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Convergence of Krylov subspace solvers with Schwarz preconditioner for the exterior Maxwell problem
Eric Darrigrand, Nabil Gmati, Rania Rais
To cite this version:
Eric Darrigrand, Nabil Gmati, Rania Rais. Convergence of Krylov subspace solvers with Schwarz pre-
conditioner for the exterior Maxwell problem. Computers & Mathematics with Applications, Elsevier,
2017, 74 (11), pp.2691-2709. �10.1016/j.camwa.2017.08.027�. �hal-01611138�
Convergence of Krylov subspace solvers with Schwarz preconditioner for the exterior Maxwell problem
Eric Darrigrand
1, Nabil Gmati
2, Rania Rais
2,31Universit´e de Rennes 1, IRMAR, Campus de Beaulieu, 35042 Rennes, France
2Universit´e de Tunis El Manar, ENIT, LAMSIN, B.P. 37, 1002 Le Belv´ed`ere, Tunisia
3Universit´e de Kairouan, ISSATK, Avenue Beit El Hekma, 3100 Kairouan, Tunisia
Dedicated to Professor Peter Monk on the occasion of his 60
thbirthday!
Abstract
The consideration of an integral representation as an exact boundary condition for the finite element resolution of wave propagation problems in exterior domain induces algorithmic difficulties. In this paper, we are interested in the resolution of an exterior Maxwell problem in 3D. As a first step, we focus on the justification of an algorithm described in literature, using an interpretation as a Schwarz method. The study of the convergence indicates that it depends significantly on the thickness of the domain of computation. This analysis suggests the use of the finite element term of Schwarz method as a preconditioner for use of Krylov iterative solvers. An analytical study of the case of a spherical perfect conductor indicates the efficiency of such approach. The consideration of the preconditioner suggested by the Schwarz method leads to a superlinear convergence of the GMRES predicted by the analytical study and verified numerically.
Keywords:finite element method; integral representation; Krylov solver; Schwarz preconditioner; exterior Maxwell problem.
Introduction
We are interested in the resolution of the 3D exterior time-harmonic Maxwell equations. The problem consists in determining the field diffracted by an obstacle. To solve the equations posed in an unbounded domain, we introduce a fictitious boundary with an artificial boundary condition which models the infinity.
Email addresses: [email protected](Eric Darrigrand1),[email protected] (Nabil Gmati2),[email protected](Rania Rais2,3).
In the literature, methods based on a local transparent boundary condition are often considered. They consist in an approximation of the Sommerfeld condition. In this context, we can consider a radiation condition at a finite distance ([7]) or the Bayliss-Gunzburger-Turkel-like conditions ([4,1,16,28]). To obtain satisfying results, the artificial boundary should be chosen far from the boundary of the obstacle. Hence, the computational domain becomes large which increases computation and memory cost. An alternative to local transparent boundary conditions is the Perfectly Matched Layer (PML) method. It consists in replacing the artificial boundary by an absorbing layer of finite elements which vanishes the reflection.
The method was introduced by Berenger and derived for the absorption of electromagnetic waves in [8].
In our paper, we rather consider an exact condition defined by an integral representation. The cho- sen strategy is named “coupling of finite elements and integral representation” and is designated by the acronym CEFRI for the French designation “Couplage El´ements Finis et Repr´esentation Int´egrale”. CE- FRI was initiated in [20] for hydrodynamic problems, and was presented and mathematically studied in the context of electromagnetism equations initially in [19]. In [30,3], a careful study of the numerical behavior of the formulation is presented for 2D Helmholtz equation and 3D Maxwell’s equations, based on numerous numerical tests. [30] also includes a study of the convergence of the iterative resolution for the 2D Helmholtz equation. Such an exact condition defined by an integral representation has been successfully applied to the context of Ultra-Weak-Variational Formulation, which is an alternative to the finite elements, where the strategy was coupled to a Fast Multipole Method [15].
