SLIDE 1 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.
SLIDE 2 Multi-domain scattering
u
in
n2 n0 n3 n1 Ω0= exterieur Ω1 Ω2 Ω3 Notations Rd = ∪n
j=0Ωj,
Ωj ∩ Ωk = ∅ for j = k Γ = ∪n
j=0Γj, Γj := ∂Ωj
κj ∈ C effective wave number in Ωj, Transmission problem (well posed) :
Find u ∈ H1 lo (∆, Ωj) su h that
∆u + κ2
j u = 0
in
Ωj, j = 0, . . . n u − u
in
in Ω0 ,
u|Γj − u|Γk = 0 ,
Concern : Solution by means of boundary integral equation methods?
SLIDE 3
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],...
SLIDE 4 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], ...
SLIDE 5 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], ... Calderón technique not possible
SLIDE 6 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], ... Calderón technique not possible Recently :
- Global multi-trace [Claeys & Hiptmair, 2011]
- Local multi-trace [Jerez & Hiptmair, 2011]
SLIDE 7 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], ... Calderón technique not possible Calderón available Recently :
- Global multi-trace [Claeys & Hiptmair, 2011]
- Local multi-trace [Jerez & Hiptmair, 2011]
SLIDE 8 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]
SLIDE 9 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]
SLIDE 10 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]
SLIDE 11 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]
- Accuracy comparable to BETI and Rumsey
- Couples adjacent only subdomains via
- "mass terms" ( ⇒ Domain Decomposition)
SLIDE 12 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]
- Accuracy comparable to BETI and Rumsey
- Couples adjacent only subdomains via
- "mass terms" ( ⇒ Domain Decomposition)
- Difficult to analyse...
- Performance of DDM strategies?
SLIDE 13 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 ElastoΦ
SLIDE 14 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 ElastoΦ
SLIDE 15 Representation theorem
Ω n H(∂Ω) :=
H+ 1
2 (∂Ω) ×
H− 1
2 (∂Ω)
Trace operators γ(v) :=
in t
∂Ω
∂nv|
in t
∂Ω
γc(v) :=
ext
∂Ω
∂nv|
ext
∂Ω
{γ} := (γ + γc)/2, [γ] := γ − γc . Potential operator : Gκ(x) = exp(ıκ|x|)/(4π|x|)
Gκ
v q
q(y) Gκ(x − y) + v(y) n(y) · (∇Gκ)(x − y) dσ(y) Theorem : If u ∈
H1 lo (Ω) with ∆u + κ2u = 0 in Ω (+ u outgoing) then Gκ
u(x)
si x ∈ Ω, si x ∈ R3 \ Ω.
SLIDE 16 Calderón’s projector
Cauchy data local to Ω : Cκ(∂Ω) := { γ(u) | ∆u + κ2u = 0
in Ω (+u
If v is solution Helmholtz equation in Ω then γ(v) ∈ Cκ(∂Ω) = ⇒ γ(v) = γ· Gκ(γ(v))
in Ω.
SLIDE 17 Calderón’s projector
Cauchy data local to Ω : Cκ(∂Ω) := { γ(u) | ∆u + κ2u = 0
in Ω (+u
If v is solution Helmholtz equation in Ω then γ(v) ∈ Cκ(∂Ω) = ⇒ γ(v) = γ· Gκ(γ(v))
in Ω.
SLIDE 18 Calderón’s projector
Cauchy data local to Ω : Cκ(∂Ω) := { γ(u) | ∆u + κ2u = 0
in Ω (+u
If v is solution Helmholtz equation in Ω then V ∈ Cκ(∂Ω) = ⇒ V = γ·
Gκ(V) in Ω.
