Local multi-trace [Jerez & Hiptmair, 2011]
Objective and literature To obtain well conditioned (or easily preconditionable) integral formulations, a natural idea consists in adapting what already exists for isolated scatterers. But usually orientation of interfaces is required for applying those techniques. On the other hand, the skeleton Γ is not a priori orientable. Local multi-trace [Jerez & Hiptmair, 2011] Already existing : • Rumsey/PMCHWT/EFIE = "Single-trace formulation" • Boundary element tearing and interconnecting method (BETI) [Steinbach & Windisch, 2010], [Langer & Steinbach, 2003], [Hsiao, Steinbach & Wendland, 2000], ... Recently : • Global multi-trace [Claeys & Hiptmair, 2011]
Objective and literature To obtain well conditioned (or easily preconditionable) integral formulations, a natural idea consists in adapting what already exists for isolated scatterers. But usually orientation of interfaces is required for applying those techniques. On the other hand, the skeleton Γ is not a priori orientable. Local multi-trace [Jerez & Hiptmair, 2011] Already existing : • Rumsey/PMCHWT/EFIE = "Single-trace formulation" • Accuracy comparable to BETI and Rumsey • Boundary element tearing and interconnecting method (BETI) [Steinbach & Windisch, 2010], [Langer & Steinbach, 2003], [Hsiao, Steinbach • Couples adjacent only subdomains via & Wendland, 2000], ... • "mass terms" ( ⇒ Domain Decomposition ) Recently : • Global multi-trace [Claeys & Hiptmair, 2011]
Objective and literature To obtain well conditioned (or easily preconditionable) integral formulations, a natural idea consists in adapting what already exists for isolated scatterers. But usually orientation of interfaces is required for applying those techniques. On the other hand, the skeleton Γ is not a priori orientable. Local multi-trace [Jerez & Hiptmair, 2011] Already existing : • Rumsey/PMCHWT/EFIE = "Single-trace formulation" • Accuracy comparable to BETI and Rumsey • Boundary element tearing and interconnecting method (BETI) [Steinbach & Windisch, 2010], [Langer & Steinbach, 2003], [Hsiao, Steinbach • Couples adjacent only subdomains via & Wendland, 2000], ... • "mass terms" ( ⇒ Domain Decomposition ) Recently : • Difficult to analyse... • Global multi-trace [Claeys & Hiptmair, 2011] • Performance of DDM strategies?
Outline I. Recap of potential theory II. Local multi-trace : 2 subdomains III. Local multi-trace : many domains IV. Relation with Optimized Schwarz Methods (OSM) V. CEMRACS project Elasto Φ
Outline I. Recap of potential theory II. Local multi-trace : 2 subdomains III. Local multi-trace : many domains IV. Relation with Optimized Schwarz Methods (OSM) V. CEMRACS project Elasto Φ
Representation theorem H + 1 H − 1 2 ( ∂ Ω) × 2 ( ∂ Ω) H ( ∂ Ω) := Trace operators � � � � v | v | in t ext ∂ Ω ∂ Ω γ ( v ) := , γ c ( v ) := , n ∂ n v | ∂ n v | ext in t Ω ∂ Ω ∂ Ω { γ } := ( γ + γ c ) / 2, [ γ ] := γ − γ c . Potential operator : G κ ( x ) = exp ( ıκ | x | ) / ( 4 π | x | ) �� v �� � ( x ) := q ( y ) G κ ( x − y ) + v ( y ) n ( y ) · ( ∇ G κ )( x − y ) d σ ( y ) G κ q ∂ Ω H 1 (Ω) with ∆ u + κ 2 u = 0 in Ω (+ u outgoing) then Theorem : If u ∈ lo � u ( x ) si x ∈ Ω , � � γ ( u ) ( x ) = G κ si x ∈ R 3 \ Ω . 0
Calderón’s projector Cauchy data local to Ω : C κ ( ∂ Ω) := { γ ( u ) | ∆ u + κ 2 u = 0 outgoing ) } . in Ω (+ u If v is solution Helmholtz equation in Ω then γ ( v ) ∈ C κ ( ∂ Ω) = ⇒ γ ( v ) = γ · G κ ( γ ( v )) in Ω .
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