Like the consideration of a local transparent boundary condition, CEFRI leads to an equivalent problem posed on a reduced bounded domain delimited by the boundary of the scatterer and the artificial boundary, where the artificial boundary condition is expressed thanks to an integral representation of the unknown on this boundary. Even if integral operators are involved, no singularity occurs because the unknown on the artificial boundary is expressed using the unknown on the boundary of the scatterer. With such an approach, no a priori condition is required on the distance between the scatterer and the artificial boundary. The main difficulty consists in the elaboration of a numerical scheme for the resolution because of the integral operators which disturb the usual properties of the discrete equations. In [21], Jin and Liu solved the discrete system by a Jacobi method in order to avoid the inversion of the integral operators. The scheme can be interpreted as a Schwarz method with total overlap. This identification has been initially considered for the Poisson and Helmholtz exterior problems in [5,6]. In our paper, the interpretation is extended to the analytical exploration of the rate of convergence of the resolution strategy for Maxwell’s equations. The theoretical analysis of the case of a spherical scatterer indicates that the method converges conditionally on the distance between the scatterer and the artificial boundary. Consequently, using the Jacobi scheme by Jin and Liu, the convergence may fail. However, this study of Schwarz method justifies the use of Krylov solvers and the choice of the preconditioner. We then analytically explore, in this paper, the convergence of a preconditioned GMRES for the resolution of the discrete system and prove the superlinear convergence of the method in the case of a spherical configuration. In [22], Jin and Liu also explored the use of a Krylov solver preconditioned by another formulation of the exterior problem to be solved. Their results are quite similar to ours but the preconditioner is different and consists of more components. In order to verify numerically our theoretical statements, we implemented the resolution strategies using the Finite Element libraryM´elina++ ([23]) which does not provide N´ed´elec elements.
For the numerical results, we then considered the regularized Maxwell equations but the mathematical study of convergence of Jin and Liu algorithm is done for both the classical and the regularized Maxwell equations.
In next section, we introduce the physical problem and explain the application of CEFRI. Section 2 is devoted to the Schwarz interpretation of the resolution strategy suggested in [21]. This consideration enables us to estimate, in Section 3, the speed of convergence of the resolution algorithm in the case of a spherical scatterer using Jin and Liu algorithm. Some test-cases illustrate the theoretical estimation. In Section 4, we investigate the convergence of a Krylov method, the GMRES, combined to the preconditioner
suggested by the previous analysis. Last section provides some numerical simulations which illustrate the convergence properties of this preconditioned application of the GMRES to CEFRI.
1. Scattering by a perfect conductor
Let us consider Ωi a bounded scatterer in R3 with Lipschitz-continuous boundary Γ and Ωe its un- bounded complementary. We are concerned with the scattering of a time-harmonic electromagnetic wave by the perfect conductor Ωi. Our purpose is to determine the total field E =Es+Einc where Einc is the incident wave and Es is the scattered field. We then consider the following scattering problem with essential boundary condition on Γ and radiation condition at infinity:
curl curlE−k2sE= 0 in Ωe, E×ne= 0 on Γ,
lim
R→∞
Z
||x||=R
||curlEs×ne−iksEs||2dγ= 0,
(1)
whereksis the wavenumber andne is the exterior unit normal. In order to use standard Lagrange finite elements for the numerical resolution, we consider an equivalent elliptic problem by adding a regularizing grad-div term in the time-harmonic Maxwell equations, as described and justified in [19]. Problem (1) is then equivalent to the following one:
curl curlE−t−1∇(divE)−ks2E= 0 in Ωe, E×ne= 0, t−1divE= 0 on Γ,
R→∞lim Z
||x||=R
||curlEs×ne−iksne×(Es×ne)||2dγ= 0,
R→∞lim Z
||x||=R
|√
t−1divEs−iksEs·ne|2dγ= 0,
(2)
where the regularization termt−1∇(divE) allows the use of a Galerkin finite element method (see [19]) and the regularization parametert−1 depends on the permittivity and the permeability of the air. The parametertis chosen positive. The choicet=∞corresponds to the initial scattering problem (1). Many different methods have been developed to solve the time-harmonic Maxwell equations in exterior domains.
In this paper, we consider the coupling between finite elements and integral representation introduced by Hazard and Lenoir in [19]. The idea consists in defining an exact boundary condition on an artificial boundary Σ surrounding the scatterer. We reduce the initial problem to an equivalent one, defined on a bounded domain Ω delimited by Γ and Σ. The reduced problem can be stated as follows
curl curlE−t−1∇(divE)−ks2E= 0 in Ω, E×nγ = 0, t−1divE= 0 on Γ,
Tν1E=Tν1(Einc− IΓt(E)) on Σ, Nν2E=Nν2(Einc− IΓt(E)) on Σ.
(3)
where nγ is the exterior unit normal of the domain Ωi on Γ. The parameters ν1 and ν2 are complex numbers with negative imaginary part. The two operatorsTν1 andNν2 are defined by Tν1E = curlE× nσ+ν1nσ×(E×nσ) and Nν2E = divE+ν2E·nσ withnσ the exterior unit normal of the domain Ω
on Σ. The condition that involvesNν2 occurs only if t6=∞. The boundary conditions on Σ are derived from an integral representation satisfied by the scattered field and identified by the following expression [19]: forx∈Ωe,
(IΓtE)(x) = Z
Γ
Gt(x, .)(curlE×nγ+t−1nγ divE)dγ
− Z
Γ
curlyGt(x, .)(E×nγ)dγ − t−1 Z
Γ
divyGt(x, .)T(E·nγ)dγ.