SLIDE 19 Calderón’s projector
Cauchy data local to Ω : Cκ(∂Ω) := { γ(u) | ∆u + κ2u = 0
in Ω (+u
Characterisation of Cauchy data The operator γ ·
Gκ : H(∂Ω) → Cκ(∂Ω) ⊂ H(∂Ω) is a continuous projector
called Calderón’s projector interior to Ω. We have V ∈ Cκ(∂Ω) ⇐ ⇒ V = γ ·
Gκ(V)
Jump formula : [γ] ·
Gκ = Id,
Calderón’s identity : ( Aκ)2 =
Id
avec
Aκ := 2{γ} · Gκ.
SLIDE 20 Calderón’s projector
Cauchy data local to Ω : Cκ(∂Ω) := { γ(u) | ∆u + κ2u = 0
in Ω (+u
γ = {γ} + 1
2[γ]
{γ} ·
Gκ(V) + 1
2[γ] ·
Gκ(V)
Characterisation of Cauchy data The operator γ ·
Gκ : H(∂Ω) → Cκ(∂Ω) ⊂ H(∂Ω) is a continuous projector
called Calderón’s projector interior to Ω. We have V ∈ Cκ(∂Ω) ⇐ ⇒ V = γ ·
Gκ(V)
Jump formula : [γ] ·
Gκ = Id,
Calderón’s identity : ( Aκ)2 =
Id
avec
Aκ := 2{γ} · Gκ.
SLIDE 21 Calderón’s projector
Cauchy data local to Ω : Cκ(∂Ω) := { γ(u) | ∆u + κ2u = 0
in Ω (+u
γ = {γ} + 1
2[γ]
{γ} ·
Gκ(V) + 1
2[γ] ·
Gκ(V)
Characterisation of Cauchy data The operator γ ·
Gκ : H(∂Ω) → Cκ(∂Ω) ⊂ H(∂Ω) is a continuous projector
called Calderón’s projector interior to Ω. We have V ∈ Cκ(∂Ω) ⇐ ⇒ V = γ ·
Gκ(V)
Jump formula : [γ] ·
Gκ = Id,
Calderón’s identity : ( Aκ)2 =
Id
avec
Aκ := 2{γ} · Gκ.
SLIDE 22 Calderón’s projector
Cauchy data local to Ω : Cκ(∂Ω) := { γ(u) | ∆u + κ2u = 0
in Ω (+u
1 2( Aκ +
Id)V
Characterisation of Cauchy data The operator γ ·
Gκ : H(∂Ω) → Cκ(∂Ω) ⊂ H(∂Ω) is a continuous projector
called Calderón’s projector interior to Ω. We have V ∈ Cκ(∂Ω) ⇐ ⇒ V = γ ·
Gκ(V)
Jump formula : [γ] ·
Gκ = Id,
Calderón’s identity : ( Aκ)2 =
Id
avec
Aκ := 2{γ} · Gκ.
SLIDE 23 Calderón’s projector
Cauchy data local to Ω : Cκ(∂Ω) := { γ(u) | ∆u + κ2u = 0
in Ω (+u
Characterisation of Cauchy data The operator γ ·
Gκ : H(∂Ω) → Cκ(∂Ω) ⊂ H(∂Ω) is a continuous projector
called Calderón’s projector interior to Ω. We have V ∈ Cκ(∂Ω) ⇐ ⇒ V = ( Aκ +
Id)V/2
Jump formula : [γ] ·
Gκ = Id,
Calderón’s identity : ( Aκ)2 =
Id
avec
Aκ := 2{γ} · Gκ.
SLIDE 24 Calderón’s projector
Cauchy data local to Ω : Cκ(∂Ω) := { γ(u) | ∆u + κ2u = 0
in Ω (+u
Characterisation of Cauchy data The operator γ ·
Gκ : H(∂Ω) → Cκ(∂Ω) ⊂ H(∂Ω) is a continuous projector
called Calderón’s projector interior to Ω. We have V ∈ Cκ(∂Ω) ⇐ ⇒ 0 = ( Aκ −
Id)V
Jump formula : [γ] ·
Gκ = Id,
Calderón’s identity : ( Aκ)2 =
Id
avec
Aκ := 2{γ} · Gκ.
