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

We want to solve a PDE in Ω using Boundary Integral Equations (BIE)

• Reformulation on ∂Ω using its fundamental solution
• Non-local integral operators (pseudo-difgerential operators)
• Dense matrices using Galerkin approximation

Figure 1: Mesh of a cavity

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Implementation

Some practical diffjculties

• Compression (H-matrices, FMM,SCSD,…)
• Parallelism and vectorization (MPI, OpenMP,…)

= ⇒ Htool library by P.-H. Tournier and P.M. (available on GitHub )

Figure 2: H-matrice for COBRA cavity

• free and
• pen-source
• ∼ 460 commits
• ∼ 6800 lines of C++

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Different points of view for DDM

Volume domain decomposition THEN boundary integral

• PMCHWT formulation
• Boundary Element Tearing and Interconnecting (BETI) method

→ boundary element counterpart of the FETI methods

• Multitrace formulation

→ the local variant is equivalent to Optimal Schwarz Method for particular parameters1

1Claeys, Dolean, and M. Gander 2019; Claeys and Marchand 2018.

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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 mesh2

• In our turn, we develop

GenEO-type preconditioners

2Hahne and Stephan 1996; Heuer 1996; Stephan 1996; Tran and Stephan 1996.

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Table of contents

• 1. Boundary Integral Equations
• 2. Domain Decomposition Methods
• 3. Preconditioners for BEM
• 4. Numerical simulations

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Boundary Integral Equations

Function spaces

Geometry

• Ω ⊂ Rd for d = 2 or d = 3, Lipschitz domain
• Γ ⊆ ∂Ω

Sobolev spaces

• H1/2(Γ) := {u|Γ | u ∈ H1/2(∂Ω)}

H1/2(Γ) := {u ∈ H1/2(∂Ω) | supp(u) ⊂ Γ}

• By duality:

H−1/2(Γ) := H1/2(Γ)∗ and H−1/2(Γ) := H1/2(Γ)∗ where H1/2(∂Ω) is defjned using local charts Associated norms

• 2

H1 2 2 L2

x y

2

x y d

1

d x y

• 2

H1 2

E

2 H1 2

where E is the extension by zero.

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Function spaces

Geometry

• Ω ⊂ Rd for d = 2 or d = 3, Lipschitz domain
• Γ ⊆ ∂Ω

Sobolev spaces

• H1/2(Γ) := {u|Γ | u ∈ H1/2(∂Ω)}

H1/2(Γ) := {u ∈ H1/2(∂Ω) | supp(u) ⊂ Γ}

• By duality:

H−1/2(Γ) := H1/2(Γ)∗ and H−1/2(Γ) := H1/2(Γ)∗ where H1/2(∂Ω) is defjned using local charts Associated norms

• ϕ2

H1/2(∂Ω) := ϕ2 L2(∂Ω) +

• ∂Ω×∂Ω

|ϕ(x) − ϕ(y)|2 |x − y|d+1 dσ(x, y)

• ϕ2
• H1/2(Γ) := EΓ(ϕ)2

H1/2(∂Ω)

where EΓ is the extension by zero.

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Boundary Integral Equations

Model problem

• L(u) = 0

in Ω ⊂ Rd + condition at infjnity if Ω is an unbounded domain L is a general linear, elliptic difgerential operator with constant coeffjcient and G the associated fundamental solution Fundamental solution L G

0 in d

Example of a fundamental solution Laplacian in

3:

G x 1 4 x for x

3 8/28

Boundary Integral Equations

Model problem

• L(u) = 0

in Ω ⊂ Rd + condition at infjnity if Ω is an unbounded domain L is a general linear, elliptic difgerential operator with constant coeffjcient and G the associated fundamental solution Fundamental solution L(G) = δ0 in Rd Example of a fundamental solution Laplacian in R3: G(x) := 1 4πx for x ∈ R3 \ {0}.