(4)
where Gt = GksI+ 1
k2sHess(Gks −Gkp) is the outgoing Green tensor associated with the differential operator curl curl−t−1∇(div)−k2sI of the regularized Maxwell equations; I is the identity matrix in R3; Hess stands for the Hessian operator; kp =√
tks; andGk is the fundamental solution of Helmholtz equation. Due to the essential condition on Γ, E×nγ = 0, the second term of the representation (4) vanishes such that:
(IΓtE)(x) = Z
Γ
Gt(x, .)(curlE×nγ+t−1nγ divE)dγ−t−1 Z
Γ
divyGt(x, .)T(E·nγ)dγ. (5) The equivalence between the problem stated on the reduced domain Ω (3) and the exterior regularized problem (2) is given by the following proposition (see [19]):
Proposition 1.1 Choose ν1, ν2 ∈ C with negative imaginary part, then the problems (2) and (3) are equivalent: Problem (2) admits at least (respectively, at most) one solution if and only if it is the same for the reduced problem (3). The following links hold:
? IfE is a solution to (2) then the restriction of E on Ωis a solution to (3).
? IfEΩ is a solution to (3) then the field E defined by
E=EΩinΩ,
E=Einc− IΓtEΩinΩe\Ω.
is a solution to (2).
To ensure existence and uniqueness of the solution, we introduce the following spaces: let H(curl,Ω) be the space of fields V satisfyingV ∈ L2(Ω)3 and curlV ∈L2(Ω)3, and H(div,Ω) the space of fields V satisfyingV ∈ L2(Ω)3 and divV ∈ L2(Ω). We then introduce Ht the Hilbert space required by our problem and defined by
Ht=
E∈H(curl,Ω)/divE∈L2(Ω), E×nγ = 0, E×nσ∈L2(Σ)3, t−1E·nσ∈L2(Σ) , (6) ift6=∞, and
H∞=
E∈H(curl,Ω),divE= 0 in Ω, E×nγ = 0, E×nσ∈L2(Σ)3 . (7) The notation (., .)tdenotes the natural scalar product on Ht:
(E, E0)t = Z
Ω
(E·E0+ curlE·curlE0+|t|−1divEdivE0)dΩ +
Z
Σ
((E×nσ)·(E0×nσ) +|t|−1(E·nσ)(E0·nσ))dσ.
IfE is in the spaceHt, then curlE×nγ+t−1nγ divE is not defined. We therefore need to extend the definition ofIΓtE to Ht by the consideration of the operator IΓt,RE defined by (see [19] for details): for allx∈Ωe
(IΓt,RE)(x) = −ks2 Z
Ω
RGt(x, .)E dΩ + Z
Ω
curlRGt(x, .)curlE dΩ + t−1
Z
Ω
divRGt(x, .)TdivE dΩ−t−1 Z
Γ
divGt(x, .)T(E·nγ)dγ ,
(8)
whereRis a linear operator that maps every smooth functionϕdefined on Γ into a smooth functionRϕ defined on Ω that satisfiesRϕ=ϕon Γ andRϕ= 0 on Σ.
The variational formulation of Problem (3) amounts to the equation (see [19]): FindE∈ Htsuch that
(At+Ct)E=Ft, (9)
where the operatorsAtandCt:Ht→ Htare defined by (AtE, E0)t =
Z
Ω
(curlE·curlE0+t−1divEdivE0−k2sE·E0)dΩ +
Z
Σ
(ν1(nσ×E)·(nσ×E0) +t−1ν2(nσ·E)(nσ·E0))dσ,
(10)
(CtE, E0)t= Z
Σ
(Tν1IΓt,RE)·E0dσ+t−1 Z
Σ
(Nν2IΓt,RE)(nσ·E0)dσ, (11) andFtis given by
(Ft, E0)t= Z
Σ
((Tν1Einc)·E0+t−1(Nν2Einc)(nσ·E0))dσ. (12) As explained in [19], Problem (9) is well posed. Moreover, the invertibility of the operatorAtis equivalently obtained by giving a proof of the uniqueness of the solution of the following problem:
curl curlE−t−1∇(divE)−ks2E= 0 in Ω, curlE×n+ν1n×(E×n) = 0 on∂Ω, divE+ν2E·n= 0 on∂Ω,
(13)
where n is the outward unit normal vector on the boundary ∂Ω, outgoing from Ω. This means that n=nσ on Σ andn=ne=−nγ on Γ. By a variational formulation, Green’s formula, and the boundary conditions on∂Ω, one can check that
Z
Ω
(curlE·curlE0+t−1divEdivE0+ks2E·E0)dΩ+ν1
Z
∂Ω
(n×E)·(n×E0)dσ+t−1ν2
Z
∂Ω
(n·E)(n·E0)dσ= 0.