SLIDE 25 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 ElastoΦ
SLIDE 26 Reformulation of the problem
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1
Aj
κj = operator associated to Ωj
Transmission conditions u0|Γ = +u1|Γ ∂n0u0|Γ = −∂n1u1|Γ Wave equations ∆u1 + κ2
1u1 = 0
dans Ω1
∆u0 + κ2
0u0 = 0
dans Ω0
u0 − u
in sortan t
SLIDE 27 Reformulation of the problem
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1
Aj
κj = operator associated to Ωj
Transmission conditions u0|Γ = +u1|Γ ∂n0u0|Γ = −∂n1u1|Γ Wave equations ∆u1 + κ2
1u1 = 0
dans Ω1
∆u0 + κ2
0u0 = 0
dans Ω0
u0 − u
in sortan t
SLIDE 28 Reformulation of the problem
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1
Aj
κj = operator associated to Ωj
Transmission conditions
∂n0u0|Γ
+1 −1
∂n1u1|Γ
∆u1 + κ2
1u1 = 0
dans Ω1
∆u0 + κ2
0u0 = 0
dans Ω0
u0 − u
in sortan t
SLIDE 29 Reformulation of the problem
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1 . .
. .
. .
Aj
κj = operator associated to Ωj
Transmission conditions
∂n0u0|Γ
+1 −1
∂n1u1|Γ
∆u1 + κ2
1u1 = 0
dans Ω1
∆u0 + κ2
0u0 = 0
dans Ω0
u0 − u
in sortan t
SLIDE 30 Reformulation of the problem
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1 . .
. .
. .
⇐ ⇒ ( A1
κ1 −
Id)U1 = 0
( A0
κ0 −
Id)U0 = F in Aj
κj = operator associated to Ωj
Transmission conditions
∂n0u0|Γ
+1 −1
∂n1u1|Γ
∆u1 + κ2
1u1 = 0
dans Ω1
∆u0 + κ2
0u0 = 0
dans Ω0
u0 − u
in sortan t
SLIDE 31 Reformulation of the problem
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1 . .
. .
. .
Aj
κj = operator associated to Ωj
Transmission conditions
∂n0u0|Γ
+1 −1
∂n1u1|Γ
κ0 −
Id A1
κ1 −
Id
U0 U1
in
SLIDE 32 Reformulation of the problem
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1
Aj
κj = operator associated to Ωj
Transmission conditions (∗) −Q·U0 − Q · U1 = 0 −Q · (∗) −Q · U0 + Q·U1 = 0 Wave equations
κ0 −
Id A1
κ1 −
Id
U0 U1
in
SLIDE 33 Reformulation of the problem
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1
Aj
κj = operator associated to Ωj
Transmission conditions (∗) −Q·U0 − Q · U1 = 0 −Q · (∗) −Q · U0 + Q·U1 = 0 Wave equations
κ0 −
Id A1
κ1 −
Id
U0 U1
in
SLIDE 34 Reformulation of the problem
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1
Aj
κj = operator associated to Ωj
Transmission conditions
−Q −Q
Id
U0 U1
κ0 −
Id A1
κ1 −
Id
U0 U1
in
SLIDE 35 Reformulation of the problem
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1
Aj
κj = operator associated to Ωj
Transmission conditions
Id
Q Q
U0 U1
κ0
A1
κ1
Id
U0 U1
in
SLIDE 36 Reformulation of the problem
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1 . .
. .
. . := U . . := F
Aj
κj = operator associated to Ωj
Transmission conditions
Id
Q Q
U0 U1
κ0
A1
κ1
Id
U0 U1
in
- Local multi-trace formulation, with α ∈ C \ {0}
[ (A −
Id) + α ( Id − Π) ] · U = F
SLIDE 37 Reformulation of the problem
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1 Local multi-trace formulation [ (A −
Id) + α ( Id − Π) ] · U = F
(#) Remarks
- Only the case α = 1 has been analyzed in [Jerez & Hiptmair, 2011] that
considered geometries with junction points and proved that (#) admits a unique solution (without establishing Garding inequality...)
