Robust coarse spaces for the boundary element method Xavier Claeys, - PowerPoint PPT Presentation

Robust coarse spaces for the boundary element method Xavier Claeys, Pierre Marchand, Frédéric Nataf September 17, 2019 — CIRM Team-projet Alpines, Inria Laboratoire J.-L. Lions, Sorbonne Université ANR project NonlocalDD 1/28

Introduction

Boundary Integral Equation Boundary Integral Equations (BIE) • Non-local integral operators (pseudo-difgerential operators) • Dense matrices using Galerkin approximation Figure 1: Mesh of a cavity 2/28 We want to solve a PDE in Ω using • Reformulation on ∂ Ω using its fundamental solution

Implementation Some practical diffjculties • Parallelism and vectorization (MPI, OpenMP,…) • free and open-source 3/28 • Compression ( H -matrices, FMM,SCSD,…) = ⇒ Htool library by P.-H. Tournier and P.M. (available on GitHub � ) • ∼ 460 commits • ∼ 6800 lines of C++ Figure 2: H -matrice for COBRA cavity

Different points of view for DDM Volume domain decomposition THEN boundary integral • PMCHWT formulation • Boundary Element Tearing and Interconnecting (BETI) method • Multitrace formulation particular parameters 1 1 Claeys, Dolean, and M. Gander 2019; Claeys and Marchand 2018. 4/28 → boundary element counterpart of the FETI methods → the local variant is equivalent to Optimal Schwarz Method for

Different points of view for DDM Boundary integral formulation THEN surface domain decomposition: Additive Schwarz Method (ASM). Figure 3: Surface decomposition for COBRA cavity • Two-level Schwarz preconditioners with coarse mesh 2 • In our turn, we develop GenEO-type preconditioners 2 Hahne and Stephan 1996; Heuer 1996; Stephan 1996; Tran and Stephan 1996. 5/28

Table of contents 1. Boundary Integral Equations 2. Domain Decomposition Methods 3. Preconditioners for BEM 4. Numerical simulations 6/28

Boundary Integral Equations

y d Function spaces x where E is the extension by zero. H 1 2 2 E H 1 2 2 • x y d 1 x 2 y L 2 Geometry Associated norms Sobolev spaces 2 7/28 • 2 H 1 2 • Ω ⊂ R d for d = 2 or d = 3, Lipschitz domain • Γ ⊆ ∂ Ω • H 1 / 2 (Γ) := { u | Γ | u ∈ H 1 / 2 ( ∂ Ω) } • � H 1 / 2 (Γ) := { u ∈ H 1 / 2 ( ∂ Ω) | supp( u ) ⊂ Γ } • By duality: � H − 1 / 2 (Γ) := H 1 / 2 (Γ) ∗ and H − 1 / 2 (Γ) := � H 1 / 2 (Γ) ∗ where H 1 / 2 ( ∂ Ω) is defjned using local charts

Function spaces Associated norms Geometry 7/28 Sobolev spaces • Ω ⊂ R d for d = 2 or d = 3, Lipschitz domain • Γ ⊆ ∂ Ω • H 1 / 2 (Γ) := { u | Γ | u ∈ H 1 / 2 ( ∂ Ω) } • � H 1 / 2 (Γ) := { u ∈ H 1 / 2 ( ∂ Ω) | supp( u ) ⊂ Γ } • By duality: � H − 1 / 2 (Γ) := H 1 / 2 (Γ) ∗ and H − 1 / 2 (Γ) := � H 1 / 2 (Γ) ∗ where H 1 / 2 ( ∂ Ω) is defjned using local charts � | ϕ ( x ) − ϕ ( y ) | 2 • � ϕ � 2 H 1 / 2 ( ∂ Ω) := � ϕ � 2 L 2 ( ∂ Ω) + d σ ( x , y ) ∂ Ω × ∂ Ω | x − y | d + 1 • � ϕ � 2 H 1 / 2 (Γ) := � E Γ ( ϕ ) � 2 � H 1 / 2 ( ∂ Ω) where E Γ is the extension by zero.

0 in Boundary Integral Equations Laplacian in 0 3 for x x 4 1 G x 3 : d Example of a fundamental solution Model problem L G Fundamental solution coeffjcient and G the associated fundamental solution L is a general linear, elliptic difgerential operator with constant 8/28 � L ( u ) = 0 in Ω ⊂ R d + condition at infjnity if Ω is an unbounded domain

Boundary Integral Equations coeffjcient and G the associated fundamental solution 1 Example of a fundamental solution Model problem Fundamental solution 8/28 L is a general linear, elliptic difgerential operator with constant � L ( u ) = 0 in Ω ⊂ R d + condition at infjnity if Ω is an unbounded domain L ( G ) = δ 0 in R d Laplacian in R 3 : for x ∈ R 3 \ { 0 } . G ( x ) := 4 π � x �

g D with V g N with W Surface potentials q and v satisfy appropriate conditions at infjnity Dirichlet (resp. Neumann) problem • Dirichlet data g D H 1 2 V q D d • Neumann data g N H 1 2 W v N • 0 in Single and double layer potential Properties v 9/28 • L q 0 and L � SL( q )( x ) := G ( x − y ) q ( y ) d σ ( y ) , Γ � DL( v )( x ) := n ( y ) · ( ∇ G )( x − y ) v ( y ) d σ ( y ) , Γ H − 1 / 2 (Γ) and x ∈ R d \ Γ . with v ∈ � H 1 / 2 (Γ) , q ∈ �

g D with V g N with W Surface potentials Dirichlet (resp. Neumann) problem N W v 1 2 H • Neumann data g N D V q H 1 2 • Dirichlet data g D 9/28 Single and double layer potential Properties � SL( q )( x ) := G ( x − y ) q ( y ) d σ ( y ) , Γ � DL( v )( x ) := n ( y ) · ( ∇ G )( x − y ) v ( y ) d σ ( y ) , Γ H − 1 / 2 (Γ) and x ∈ R d \ Γ . with v ∈ � H 1 / 2 (Γ) , q ∈ � • L ◦ SL( q ) = 0 and L ◦ DL( v ) = 0 in R d \ Γ • SL( q ) and DL( v ) satisfy appropriate conditions at infjnity