Then, choosingE0 =E, we getn×E= 0 andn·E= 0 on ∂Ω. Considering again the boundary condi- tions: curlE×n= 0, divE = 0 on ∂Ω combined to the integral representation of the field solving the first equation of Problem (13) inside a bounded domain ([19]), leads toE= 0 in Ω and proves the property:
Property 1.2 The operator At defined by the expression (10) is invertible.
Remark 1.3 If t6=∞, generallyHt6⊂H1(Ω)3 [17]. IfΩ is a Lipschitz domain with no re-entrant cor- ners, then Ht∩H1(Ω)3 is dense on Ht [10]. Thus, we can discretize (9) using Lagrange finite elements.
But if the domain has re-entrant corners or edges,Ht∩H1(Ω)3 is a closed proper subspace ofHt([14]).
Therefore,Hthas no dense subspace included inH1(Ω)3. In this case, the discretization by Lagrange finite elements of the regularized Maxwell equations leads to a solution that does not converge to the physical
solution ([11]). In fact in this latter configuration, we can use a weighted regularization method ([13,12]).
For the numerical resolution of Problem (2), we restrict ourselves to the case of obstacles without geo- metrical singularities and such that Ωhas no re-entrant corners.
Remark 1.4 The solution E∈ Ht of Problem (9) satisfiescurl curlE−t−1∇(divE)∈L2(Ω)3. Hence, the quantity curlE×nγ+t−1nγ divE is defined in H1/2(Γ)3 by means of Green’s formula. For such a solution E, the integral representation (5) is then as valid as the integral representation (8).
In the case of the classical Maxwell equations, the weak form of the problem obtained by the application of CEFRI can be derived in a similar way. We just give below the equivalent resulting problem and its variational formulation. By the application of CEFRI, Problem (1) is equivalent to the following problem defined on the bounded domain Ω:
curl curlE−k2sE= 0 in Ω, E×nγ = 0 on Γ,
Tν1E=Tν1(Einc− IΓ∞E) on Σ,
(14)
where IΓ∞ is defined in (5) witht=∞. The variational formulation of Problem (14) can be written as:
FindE ∈ H∞
(A∞+C∞)E=F∞, (15)
The Hilbert spaceH∞is given by (6) fort=∞. The operatorsA∞,C∞andF∞are respectively defined in (10), (11) and (12) takingt=∞.
2. Schwarz method interpretation
In this section, we first present a resolution strategy for the regularized Maxwell equations. Then, the case of the classical Maxwell equations is briefly considered.
To solve System (9), Jin and Liu [21] suggested to considerCt, the term containing the integral repre- sentation, in the right hand side. An application of the fixed point algorithm leads to findingEn+1 such that
curl curlEn+1−t−1∇(divEn+1)−ks2En+1= 0 in Ω, En+1×nγ= 0, t−1divEn+1= 0 on Γ,
Tν1En+1=Tν1(Einc− IΓtEn) on Σ, Nν2En+1=Nν2(Einc− IΓtEn) on Σ.
(16)
In this paper, we focus on an original mathematical justification of convergence of the algorithm expressed by Jin and Liu. We interpret the algorithm based on (16) as a Schwarz method. This interpretation has been initially considered for Poisson and Helmholtz problems in [5,6]. The strategy is designated as Total Overlapping Schwarz Method. Indeed the overlapping area is the total domain Ω. We hereby extend this work to the case of Maxwell’s equations: it consists in replacing equivalently the problem (16) by the two following subproblems. The first one is a transmission problem inR3:
curl curlE2n+1−t−1∇(divE2n+1)−k2sE2n+1= 0 in Ωi∪Ωe, nγ×[E2n+1] = 0, nγ×[curlE2n+1] =−nγ×curlE2non Γ, [t−1divE2n+1] = 0, nγ·[t−1E2n+1] =−nγ·t−1E2non Γ,
R→∞lim Z
||x||=R
||curlE2n+1s ×ne−iksne×(Es2n+1×ne)||2dγ= 0,
R→∞lim Z
||x||=R
|√
t−1divE2n+1s −iksE2n+1s ·ne|2dγ= 0.
(17)
The second one consists in findingE2n+2 such that
curl curlE2n+2−t−1∇(divE2n+2)−ks2E2n+2= 0 in Ω, E2n+2×nγ= 0, t−1divE2n+2= 0 on Γ,
Tν1E2n+2=Tν1E2n+1on Σ, Nν2E2n+2=Nν2E2n+1on Σ.
(18)
The solutionE2n+1of (17) has an explicit expression given by an integral representation. By inserting this representation in the right-hand side of the second and third boundary conditions of (18) we effectively obtain the solution of (16). At the iterationn, Schwarz algorithm (Jin and Liu’s scheme) is defined by
AtEn+1=−CtEn+Ft. (19) Numerically, we use the scheme suggested by Jin and Liu and do not use Subproblems (17) and (18).
