Optimized Schwarz methods for the time-harmonic Maxwell equations with damping

Transcription

1 Optimized Schwarz methods for the time-harmonic Maxwell equations with damping Mohamed El Bouajaji, Victorita Dolean, Martin Gander, Stephane Lanteri To cite this version: Mohamed El Bouajaji, Victorita Dolean, Martin Gander, Stephane Lanteri. Optimized Schwarz methods for the time-harmonic Maxwell equations with damping. 0. <hal > HAL Id: hal Submitted on 3 Nov 0 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 OPTIMIZED SCHWARZ METHODS FOR THE TIME-HARMONIC MAXWELL EQUATIONS WITH DAMPING M. EL BOUAJAJI, V. DOLEAN, M.J. GANDER AND S. LANTERI Abstract. In a previous paper, two of the authors have proposed and analyzed an entire hierarchy of optimized Schwarz methods for Maxwell s equations both in the time-harmonic and time-domain case. The optimization process has been perfomed in a particular situation where the electric conductivity was neglected. Here, we take into account this physical parameter which leads to a fundamentally different analysis and a new class of algorithms for this more general case. From the mathematical point of view, the approach is different, since the algorithm does not encounter the pathological situations of the zero-conductivity case and thus the optimization problems are different. We analyze one of the algorithms in this class in detail and provide asymptotic results for the remaining ones. We illustrate our analysis with numerical results. Key words. Schwarz algorithms, optimized transmission conditions, Maxwell s equations AMS subject classifications. 65M55, 65F0, 65N. Introduction. Finding robust solvers for indefinite problems, such as Helmholtz and the time-harmonic version of Maxwell s equations, has always been a challenging research topic. From the pioneering work of Desprès [5], where the first provably convergent domain decomposition algorithm for Helmholtz equations was proposed and then extended to Maxwell s equations in [6], other studies followed with the declared purpose of improving the performance of these algorithms. The first attempts to obtain better algorithms for this kind of equations were given in [4] and then [3], where the first inspiring ideas of optimized Schwarz methods can be found. Then, in [5] and in particular in [4], the advantange of the optimization process was used at its best for the Helmholtz equations to design an algorithm without overlap; for the second order Maxwell system see []. But, for the first time, an entire hierarchy of optimized transmission conditions for the first order Maxwell s equations was given in [8], with complete asymptotic results. Applications on real life problems using a Discontinunous Galerkin method can be found in [9, 0]. For finite-element based non-overlapping and non-conforming domain decomposition methods for the computation of multiscale electromagnetic radiation and scattering problems we refer to [6, 8, 7]. This paper is organized as follows: in Section, we introduce the mathematical formulation of the problem to be solved, and in Section 3, the classical and optimized Schwarz algorithms are presented. These algorithms are analyzed by computing their convergence factor in Fourier space in Section 4. The main part of the paper and the core of this study is given in Section 5, where an optimization problem is solved for one of the optimized Schwarz methods in all detail, and asymptotic results are given for all optimized Schwarz algorithms considered in this paper. INRIA SOPHIA ANTIPOLIS-MÉDITERRANÉE, 0690 SOPHIA ANTIPOLIS CEDEX, FRANCE. UNIV. DE NICE SOPHIA-ANTIPOLIS, LABORATOIRE J.-A. DIEUDONNÉ, NICE, FRANCE. DOLEAN@UNICE.FR SECTION DE MATHÉMATIQUES, UNIVERSITÉ DE GENÈVE, CP 64, GENÈVE, MARTIN.GANDER@MATH.UNIGE.CH INRIA SOPHIA ANTIPOLIS-MÉDITERRANÉE, 0690 SOPHIA ANTIPOLIS CEDEX, FRANCE.

3 . Presentation of the problem. The system of Maxwell s equations describes mathematically the propagation of electromagnetic waves ε E t curlh+σe = J, µ H t +curle = 0, (.) where E and H denote the electric and magnetic field, ε is the electric permittivity and µ the magnetic permeability. Here σ denotes the conductivity and J the applied current density. We are interested in solving the time-harmonic Maxwell equations, which are obtained from their time-domain counterpart (.) by assuming that the electric field E and the magnetic field H follow a harmonic dependence on time (as a result of the time-harmonic dependence of the current density J(x, t) = Re(J(x) exp(iωt))), E(x, t) = Re(E(x) exp(iωt)), H(x, t) = Re(H(x) exp(iωt)), where the positive real parameter ω is the pulsation of the harmonic wave. The unknow complex-valued vector fields E and H are the solutions of the time-harmonic Maxwell s equations iωεe curl H +σe = J, iωµh +curl E = 0. (.) Equations (.) are solved in a bounded domain Ω. On the boundary Ω = Γ a Γ m, the following boundary conditions are imposed: a perfect electric conductor condition on Γ m : n E = 0, an impedance condition on Γ a : Bn(E,H) = Bn(E inc,h inc ), where the vector (E inc, H inc ) represents an incident electromagnetic wave and n denotes the unit outward normal at any point of Ω. The impedance operator is given by Bn(E,H) = n E Z n (H n). Note that this condition is equivalent to imposing Dirichlet conditions on the incoming characteristics if we consider the hyperbolic nature of the underlying time-dependent system, and can be seen also as an approximation of a transparent boundary condition. 3. Schwarz Methods for Maxwell s Equations. We introduce here a domain decomposition method based on a Schwarz algorithm. The computational domain Ω with boundary Ω is partitioned into N subdomains Ω i, i =,..,N, such that Ω = N i= Ω i. We set Γ ij = Ω i Ω j \ Ω where Ω i is the boundary of Ω i. For a simple decomposition of the domain into two subdomains(as shown in Figure 3.), the classicalschwarzalgorithmconsistsincomputing(e j,n+,h j,n+ )from(e j,n,h j,n ), for j =, by the iterative process iωεe,n +curl H,n σe,n = J in Ω, iωµh,n +curl E,n = 0 in Ω, Bn (E,n,H,n ) = Bn (E,n,H,n ) on Γ, iωεe,n +curl H,n σe,n = J in Ω, iωµh,n +curl E,n = 0 in Ω, Bn (E,n,H,n ) = Bn (E,n,H,n ) on Γ. (3.) For the classical Schwarz method, we see that at the interfaces (artificial boundaries between subdomains), characteristic information is exchanged. This type of