Surface potentials Single and double layer potential Dirichlet (resp. Neumann) problem Properties 9/28 � SL( q )( x ) := G ( x − y ) q ( y ) d σ ( y ) , Γ � DL( v )( x ) := n ( y ) · ( ∇ G )( x − y ) v ( y ) d σ ( y ) , Γ H − 1 / 2 (Γ) and x ∈ R d \ Γ . with v ∈ � H 1 / 2 (Γ) , q ∈ � • L ◦ SL( q ) = 0 and L ◦ DL( v ) = 0 in R d \ Γ • SL( q ) and DL( v ) satisfy appropriate conditions at infjnity • Dirichlet data g D ∈ H 1 / 2 (Γ) = ⇒ V ( q ) = g D with V = γ D ◦ SL • Neumann data g N ∈ H − 1 / 2 (Γ) = ⇒ W ( v ) = g N with W = γ N ◦ DL

f v H a u h v h f v h H i i Considered problem v h such that s H s where h h h 1 N . Hypothesis: a is symmetric positive defjnite H s • Discretization using the Boundary Element Method (BEM): fjnd u h H s • Variational formulation: fjnd u H s such that a u v s v We want to solve a Boundary Integral Equation of the fjrst kind H s where s 1 2 and f H s . 10/28 defjned on Γ .

a u h v h f v h H i i Considered problem Hypothesis: a is symmetric positive defjnite N . 1 h where h v h H s s such that We want to solve a Boundary Integral Equation of the fjrst kind H s h u h • Discretization using the Boundary Element Method (BEM): fjnd 10/28 defjned on Γ . • Variational formulation: fjnd u ∈ � H s (Γ) such that ∀ v ∈ � a ( u , v ) = � f , v � H − s (Γ) × � H s (Γ) , H s (Γ) , where s = ± 1 / 2 and f ∈ H − s (Γ) .

Considered problem We want to solve a Boundary Integral Equation of the fjrst kind Hypothesis: a is symmetric positive defjnite • Discretization using the Boundary Element Method (BEM): fjnd 10/28 defjned on Γ . • Variational formulation: fjnd u ∈ � H s (Γ) such that ∀ v ∈ � a ( u , v ) = � f , v � H − s (Γ) × � H s (Γ) , H s (Γ) , where s = ± 1 / 2 and f ∈ H − s (Γ) . u h ∈ V h ⊂ � H s (Γ) such that a ( u h , v h ) = � f , v h � H − s (Γ) × � H s (Γ) , ∀ v h ∈ V h , where V h = Span( ϕ i , i = 1 . . . N ) .

Considered problem We want to solve a Boundary Integral Equation of the fjrst kind Hypothesis: a is symmetric positive defjnite • Discretization using the Boundary Element Method (BEM): fjnd 10/28 defjned on Γ . • Variational formulation: fjnd u ∈ � H s (Γ) such that ∀ v ∈ � a ( u , v ) = � f , v � H − s (Γ) × � H s (Γ) , H s (Γ) , where s = ± 1 / 2 and f ∈ H − s (Γ) . u h ∈ V h ⊂ � H s (Γ) such that a ( u h , v h ) = � f , v h � H − s (Γ) × � H s (Γ) , ∀ v h ∈ V h , where V h = Span( ϕ i , i = 1 . . . N ) .

Considered problem Remarks • Laplace equation on screens, Laplace equation with Dirichlet con- ditions on closed surface, Modifjed Helmholtz… • Example of analytical expression for Laplacian in 3D: 1 • Condition number for the linear system associated with the pre- ceding bilinear form and obtained with fjnite element: 11/28 � � � V ( q ) , ϕ � = 4 π � x − y � q ( y ) ϕ ( x ) d s y d s x Γ Γ κ ( V ) ≤ Ch − 1 .

� Expensive for dense matrices (complexity in O N 3 ) � Only matrix-vector products ( O N 2 or quasi linear Context LU decomposition Pf preconditioning techniques: PA h u h � But ill-conditioned, especially when the mesh is refjned complexity with compression) � Less intrusive • Iterative methods: � Possibility to use Algebraic system � Factorisation can be stored for multi-rhs • Direct methods: Solvers hierarchical matrices, Sparse Cardinal Sine Decomposition,…) 12/28 A h u h = f , with u h ∈ R d A h a dense matrix usually compressed (Fast Multipole Method,

Context Algebraic system Pf preconditioning techniques: PA h u h � But ill-conditioned, especially when the mesh is refjned complexity with compression) � Less intrusive • Iterative methods: 12/28 � Factorisation can be stored for multi-rhs • Direct methods: Solvers hierarchical matrices, Sparse Cardinal Sine Decomposition,…) A h u h = f , with u h ∈ R d A h a dense matrix usually compressed (Fast Multipole Method, � Expensive for dense matrices (complexity in O ( N 3 ) ) � Possibility to use H − LU decomposition � Only matrix-vector products ( O ( N 2 ) or quasi linear