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Surface potentials

Single and double layer potential SL(q)(x) :=

• Γ

G(x − y)q(y) dσ(y), DL(v)(x) :=

• Γ

n(y) · (∇G)(x − y)v(y)dσ(y), with v ∈ H1/2(Γ), q ∈ H−1/2(Γ) and x ∈ Rd \ Γ. Properties

• L

q 0 and L v 0 in

d

• q and

v satisfy appropriate conditions at infjnity Dirichlet (resp. Neumann) problem

• Dirichlet data gD

H1 2 V q gD with V

D

• Neumann data gN

H

1 2

W v gN with W

N 9/28

Surface potentials

Single and double layer potential SL(q)(x) :=

• Γ

G(x − y)q(y) dσ(y), DL(v)(x) :=

• Γ

n(y) · (∇G)(x − y)v(y)dσ(y), with v ∈ H1/2(Γ), q ∈ H−1/2(Γ) and x ∈ Rd \ Γ. Properties

• L ◦ SL(q) = 0 and L ◦ DL(v) = 0 in Rd \ Γ
• SL(q) and DL(v) satisfy appropriate conditions at infjnity

Dirichlet (resp. Neumann) problem

• Dirichlet data gD

H1 2 V q gD with V

D

• Neumann data gN

H

1 2

W v gN with W

N 9/28

Surface potentials

Single and double layer potential SL(q)(x) :=

• Γ

G(x − y)q(y) dσ(y), DL(v)(x) :=

• Γ

n(y) · (∇G)(x − y)v(y)dσ(y), with v ∈ H1/2(Γ), q ∈ H−1/2(Γ) and x ∈ Rd \ Γ. Properties

• L ◦ SL(q) = 0 and L ◦ DL(v) = 0 in Rd \ Γ
• SL(q) and DL(v) satisfy appropriate conditions at infjnity

Dirichlet (resp. Neumann) problem

• Dirichlet data gD ∈ H1/2(Γ) =

⇒ V(q) = gD with V = γD ◦ SL

• Neumann data gN ∈ H−1/2(Γ) =

⇒ W(v) = gN with W = γN ◦ DL

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Considered problem

We want to solve a Boundary Integral Equation of the fjrst kind defjned on Γ.

• Variational formulation: fjnd u

Hs such that a u v f v H

s

Hs

v Hs where s 1 2 and f H

s

.

• Discretization using the Boundary Element Method (BEM): fjnd

uh

h

Hs such that a uh vh f vh H

s

Hs

vh

h

where

h i i

1 N . Hypothesis: a is symmetric positive defjnite

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Considered problem

We want to solve a Boundary Integral Equation of the fjrst kind defjned on Γ.

• Variational formulation: fjnd u ∈

Hs(Γ) such that a(u, v) = f, vH−s(Γ)×

Hs(Γ),

∀v ∈ Hs(Γ), where s = ±1/2 and f ∈ H−s(Γ).

• Discretization using the Boundary Element Method (BEM): fjnd

uh

h

Hs such that a uh vh f vh H

s

Hs

vh

h

where

h i i

1 N . Hypothesis: a is symmetric positive defjnite

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Considered problem

We want to solve a Boundary Integral Equation of the fjrst kind defjned on Γ.

• Variational formulation: fjnd u ∈

Hs(Γ) such that a(u, v) = f, vH−s(Γ)×

Hs(Γ),

∀v ∈ Hs(Γ), where s = ±1/2 and f ∈ H−s(Γ).

• Discretization using the Boundary Element Method (BEM): fjnd

uh ∈ Vh ⊂ Hs(Γ) such that a(uh, vh) = f, vhH−s(Γ)×

Hs(Γ),

∀vh ∈ Vh, where Vh = Span(ϕi, i = 1 . . . N). Hypothesis: a is symmetric positive defjnite

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Considered problem

We want to solve a Boundary Integral Equation of the fjrst kind defjned on Γ.

• Variational formulation: fjnd u ∈

Hs(Γ) such that a(u, v) = f, vH−s(Γ)×

Hs(Γ),

∀v ∈ Hs(Γ), where s = ±1/2 and f ∈ H−s(Γ).

• Discretization using the Boundary Element Method (BEM): fjnd

uh ∈ Vh ⊂ Hs(Γ) such that a(uh, vh) = f, vhH−s(Γ)×

Hs(Γ),

∀vh ∈ Vh, where Vh = Span(ϕi, i = 1 . . . N). Hypothesis: a is symmetric positive defjnite

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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:

V(q), ϕ =

• Γ
• Γ

1 4πx − yq(y)ϕ(x) dsy dsx

• Condition number for the linear system associated with the pre-

ceding bilinear form and obtained with fjnite element: κ(V) ≤ Ch−1.