- The case of α arbitrary has been proposed in the DDXXI proceedings
[Jerez,Hiptmair,Lee,Peng, 2012] dedicated to domain decomposition, for a slightly different boundary value problem.
SLIDE 38 Spectrum of the multi-trace operator ?
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1 Local multi-trace formulation [ (A −
Id) + α ( Id − Π) ] · U = F
(#) Lemma : Π2 =
Id
Q Q
Q Q
Q2 =
Id
Q2 =
Id
- Lemma : AΠ + ΠA = 0 si κ0 = κ1
A · Π =
κ0
A1
κ1
Q Q
κ0Q
A1
κ1Q
A1
κ1
−Q
A0
κ0
−Q
κ0
A1
κ1
SLIDE 39 Spectrum of the multi-trace operator ?
n0 n1 Ω1 Ω0 Γ := ∂Ω0 = ∂Ω1 = 0 Local multi-trace formulation [ (A −
Id) + α ( Id − Π) ] · U = F
(#) Lemma : i) Π2 =
Id
ii) AΠ + ΠA = 0, si κ0 = κ1 S( (A −
Id) + α ( Id − Π) ) = S(A − αΠ) − 1 + α
(A − αΠ)2 = A2 + α2Π2 − α(AΠ + ΠA) = (1 + α2) Id Result : if κ0 = κ1 S( (A −
Id) + α ( Id − Π) ) = −1 + α ±
√ 1 + α2 = ⇒
SLIDE 40 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 ElastoΦ
SLIDE 41 Back to the general case...
n2 n0 n3 n1 Ω0 Ω1 Ω2 Ω3 Transmission problem
Find u ∈ H1 lo (∆, Ωj) tel que
∆u + κ2
j u = 0
dans
Ωj, j = 0, . . . n u − u
in
in Ω0 ,
u|Γj − u|Γk = 0 , ∂nju|Γj + ∂nku|Γk = 0 ,
Γj ∩ Γk, ∀j, k
.
Notations γj =
d
γj
n
nj = normal to Γj. Additional assumption : . no junction point
SLIDE 42 Back to the general case...
Ω3 Ω2 Ω1 Ω0 Transmission problem
Find u ∈ H1 lo (∆, Ωj) tel que
∆u + κ2
j u = 0
dans
Ωj, j = 0, . . . n u − u
in
in Ω0 ,
u|Γj − u|Γk = 0 , ∂nju|Γj + ∂nku|Γk = 0 ,
Γj ∩ Γk, ∀j, k
.
Notations γj =
d
γj
n
nj = normal to Γj. Additional assumption : . no junction point
SLIDE 43 Multi/single-trace space
Multi-trace space : H(Γ) := H(Γ0) × · · · × H(Γn) avec H(Γj) =
H
1 2 (Γj) ×
H− 1
2 (Γj).
Duality on H(Γ) : U, V =
n
uj pj
vj qj
Γj
=
n
ujqj − pjvjdσ,
SLIDE 44 Multi/single-trace space
Multi-trace space : H(Γ) := H(Γ0) × · · · × H(Γn) avec H(Γj) =
H
1 2 (Γj) ×
H− 1
2 (Γj).
Duality on H(Γ) : U, V =
n
uj pj
vj qj
Γj
=
n
ujqj − pjvjdσ, Single-trace space : X(Γ) =
adh
(γj(v))j=0...n
H1(Rd), ∆v ∈ L2(Rd)
for H(Γ) X(Γ) = elements of H(Γ) satisfying the transmission conditions.
SLIDE 45 Multi/single-trace space
Multi-trace space : H(Γ) := H(Γ0) × · · · × H(Γn) avec H(Γj) =
H
1 2 (Γj) ×
H− 1
2 (Γj).