The intermediate problems (17) and (18) are used for theoretical justifications. This allows us to derive convergence estimations that cannot be derived directly from System (16).
Since the operatorAtis invertible, Equation (19) is equivalent to
En+1=−A−1t CtEn+A−1t Ft. (20) Both iterative schemes (19) and (20) converge if and only if the spectral radius ofAt−1Ctis strictly lower than one. This statement shows At as a natural preconditioning of Schwarz method. In the sequel, we designateAtas Schwarz preconditioner.
In the case of the classical Maxwell equations, an equivalent strategy can be applied and consists in the consideration of the two following subproblems: The first one consists in findingE2n+1such that
curl curlE2n+1−k2sE2n+1= 0 in Ωi∪Ωe, nγ×[E2n+1] = 0 on Γ,
nγ×[curlE2n+1] =−nγ×curlE2n on Γ,
R→∞lim Z
||x||=R
||curlE2n+1s ×ne−iksE2n+1s ||2dγ= 0.
(21)
The other one consists in the following problem: FindE2n+2 such that
curl curlE2n+2−ks2E2n+2= 0 in Ω, E2n+2×nγ = 0 on Γ,
Tν1E2n+2=Tν1E2n+1on Σ.
(22)
Then, at the iterationn, Schwarz algorithm for the classical Maxwell equations is defined by
A∞En+1=−C∞En+F∞. (23) Since the operatorA∞ is invertible, then
En+1=−A−1∞C∞En+A−1∞F∞. (24) The same statement as for the regularized equations occurs by replacingAtandCtwithA∞ andC∞.
3. Analytical estimation of the convergence for a spherical scatterer
In this section, we investigate an analytical characterization of the convergence of Schwarz method in a spherical configuration where the scatterer Ωi is a perfectly conducting ball. The analysis is first pre- sented for the regularized Maxwell equations. Then, using the same approach, we give the corresponding results for the classical Maxwell’s equations. The calculation was initially done for the classical Maxwell equations in a way which was not applicable to the regularized equations. At the end of the section, we also reproduce the proof of the rate of convergence in the case of the classical equations in the initial way, which leads to the same convergence estimates.
As a first step, we introduce specific functions based on the Bessel and Hankel functions Jl(r) = jl(r) +rjl0(r) andHl(r) =hl(r) +rh0l(r) wherejl(resp.hl) is the spherical Bessel function (resp. spheri- cal Hankel function of the first kind) of degreel. The following derivation also uses the spherical harmonics Ylm,m=−l, ..., l, of order l >0.
The special functions of the spherical configuration offer specific properties ([9,24]), with ˆx=x/|x|:
Proposition 3.1 Let us consider the regularized Maxwell equation
curl curl E−t−1∇(div E)−k2sE = 0. (25) Let Ylm,m=−l, ..., l, be the spherical harmonics of order l >0. The functions
Mlm(x) = curl{x jl(ks|x|)Ylm(ˆx)}, Mflm = 1 iks
curlMlm, Mclm(x) = 1 kp
∇(jl(ks|x|)Ylm(ˆx)) are solutions of (25) in R3and the functions
Nlm(x) = curl{x hl(ks|x|)Ylm(ˆx)}, Nelm = 1 iks
curlNlm, Nblm(x) = 1 kp
∇(hl(ks|x|)Ylm(ˆx)) are radiating solutions of (25) inR3\ {0}.
Remark 3.2 For the classical Maxwell equations, a similar result occurs, involving onlyMlm, Nlm,Mflm
and Nelm ([27]). For the regularized Maxwell equations, the result is based on some work by Morse and Feshbach ([26]) andMclm andNblm contribute to the irrotational part of the field.
Proposition 3.3 If Ylm denote the spherical harmonics of order l > 0 and m = −l, ..., l, then the tangential fields on the unit sphereS
Ulm= 1
pl(l+ 1)∇SYlm and Vlm=n×Ulm
are called vector spherical harmonics of orderl. The notation∇S denotes the surface gradient on the unit sphere S and n is the outward unit normal to S. The fields Ulm andVlm, l=1,2,3,..., form a complete orthonormal system in the Hilbert space
T2(S) =
a:S →C3such thata∈L2(S)3, a.n= 0 of square integrable tangential fields onS equipped with the usualL2 inner product.
In this analytical investigation, we consider the scatterer to be a ball of radiusR∗, we suppose that the artificial boundary Σ is a sphere concentric to Γ with radius R > R∗ and we perform the proof in the case whereν1=ν2=−iks.