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Context

Algebraic system Ahuh = f, with uh ∈ Rd Ah a dense matrix usually compressed (Fast Multipole Method, hierarchical matrices, Sparse Cardinal Sine Decomposition,…) Solvers

• Direct methods:

Factorisation can be stored for multi-rhs Expensive for dense matrices (complexity in O N3 ) Possibility to use LU decomposition

• Iterative methods:

Less intrusive Only matrix-vector products (O N2 or quasi linear complexity with compression) But ill-conditioned, especially when the mesh is refjned preconditioning techniques: PAhuh Pf

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Context

Algebraic system Ahuh = f, with uh ∈ Rd Ah a dense matrix usually compressed (Fast Multipole Method, hierarchical matrices, Sparse Cardinal Sine Decomposition,…) Solvers

• Direct methods:

Factorisation can be stored for multi-rhs Expensive for dense matrices (complexity in O(N3)) Possibility to use H − LU decomposition

• Iterative methods:

Less intrusive Only matrix-vector products (O(N2) or quasi linear complexity with compression) But ill-conditioned, especially when the mesh is refjned preconditioning techniques: PAhuh Pf

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Context

Algebraic system Ahuh = f, with uh ∈ Rd Ah a dense matrix usually compressed (Fast Multipole Method, hierarchical matrices, Sparse Cardinal Sine Decomposition,…) Solvers

• Direct methods:

Factorisation can be stored for multi-rhs Expensive for dense matrices (complexity in O(N3)) Possibility to use H − LU decomposition

• Iterative methods:

Less intrusive Only matrix-vector products (O(N2) or quasi linear complexity with compression) But ill-conditioned, especially when the mesh is refjned = ⇒ preconditioning techniques: PAhuh = Pf

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Domain Decomposition Methods

Example of decomposition

• Γ1

Γ1 ⊂ Γ1

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Example of decomposition

• Γ1

Γ1 ⊂ Γ1

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Example of decomposition

• Γ1

Γ1 ⊂ Γ1

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Example of decomposition

• Γ1

Γ1 ⊂ Γ1

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Example of decomposition

• Γ1

Γ1 ⊂ Γ1

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Notations

Subdomains

• dofh,p ⊂ {1, . . . , N}

Γp := ∪j∈dofh,p supp(ϕj)

• Γp :=

Γp \ ∪j/

∈dofh,p supp(ϕj) ⊂

Γp Decomposition

• Number of unknowns in the subdomain p: Np,
• Extension by zero: RT

p ∈ RN×Np,

• Restriction matrices: Rp
• Partition of unity: diagonal matrices Dp ∈ RNp×Np s.t.

n

• p=1

RT

pDpRp = Id. 14/28

Preconditioners for BEM

Additive Schwarz preconditioner

Additive Schwarz Preconditioner3 PASM = RT

0(R0AhRT 0)−1R0 + n

• p=1

RT

p(RpAhRT p)−1Rp

• Z = RT

0 ∈ RN×N0, an interpolation operator from the coarse space

to the fjnite element space

• The coarse space Vh,0 is spanned by the columns of Z

3Widlund and Dryja 1987.

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Additive Schwarz Preconditioner and Fictitious Space Lemma

Hypotheses of the Fictitious Space lemma4 (H1) n

p=0 RT pup h2 Ah ≤ cR

n

p=0RT pup h2 Ah

∀(up

h)n p=0 ∈ n p=0 CNp,

(H2) For uh ∈ Vh, how can we defjne (up

h)n p=0 ∈ n p=0 CNp s.t.

uh = n

p=0 RT pup h and

cT

n

• p=0

RT

pup h2 Ah ≤ uh2 Ah,

Result cond2(PASMAh) ≤ cR cT .

4Nepomnyaschikh 1992.

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Lemmas

Lemma (Sauter and Schwab 2011, Lemma 4.1.49 (b)) For (up)1≤p≤n ∈ n

p=1

H1/2( Γp), we have the following inequality:

• n
• p=1

E

Γp(up)

• 2
• H1/2(Γ)
• n
• p=1

up2

• H1/2(

Γp) .