Duality on H(Γ) : U, V =
n
uj pj
vj qj
Γj
=
n
ujqj − pjvjdσ, Single-trace space : X(Γ) =
adh
(γj(v))j=0...n
H1(Rd), ∆v ∈ L2(Rd)
for H(Γ) X(Γ) = elements of H(Γ) satisfying the transmission conditions. Lemma : For U ∈ H(Γ), we have : U ∈ X(Γ) ⇐ ⇒ U, V = 0 ∀V ∈ X(Γ).
SLIDE 46 Transmission operator
For U = (uj, pj)n
j=0, V = (vj, qj)n j=0 ∈ H(Γ) set
Π(U) = V ⇐ ⇒ vj = uk qj = −pk
sur Γj ∩ Γk
Elementary properties i) Π2 =
Id,
ii) Π : H(Γ) → H(Γ) continuous (false with jonctions...), iii) U ∈ X(Γ) ⇐ ⇒ Π(U) = U, iv) Π(U), V = Π(V), U ∀U, V ∈ H(Γ).
SLIDE 47 Reformulation of the problem
⇐ ⇒ ( Id − Π)U = 0 (∗) ⇐ ⇒ ( A0
κ0 −
Id)(U0 − U in
0 ) = 0
⇐ ⇒ ( A1
κ1 −
Id)U1 = 0
. . . . . ⇐ ⇒ ( An
κn −
Id)Un = 0
Id)U = F
(∗∗) with A =
diag
j=0...n
{
Aj
κj }
u|Γj − u|Γk = 0 , ∂nju|Γj + ∂nku|Γk = 0
,
Γj ∩ Γk, ∀j, k
.
Find u ∈ H1 lo (∆, Ωj) su h that
∆u + κ2
j u = 0
in
Ωj, j = 0, . . . n u − u
in
in Ω0 ,
SLIDE 48 Reformulation of the problem
⇐ ⇒ ( Id − Π)U = 0 (∗) ⇐ ⇒ ( A0
κ0 −
Id)(U0 − U in
0 ) = 0
⇐ ⇒ ( A1
κ1 −
Id)U1 = 0
. . . . . ⇐ ⇒ ( An
κn −
Id)Un = 0
Id)U = F
(∗∗) with A =
diag
j=0...n
{
Aj
κj }
u|Γj − u|Γk = 0 , ∂nju|Γj + ∂nku|Γk = 0
,
Γj ∩ Γk, ∀j, k
.
Find u ∈ H1 lo (∆, Ωj) su h that
∆u + κ2
j u = 0
in
Ωj, j = 0, . . . n u − u
in
in Ω0 ,
Local multi-trace formulation . .(∗∗) + α · (∗) Find U ∈ H(Γ) such that [ (A −
Id) + α( Id − Π) ]U = F.
SLIDE 49 Spectral analysis
Theorem : Let p = diameter of the adjacency graph of the subdomain
- partition. If κ0 = · · · = κn then (AΠ + ΠA)p+1 = 0.
Ω1 Ω2 Ω3 Ω4 Ω0 2 1 3 4
Corollary : if κ0 = · · · = κn, then S
Id) + α( Id − Π)
√ 1 + α2}
Ref : X.Claeys, Essential spectrum of local multi-trace boundary integral operators, to appear in IMA J. Appl. Math. Ref : A.Ayala, X.Claeys, V.Dolean and M.Gander Closed form inverse of local multi-trace operators, Proceedings of the DDXXIII conference.
SLIDE 50 Well posedness
Construction of a parametrix : in the case κ0 = κ1 = · · · = κn, with the previous calculus, we have : [ (A − αΠ)−(1 − α) Id ] · [ (A − αΠ)+(1 − α) Id ] . . = 2α
Id − α(AΠ + ΠA) = 2α( Id + nilp
t )
Proposition With arbitrary κj, the local multi-trace operator (A −
Id) + α( Id − Π) is
Fredholm with index 0 for α = 0. A classical argument based on radition conditions provides injectivity for arbitrary κj for α = 0. This yields : Corollaire With the κj’s arbitrary, (A −
Id) + α( Id − Π) is invertible for α = 0.