3.1. Characterization of the convergence for the regularized equations
The exact solution of the regularized exterior Maxwell problem can be written thanks to the functions introduced in Propositions 3.1-3.3 (see for example [9]):
E=
∞
X
l=1 l
X
m=−l
αlmNlm+βlmNelm+γlmNblm
+Einc.
The error occuring on the fieldEusing Schwarz method is denoted by (wn)nand defined by the following expression:
w2n+1=
E2n+1−E in Ωe, E2n+1 in Ωi,
and w2n+2=E2n+2−E in Ω,
whereE2n+1 andE2n+2 are respectively solutions to (17) and (18). Then, the error on the fieldEcan be written in terms ofMlm,Nlm, Mflm,Nelm,Mclm andNblm :
w2n+1=
∞
X
l=1 l
X
m=−l
a2n+1lm Mlm+ea2n+1lm Mflm+ba2n+1lm Mclm
, |x| ≤R∗,
∞
X
l=1 l
X
m=−l
b2n+1lm Nlm+eb2n+1lm Nelm+bb2n+1lm Nblm
, |x|> R∗,
w2n+2=
∞
X
l=1 l
X
m=−l
c2n+2lm Mlm+d2n+2lm Nlm+ec2n+2lm Mflm+de2n+2lm Nelm+bc2n+2lm Mclm+db2n+2lm Nblm
. As a first step, we interpret the boundary conditions on Γ and on Σ. Some conditions give relations between coefficients such that the number of significant coefficients is reduced:
? the coefficients c2n+2lm ,ec2n+2lm ,bc2n+2lm
are uniquely expressed thanks to
d2n+2lm ,de2n+2lm ,db2n+2lm (let us remark that the same link is valid at the level iteration 2n);
? an equivalent statement occurs between a2n+1lm ,ea2n+1lm ,ba2n+1lm and
b2n+1lm ,eb2n+1lm ,bb2n+1lm . The other conditions give the ingredients for the characterization of the convergence:
? a relation expresses
b2n+1lm ,eb2n+1lm ,bb2n+1lm from
d2nlm,de2nlm,db2nlm
;
? a relation gives
d2n+2lm ,de2n+2lm ,db2n+2lm from
b2n+1lm ,eb2n+1lm ,bb2n+1lm .
A strategy to characterize the convergence consists in introducing the quantity Λn =
Tν1(w2n) Nν2(w2n)
and study the relation between Λn and Λn+1 which corresponds equivalently to the relation between the vectors of coefficients
d2n+2lm ,de2n+2lm ,db2n+2lm and
d2nlm,de2nlm,db2nlm
. Let us denote Ker the operator which maps
d2nlm,de2nlm,db2nlm onto
d2n+2lm ,de2n+2lm ,db2n+2lm
. The previous statements lead to the following result:
using the notations
Hl(ks|x|) =hl(ks|x|) +ks|x|h0l(ks|x|), Jl(ks|x|) =jl(ks|x|) +ks|x|jl0(ks|x|), αl(ksR) =Jl(ksR)−iksRjl(ksR), βl(ksR) =Hl(ksR)−iksRhl(ksR), H
√ t
l =hl(ksR)−i√
th0l(ksR), J
√ t
l =jl(ksR) +i√
tjl0(ksR), γl∗= Hl(ksR∗)
Jl(ksR∗), δ∗l = hl(ksR∗) jl(ksR∗), we have the relation
Ar
d2n+2lm
de2n+2lm db2n+2lm
=Br
d2nlm
de2nlm db2nlm
(26)
where
Ar=
0 βl(ksR)−γl∗αl(ksR) − i
√t(γl∗jl(ksR)−hl(ksR))
βl(ksR)−γl∗αl(ksR) 0 0
0 −√
tl(l+ 1)
R (hl(ksR)−γl∗jl(ksR)) kp
t −H
√t l +γl∗J
√t l
and
Br=
0 βl(ksR) i
√thl(ksR)
βl(ksR) 0 0
0 −√
tl(l+ 1)
R hl(ksR)−kp
t
hl(ksR) +i√
th0l(ksR)
.
As a consequence, the rate of convergence of the Total Overlapping Schwarz method is given by the spectral radius of A−1r Br. The method converges when this spectral radius is strictly lower than 1. In the case of the classical Maxwell equations, we give the asymptotic behavior of the spectral radius with respect to the degreel (see Sections 3.2 and 3.3). In the current context of the regularized equations, we just illustrate the behavior ofρ(Ker) the spectral radius ofKer=A−1r Brwith respect to the geometrical,
physical and regularization parameters: the thicknessR−R∗, wavenumberks, wavelength λs = 2π/ks
and regularization parametert.