Proof for (H1)

• n
• p=0

RT

pup h

• 2

Ah

RT

0u0 h2 Ah +

• n
• p=1

RT

pup h

• 2

Ah

• n

p=1

• RT

pup h

• 2

Ah

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Spectral Coarse Space: GenEO

• Assume there exists (Bp)n

p=1 ∈ (CNp×Np)n, s.t. n

• p=1

(BpRpuh, Rpuh) ≤ uh2

Ah

• n

p 0 RT pup h 2 Ah

uh 2

Ah n p 1 RT pup h 2 Ah

• Idea of the GenEO coarse space: a suffjcient condition is

AhRT

pup h RT pup h

BpRpuh Rpuh We introduce the following eigenvalue problem: fjnd

p k vp k s.t.

DpRpAhRT

pDpvp k p kBpvp k 18/28

Spectral Coarse Space: GenEO

• Assume there exists (Bp)n

p=1 ∈ (CNp×Np)n, s.t. n

• p=1

(BpRpuh, Rpuh) ≤ uh2

Ah

• n

p=0RT pup h2 Ah uh2 Ah + n p=1RT pup h2 Ah

• Idea of the GenEO coarse space: a suffjcient condition is

AhRT

pup h RT pup h

BpRpuh Rpuh We introduce the following eigenvalue problem: fjnd

p k vp k s.t.

DpRpAhRT

pDpvp k p kBpvp k 18/28

Spectral Coarse Space: GenEO

• Assume there exists (Bp)n

p=1 ∈ (CNp×Np)n, s.t. n

• p=1

(BpRpuh, Rpuh) ≤ uh2

Ah

• n

p=0RT pup h2 Ah uh2 Ah + n p=1RT pup h2 Ah

• Idea of the GenEO coarse space5: a suffjcient condition is

(AhRT

pup h, RT pup h) ≤ τ(BpRpuh, Rpuh).

We introduce the following eigenvalue problem: fjnd

p k vp k s.t.

DpRpAhRT

pDpvp k p kBpvp k

5Spillane et al. 2011, 2014.

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Spectral Coarse Space: GenEO

• Assume there exists (Bp)n

p=1 ∈ (CNp×Np)n, s.t. n

• p=1

(BpRpuh, Rpuh) ≤ uh2

Ah

• n

p=0RT pup h2 Ah uh2 Ah + n p=1RT pup h2 Ah

• Idea of the GenEO coarse space5: a suffjcient condition is

(AhRT

pup h, RT pup h) ≤ τ(BpRpuh, Rpuh).

We introduce the following eigenvalue problem: fjnd (λp

k, vp k) s.t.

DpRpAhRT

pDpvp k = λp kBpvp k,

5Spillane et al. 2011, 2014.

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Spectral Coarse Space: GenEO

We defjne Zp,τ = ker(Bp) ∪ Span(vp

k | λp k > τ), Πp, the projector on Zp,τ

and, Vh,0 = Span(RT

pDpvp h | 1 ≤ p ≤ N, vp h ∈ Zp,τ)

RT

0 = Zτ ∈ RN×N0 be a column matrix so that Vh,0 is spanned by its

columns and N0 = dim(Vh,0). u0

h

R0RT

1R0 n p 1

RT

pDp pRpu

and up

h

Dp Id

p Rpuh

1 p n

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Spectral Coarse Space: GenEO

We defjne Zp,τ = ker(Bp) ∪ Span(vp

k | λp k > τ), Πp, the projector on Zp,τ

and, Vh,0 = Span(RT

pDpvp h | 1 ≤ p ≤ N, vp h ∈ Zp,τ)

RT

0 = Zτ ∈ RN×N0 be a column matrix so that Vh,0 is spanned by its

columns and N0 = dim(Vh,0). u0

h = (R0RT 0)−1R0

 

n

• p=1

RT

pDpΠpRpu

  and up

h = Dp(Id − Πp)Rpuh,

∀1 ≤ p ≤ n

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Spectral Coarse Space: GenEO

= ⇒

n

• p=0

RT

pup h = uh

and

n

• p=0

RT

pup h2 Ah τuh2 Ah

Theorem With the previous coarse space, there exists C 0 independent of the meshsize and the number of subdomains such that

2 PASMAh

C

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Spectral Coarse Space: GenEO

= ⇒

n

• p=0

RT

pup h = uh

and

n

• p=0

RT

pup h2 Ah τuh2 Ah

Theorem With the previous coarse space, there exists CΓ > 0 independent of the meshsize and the number of subdomains such that cond2(PASMAh) < CΓτ.