SLIDE 51
Block-Jacobi DDM
Global solver cf [Jerez,Hiptmair,Lee,Peng, 2013] [ A + (α − 1) Id ] U(k+1) − αΠ U(k) = F
SLIDE 52
Block-Jacobi DDM
Global solver cf [Jerez,Hiptmair,Lee,Peng, 2013] [ A + (α − 1) Id ] U(k+1) = αΠ U(k) + F
SLIDE 53
Block-Jacobi DDM
Global solver cf [Jerez,Hiptmair,Lee,Peng, 2013] U(k+1) = α[ A + (α − 1) Id ]−1Π U(k) + F
SLIDE 54 Block-Jacobi DDM
Global solver cf [Jerez,Hiptmair,Lee,Peng, 2013] U(k+1) = α[ A + (α − 1) Id ]−1Π U(k) + F . .
1 α(2 − α)[ A + (1 − α) Id ]
SLIDE 55
Block-Jacobi DDM
Global solver cf [Jerez,Hiptmair,Lee,Peng, 2013] U(k+1) = 1 2 − α[ A + (1 − α) Id ]Π U(k) + F
SLIDE 56 Block-Jacobi DDM
Global solver cf [Jerez,Hiptmair,Lee,Peng, 2013] U(k+1) = 1 2 − α[ A + (1 − α) Id ]Π U(k) + F . .
=
Jα
Proposition In the case κ0 = · · · = κn, we have S( Jα) =
2 − α
1) For arbitrary κj, this yields the essential spectrum. 2) The value α = 0 appears as optimal :
Jα=0 is nilpotent (and remains
meaningful!) and corresponds to the Jacobi iteration for the problem : [(A −
Id) + 2( Id − Π)]·Π(
U) = −2 F.
SLIDE 57 Numerical evidences S
Id) + α( Id − Π)
Ω1 Ω0 radius = 1 Ω1 Ω0 side = 1
0.5 1 1.5 2
0.5 1 1.5 2
α = 1., κ0 = κ1 = κ2 = 1
0.5 1 1.5 2
0.5 1 1.5 2
α = 1., κ0 = κ1 = κ2 = 1
SLIDE 58 Numerical evidences S
Id) + α( Id − Π)
Ω1 Ω0 Ω2
SLIDE 59 Numerical evidences S
Id) + α( Id − Π)
0.5 1 1.5 2
0.5 1 1.5 2
α = 0.5, κ0 = κ1 = κ2 = 1
0.5 1 1.5 2
0.5 1 1.5 2
α = 0.5, κ0 = 5, κ1 = 1, κ2 = 2
0.5 1 1.5 2
0.5 1 1.5 2
α = 1, κ0 = κ1 = κ2 = 1
0.5 1 1.5 2
0.5 1 1.5 2
α = 1, κ0 = 5, κ1 = 1, κ2 = 2
SLIDE 60 Convergence history of Block-Jacobi iterations
1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1 100 50 100 150 200 250 0.5 0.25 0.
Quadratic norm of the block-Jacobi residual for κ0 = i, κ1 = 3i/4, κ2 = 3i/2
α =
SLIDE 61 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 ElastoΦ
SLIDE 62 Factorization of Calderón projectors
Lemma : For
T+ : V+ → H and T− : H → V− continuous linear maps
between Banach spaces, and
T− T+ : V+ → V− bijective, P := T+ · ( T− T+)−1 · T− : H → H
is a projector with
N( P) = N( T−) and R( P) = R( T+).