10 20 30
l(mode) 0
0.2 0.4 0.6
ρ(eKr)
ρ(Ker) versusRwithks= 2.5,t= 1 R= 1 +λs/10 R= 1 +λs/5 R= 1 +λs
R= 1 + 5λs
10 20 30
l(mode) 0
0.2 0.4 0.6 0.8 1
ρ(eKr)
ρ(Ker) versusRwithks= 5,t= 1 R= 1 +λs/10 R= 1 +λs/5 R= 1 +λs
R= 1 + 5λs
0 20 40 60
l(mode) 0
0.5 1 1.5
ρ(eKr)
ρ(Ker) versusRwithks= 10,t= 1 R= 1 +λs/10 R= 1 +λs/5 R= 1 +λs
R= 1 + 5λs
Figure 1. The spectral radius ofKervs the thickness of the domain, forks= 2.5 (left),ks= 5 (center) andks= 10 (right), witht= 1.
Figure 1 indicates for the first values of l how ρ(Ker) is behaving with respect to the thickness of the computational domain Ω whentis given equal to 1 and when kstakes three different values (2.5, 5, 10).
For all these values ofks, one can observe that larger is R−R∗, smaller is the spectral radius. Figure 2
10 20 30
l(mode) 0
0.2 0.4 0.6 0.8
ρ(eKr)
ρ(Ker) versuskswithR= 1 +λs/5,t= 1 ks= 1.25 ks= 2.5 ks= 5 ks= 10
10 20 30
l(mode) 0
0.1 0.2 0.3 0.4
ρ(eKr)
ρ(Ker) versuskswithR= 1 +λs,t= 1
ks= 1.25 ks= 2.5 ks= 5 ks= 10
10 20 30
l(mode) 0
0.05 0.1 0.15
ρ(eKr)
ρ(Ker) versuskswithR= 1 + 5λs,t= 1
ks= 1.25 ks= 2.5 ks= 5 ks= 10
Figure 2. The spectral radius ofKervs the wavenumberks, forR=R∗+λs/5 (left),R=R∗+λs(center) andR=R∗+ 5λs
(right), witht= 1.
indicates for the first values oflhowρ(Ker) is behaving with respect to the wavenumberkswhentis given equal to 1 and whenR−R∗ takes different values with respect to the wavelength (λs/5,λs, 5λs). For all these values ofR−R∗, we can see that the quantityρ(Ker) is rather deteriorated when the wavenumber increases. Figures 3-4 give for the first values oflthe behavior ofρ(Ker) with respect to the regularization parametert when the wavenumber ks takes the value 5, and R−R∗ takes the values (λs/5, λs, 5λs).
These observations advise a choice oftof order of the unity. The same behavior was observed forks= 2.5 andks= 10.
3.2. Rate of convergence for the classical equations
We now adapt the results of Section 3.1 to the classical Maxwell equations. In this context, thanks to Remark 3.2, the calculation does not involveMclm andNblm anymore. By the same strategy as the one in Section 3.1, the rate of convergence is given by the spectral radius of matrixA−1c Bc where the matrixAc
(respectivelyBc) is one diagonal 2×2 block of the matrix Ar (respectivelyBr) and
10 20 30 l(mode) 0
0.5 1 1.5 2 2.5
ρ(eKr)
ρ(Ker) versustwithR= 1 +λs/5,ks= 5 t= 1 t= 2 t= 10 t= 100 t= 1000
10 20 30
l(mode) 0
0.5 1 1.5
ρ(eKr)
ρ(Ker) versustwithR= 1 +λs,ks= 5
t= 1 t= 2 t= 10 t= 100 t= 1000
10 20 30
l(mode) 0
1 2 3 4
ρ(eKr)
ρ(Ker) versustwithR= 1 + 5λs,ks= 5
t= 1 t= 2 t= 10 t= 100 t= 1000
Figure 3. The spectral radius ofKervs the regularization parametert, forR=R∗+λs/5 (left),R=R∗+λs(center) and R=R∗+ 5λs(right), withks= 5.
10 20 30
l(mode) 0
0.5 1
ρ(eKr)
ρ(Ker) versustwithR= 1 +λs/5,ks= 5 t= 1 t= 0.5 t= 0.1 t= 0.01
10 20 30
l(mode) 0
1 2 3
ρ(eKr)
ρ(Ker) versustwithR= 1 +λs,ks= 5
t= 1 t= 0.5 t= 0.1 t= 0.01
10 20 30
l(mode) 0
0.5 1 1.5 2 2.5
ρ(eKr)
ρ(Ker) versustwithR= 1 + 5λs,ks= 5
t= 1 t= 0.5 t= 0.1 t= 0.01
Figure 4. The spectral radius ofKervs the regularization parametert, forR=R∗+λs/5 (left),R=R∗+λs(center) and R=R∗+ 5λs(right), withks= 5.