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Concrete coarse spaces: three possibilities

For the hypersingular operator W (s = 1/2): (i) Continuous injection

n

• p=1

uh|Γp2

L2(Γp) uh2 L2(Γ) uh2

• H1/2(Γ) ≃ uh2

Ah,

(ii) Inverse inequality6:

n

• p=1

hT ∇u|Γp2

L2(Γp) hT ∇uh2 L2(Γ) uh2

• H1/2(Γ) ≃ uh2

Ah,

where T is the mesh and hT |T = |T|1/(d−1) for every mesh element T. (iii) H1/2 − localization

n

• p=1

V−1

p uh, uh H−1/2(Γp)×H1/2(Γp)

• (MpV−1

p Mpuh,uh)≤

?

• n
• p=1

uh|Γp2

H1/2(Γ) uh2

• H1/2(Γ) ≃ uh2

Ah,

6Aurada et al. 2017.

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Numerical simulations

2D test case

• Problem: −∆u + κ2u = 0 with γNu = 100 ∗ (x − 1.5)2
• Ah is compressed using a H-matrix
• Parameters: 24000 P1 elements, τ = 60
• Supercomputer: Occigen (64th in TOP500)
• Solution for κ = 1

Figure 4: Solution of −∆u + κ2u = 0 with γNu = 100 ∗ (x − 1.5)2

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Spectra

1 2 3 4 5 6 7 0.5 1 Number of the subdomain p |λp

k|

GenEO single layer 1 2 3 4 5 6 7 5 10 15 Number of the subdomain p |λp

k|

GenEO mass 1 2 3 4 5 6 7 2 4 6 Number of the subdomain p |λp

k|

GenEO stifgness 1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 Number of the subdomain p |λp

k|

GenEO Slobodeckij

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Strong scaling 2D

20 40 60 80 100 120 140 20 40 60 80 100 120 Number of subdomains Number of iterations Strong scaling in 2D with CG Stifgness Slobodeckij Single layer One level 20 40 60 80 100 120 140 20 40 60 80 Number of subdomains Number of iterations Strong scaling in 2D with GMRes Stifgness Slobodeckij Single layer One level

Figure 5: Number of iterations in function of the number of subdomain with CG (right) and GMRes (left) for κ = 0.1

Remark: 656 iterations without preconditioner for CG and 450 for GMRes.

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Strong scaling 2D

20 40 60 80 100 120 140 2 4 6 8 10 Number of subdomains Mean local contribution Strong scaling in 2D 20 40 60 80 100 120 140 100 150 200 250 Number of subdomains Size of the coarse space Strong scaling in 2D

Figure 6: Mean local contribution (left) and size of the coarse space (right) in function of the number of subdomains for κ = 0.1

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Implementation

C++ libraries available on GitHub :

• https://github.com/xclaeys/BemTool by X. Claeys,
• BEM library for Laplace, Helmholtz, Yukawa and Maxwell

matrices in 2D and 3D.

• https://github.com/PierreMarchand20/htool by P.-H.

Tournier and P. M.,

• Hierarchical matrices (H−matrices),
• Parallelized assembly, H−matrice/vector and

H−matrice/matrice products using MPI and OpenMP,

• DDM preconditioners with HPDDMs.
• https://github.com/hpddm/hpddm by P. Jolivet and F.

Nataf,

• Various iterative solvers and block solvers,
• DDM preconditioners,
• Support for dense/compressed matrices recently added.

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Summary

• Theoretical results about a new coarse space for DDM

preconditioner applied to BEM matrices for the hypersingular

• perator
• Implementation available using Htool and HPDDM (which can be

interfaced with your own code) = ⇒ article to be submitted and Htool integrated in Freefem++

27/28

Outlook

• Numerical approximation of the fractional Laplacian7
• Other coarse spaces (SHEM)8: work with G. Ciaramella from

University of Konstanz

• Bonus: try block solvers for problems with multiple right-hand

side

Thank you for your attention!

7Ainsworth and Glusa 2018; Bonito et al. 2018.

• 8M. J. Gander, Loneland, and Rahman 2015.