For
Pj = ( Id + Aj)/2 the Calderón projector of Ωj, this rewrites Pj =
DtNj
DtNj,c)−1 · [− DtNj,c Id ]
nj Ωj Rd \ Ωj Interior DtN map
DtNj(v) := ∂njφj|Γj where
∆φj + κ2
j φj = 0 in Ωj
φj|Γj = v (+RC) Exterior DtN map
DtNj,c(v) := ∂njψj|Γj where
∆ψj + κ2
j ψj = 0 in Rd \ Ωj
ψj|Γj = v (+RC)
SLIDE 63 Factorization of Calderón projectors
Lemma : For
T+ : V+ → H and T− : H → V− continuous linear maps
between Banach spaces, and
T− T+ : V+ → V− bijective, P := T+ · ( T− T+)−1 · T− : H → H
is a projector with
N( P) = N( T−) and R( P) = R( T+).
For
Pj = ( Id + Aj)/2 the Calderón projector of Ωj, this rewrites Pj =
DtNj
DtNj,c)−1 · [− DtNj,c Id ] P = ( Id + A)/2 = diag
j=0...n
( Pj) =
T+ · ( T− T+)−1 T−
In particular :
T− P = T− where T− = diag
j=0...n
( [− DtNj,c
Id ] )
SLIDE 64 Reformulation of block Jacobi MTF
We focus here on the block Jacobi DDM algorithm with α = 0. The wavenumbers κj, j = 0 . . . n may differ. It follows the recurrence U(p+1) = 1 2( Id +
A) · Π · Up +
F With
P = ( Id + A)/2 = Calderón projectors, previous identity T− P = T−
yields
T−(U(p+1)) = T− · Π(Up) + T−(
F) (∗) With U(p) = (U(p)
j
)j=0...n define φ(p)
j
(x) := Gj
κj(U(p) j
)(x) the associated volumic
- solutions. Then (∗) rewrites
Optimized Schwarz Method −∂njφ(p+1)
j
+
DtNj,c(φ(p+1)
j
) = ∂nkφ(p)
k
+
DtNj,c(φ(p)
k ) + ˜
fj
Ref : X. Claeys, V. Dolean et M. Gander, An introduction to multitrace formulations and associated domain decomposition solvers, submitted. Ref : Master thesis of P.Marchand
SLIDE 65
. .
Conclusion
SLIDE 66 Bibliography
- R. Hiptmair and C. Jerez-Hanckes. Multiple traces boundary integral formulation for
Helmholtz transmission problems. Adv. Comput. Math., 37(1) : 39-91,2012.
- X. Claeys, R. Hiptmair, and C. Jerez-Hanckes. Multi-trace boundary integral
- equations. In Direct and Inverse Problems in Wave Propagation and Applications. I.
Graham, U. Langer, M. Sini, M. Melenk, 2012.
- R. Hiptmair, C. Jerez-Hanckes, J. Lee, and Z. Peng. Domain decomposition for
boundary integral equations via local multi-trace formulations. Proc. DD XXI, Technical Report 2013-08, SAM ETH Zürich, 2013.
- V. Dolean et M. Gander, Multitrace formulations and Dirichlet-Neumann algorithms,
- Proc. DD XXII.
- X. Claeys, V. Dolean et M. Gander, An introduction to multitrace formulations and
associated domain decomposition solvers, available on HAL.
- X. Claeys, Essential spectrum of local multi-trace boundary integral operators, to
appear in IMA J. Appl. Math.
- A.Ayala, X.Claeys, V.Dolean and M.Gander Closed form inverse of local multi-trace
- perators, Proc. DDXXIII.
SLIDE 67 Future perspectives
Simple general message : The local multi-trace formulation is a rewriting of the Optimized Schwarz Method by means of boundary integral operators. Potential future directions of research :
- Quasi-local transmission operators for devising Optimized Schwarz (OSM)
DDM strategies.
- Implementation of multi-trace formulations on more realistic test cases in
conjunction with classical DDM schemes (Jacobi, Gauss-Seidel, Krylov,...).
- Use of global MTF to precondition classical DDM applied to local MTF.