A−1c Bc=
1−hl(ksR∗) jl(ksR∗)
Jl(ksR)−iksRjl(ksR) Hl(ksR)−iksRhl(ksR)
−1
0 0
1−Hl(ksR∗) Jl(ksR∗)
Jl(ksR)−iksRjl(ksR) Hl(ksR)−iksRhl(ksR)
−1
. (27)
For the classical equations, the Total Overlapping Schwarz method converges if the spectral radius ρ(A−1c Bc) of A−1c Bc is strictly lower than 1. For R∗ = 1, the asymptotic behavior of the spherical Bessel functions leads to the asymptotic estimation: for largel,
ρ(A−1c Bc)∼ 1
R2l−1 , i= 1,2. (28)
As a consequence, there might be a finite number of modes (corresponding to some values ofl) outside of the unit disk for any value ofRdepending on the wavenumber. Illustrations are given in Section 3.3 where the same result is obtained using a different strategy. The figures show the behavior of the eigenvalues of the iteration matrix with respect to the wavelengthks and radiusR.
3.3. Rate of convergence for the classical equations – the initial derivation
Here, we propose a second strategy to calculate the rate of convergence of Schwarz method for the classical equations. This strategy was indeed the initial one but not extendible to the regularized equations.
For alln, we defineΛen=Tν1(w2n). In order to estimate the convergence of the error (w2n)n, we study the convergence of (eΛn)n considering thatΛen+1=KecΛen withKec a map to be determined fromT2(Σ) to
T2(Σ), whereT2(Σ) is defined in Proposition 3.3. The quantityΛen can be expressed on the basis of the vector spherical harmonics as follows (again due to Proposition 3.3), using the coefficients introduced in Section 3.1:
Λen =Tν1(w2n) =
∞
X
l=1 l
X
m=−l
c2nlmTν1(Mlm) +d2nlmTν1(Nlm) +ec2nlmTν1(fMlm) +de2nlmTν1(Nelm)
=X
l,m
λ1,nlm Ulm+λ2,nlm Vlm.
The relations between Λen andΛen+1 given by conditions on Σ and Γ induce the following statement on the componentsλ1,nlm andλ2,nlm
λ1,n+1lm =τ1,lmλ1,nlm, λ2,n+1lm =τ2,lmλ2,nlm, where
τ1,lm= d˜2nlm
eb2n+1lm + c˜2nlm eb2n+1lm
iksJl(ksR) +k2sRjl(ksR) iksHl(ksR) +ks2Rhl(ksR)
!−1
,
τ2,lm= d2nlm
b2n+1lm + c2nlm b2n+1lm
Jl(ksR)−iksRjl(ksR) Hl(ksR)−iksRhl(ksR)
−1 .
These expressions clearly indicate that Kec is linear and has a diagonal representation in the basis {Ulm, Vlm, l∈N, m=−l, ..., l} ofT2(Σ). The relation between (b2n+1lm ,eb2n+1lm ) and (d2n+1lm ,de2n+1lm ) leads to
τ1,lm=
1−Hl(ksR∗) Jl(ksR∗)
Jl(ksR)−iksRjl(ksR) Hl(ksR)−iksRhl(ksR)
−1 , τ2,lm=
1−hl(ksR∗) jl(ksR∗)
Jl(ksR)−iksRjl(ksR) Hl(ksR)−iksRhl(ksR)
−1
.
(29)
The spectrum of Kec consists of the eigenvalues τ1,lm and τ2,lm. Their explicit derivation indicates that these quantities are independent ofm. Let us denote themτ1,l andτ2,l in the sequel. These eigenvalues define the rate of convergence of the Schwarz method. Then, the convergence of the Total Overlapping Schwarz method is ensured if|τi,l|<1, for alli= 1,2, for alll ∈N. Relations (27) and (29) are similar which indicates that the result is given by the estimation (28).
Let us now illustrate the eigenvalues τ1,l and τ2,l for different values ofR. Figures 5, 6 indicate how τ1,: and τ2,: behave with respect to the thickness of the computational domain Ω when ks takes three different values (2.5, 5, 10). Figures 7, 8 indicate howτ1,: andτ2,: behave with respect to the wavenumber kswhenR−R∗ takes different values with respect to the wavelength (λs/5,λs, 5λs).
3.4. Motivation of Schwarz preconditioner
Due to the observed limitations (the requirements on the thickness of the computational domain, the behavior with respect to the wavenumber), a Krylov method appears as a relevant alternative to the algorithm defined by (16): due to the properties of Krylov solvers demonstrated in [18], the convergence of a Krylov method is ensured for the resolution of Problem (9) using At as a preconditioner. Such a strategy consists in solving the system
(I+A−1t Ct)E=A−1t Ft (30)