28/28

Outlook

• Numerical approximation of the fractional Laplacian7
• Other coarse spaces (SHEM)8: work with G. Ciaramella from

University of Konstanz

• Bonus: try block solvers for problems with multiple right-hand

side

Thank you for your attention!

7Ainsworth and Glusa 2018; Bonito et al. 2018.

• 8M. J. Gander, Loneland, and Rahman 2015.

28/28

Weak scaling 2D

10 20 30 40 50 60 70 20 40 60 80 Number of subdomains Number of iterations Weak scaling in 2D with CG Stifgness Slobodeckij Single layer One level 10 20 30 40 50 60 70 20 40 60 Number of subdomains Number of iterations Weak scaling in 2D with GMRes Stifgness Slobodeckij Single layer One level

Figure 7: Number of iterations in function of the number of subdomain with CG (right) and GMRes (left) for κ = 0.1

Remark: We have 750 degrees of freedom for each subdomain.

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Weak scaling 2D

10 20 30 40 50 60 70 3.2 3.4 3.6 3.8 4 Number of subdomains Mean local contribution Weak scaling in 2D with CG 10 20 30 40 50 60 70 50 100 150 200 Number of subdomains Size of the coarse space Weak scaling in 2D with CG

Figure 8: Mean local contribution (left) and size of the coarse space (right) in function of the number of subdomains for κ = 0.1

28/28

References

References i

Bibliography Ainsworth, Mark and Christian Glusa (2018). “Towards an Effjcient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains”. In: Contemporary Computational Mathematics

• A Celebration of the 80th Birthday of Ian Sloan. Springer

International Publishing, pp. 17–57. doi: 10.1007/978-3-319-72456-0_2. Aurada, M. et al. (Apr. 2017). “Local inverse estimates for non-local boundary integral operators”. In: Mathematics of Computation 86.308, pp. 2651–2686. doi: 10.1090/mcom/3175. Bonito, Andrea et al. (Mar. 2018). “Numerical methods for fractional difgusion”. In: Computing and Visualization in Science 19.5-6,

• pp. 19–46. doi: 10.1007/s00791-018-0289-y.

References ii

Claeys, Xavier, V. Dolean, and M.J. Gander (Jan. 2019). “An introduction to multi-trace formulations and associated domain decomposition solvers”. In: Applied Numerical Mathematics 135, pp. 69–86. doi: 10.1016/j.apnum.2018.07.006. Claeys, Xavier and Pierre Marchand (Nov. 2018). “Boundary integral multi-trace formulations and Optimised Schwarz Methods”. working paper or preprint. url: https://hal.inria.fr/hal-01921113. Gander, Martin J., Atle Loneland, and Talal Rahman (Dec. 16, 2015). “Analysis of a New Harmonically Enriched Multiscale Coarse Space for Domain Decomposition Methods”. In: arXiv: http://arxiv.org/abs/1512.05285v1 [math.NA].

References iii

Hahne, Manfred and Ernst P Stephan (Mar. 1, 1996). “Schwarz iterations for the effjcient solution of screen problems with boundary elements”. In: Computing 56, pp. 61–85. issn: 1436-5057. doi: 10.1007/BF02238292. url: http://dx.doi.org/10.1007/BF02238292. Heuer, Norbert (Sept. 1996). “Effjcient Algorithms for the p-Version of the Boundary Element Method”. In: Journal of Integral Equations and Applications 8.3, pp. 337–360. doi: 10.1216/jiea/1181075956. url: http://dx.doi.org/10.1216/jiea/1181075956.

References iv

Nepomnyaschikh, S. V. (1992). “Decomposition and fjctitious domains methods for elliptic boundary value problems”. In: Fifth International Symposium on Domain Decomposition Methods for Partial Difgerential Equations. Philadelphia, PA: Society for Industrial and Applied Mathematics, pp. 62–72. Sauter, Stefan A. and Christoph Schwab (2011). Boundary element

• methods. Vol. 39. Springer Series in Computational Mathematics.

Translated and expanded from the 2004 German original. Springer-Verlag, Berlin, pp. xviii+561. isbn: 978-3-540-68092-5. doi: 10.1007/978-3-540-68093-2. url: http://dx.doi.org/10.1007/978-3-540-68093-2.

References v

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