4 3 Γ n Ω Ω Ω n Γ Fig. 3.. Domain decomposition into two subdomains ρ cla Classical Schwarz without overlap Classical Schwarz with overlap k Fig. 3.. Convergence factor ρ cla of the classical Schwarz method as a function of k, for L = 0, ω = π, σ = and µ = ε =. conditions have been proposed for the first time in [6] for the second order version of the time-harmonic Maxwell equations. In the constant coefficient case, an estimate of the convergence factor of this algorithm has been obtained in [8] using Fourier analysis. The convergence factor (the reduction of the error between two successive interations) for each Fourier mode k is given by k ω ρ cla (k, ω,σ,z,l) = +i ωσz i ω k ω +i ωσz +i ω e k ω +i ωσzl, where ω = ω εµdenotestheadimensionalizedfrequencyandlistheoverlapbetween domains. From this result, we see that if σ > 0, the method converges also without overlap (L = 0). But for the high frequencies, the convergence factor is close to one and therefore the algorithm is not very efficient, see Figure 3.. The convergence can be improved by modifying the transmission conditions in

5 4 the classical Schwarz algorithm (3.), namely iωεe,n +curl H,n σe,n =J in Ω, iωµh,n +curl E,n =0 in Ω, (Bn +S Bn )(E,n,H,n )=(Bn +S Bn )(E,n,H,n )on Γ, iωεe,n +curl H,n σe,n =J in Ω, iωµh,n +curl E,n =0 in Ω, (Bn +S Bn )(E,n,H,n )=(Bn +S Bn )(E,n,H,n )on Γ, (3.) where S j, j =, are tangential, possibly pseudo-differential operators. For σ = 0, different choices of S j, j =, have been developed in [8] and lead to different optimized Schwarz algorithms for Maxwell s equations. In the next section, we develop different choices of S j for the case σ > 0. This leads to optimization problems which are very different from the case σ = 0. The min-max problems can now have interior maxima, which complicates the mathematical analysis substantially, but their solution is necessary to obtain optimized transmission conditions in the case σ > Convergence Analysis for Non-Zero Electric Conductivity. We present here an analysis of algorithm (3.) for the case where the electric conductivity is nonzero, σ > 0. In [8], it was proved that if the operators S l, l =, have the Fourier symbol F(S l ) = [ k y kz λσz k y k z (λ+i ω)(λ+i ω +σz) k y k z kz ky λσz ], (4.) where λ = k ω +i ωσz, then the algorithm (3.) converges in two iterations. It was also shown that these symbols can be written in different forms, F(S l ) = (λ+i ω)(λ+i ω+σz) M = = k λσz k +λσz λ i ω λ+i ω M λ i ω σz λ+i ω+σz M = (λ i ω)(λ i ω σz) M, where M and M are given by [ k M = y kz λσz k y k z k y k z kz ky λσz ] [ k, M = y kz +λσz k y k z k y k z kz ky +λσz ], and λ = k ω +iωσz. This motivates different approximations of the transparent conditions in the context of optimized Schwarz methods. In the following we will denote by M s and M s the matrices M and M where we replaced the non-local operator λ by a constant s. Corollary 4.. For different symbols of the operators S j (which approximate the non local operators 4.) we get the following convergence factors: Algorithm If S j, j =,, have the symbols σ j = F(S j ) = 0, (4.) then the Schwarz algorithm (3.) has the convergence factor k ω ρ ( k, ω,σ,z,l) = +i ωσz i ω e k ω +i ωσz +i ω This algorithm is the classical Schwarz algorithm. k ω +i ωσzl. (4.3)

6 Algorithm If S j, j =,, have the Fourier symbols σ j = F(S j ) = s i ω (s+i ω)( k +sσz) M s, s C, (4.4) then the Schwarz algorithm (3.) has the convergence factor k ω ρ ( k, ω,σ,z,l,s) = +i ωσz s k ω +i ωσz +s Algorithm 3 If S j, j =,, have the Fourier symbols σ j = F(S j ) = e k ω +i ωσzl. 5 (4.5) k ω +i ωσz +(i ω +σz)s M s, s C, (4.6) then (3.) has the convergence factor k ω ρ 3 ( k, ω,σ,z,l,s) = +i ωσz i ω k ω +i ωσz i ω ρ ( k, ω,σ,z,l,s). (4.7) Algorithm 4 If S j, j =,, have the Fourier symbols σ j = F(S j ) = s j i ω (s j +i ω)( k +s j σz) M s j, s j C, (4.8) then the Schwarz algorithm (3.) has the convergence factor ρ 4 ( k, ω,σ,z,l,s,s ) = k ω +i ωσz s l e k ω +i ωσz +s l l= Algorithm 5 If S j, j =,, have the Fourier symbols σ j = F(S j ) = k ω +i ωσzl (4.9) k ω +i ωσz +(i ω +σz)s j M sj, s j C, (4.0) then the Schwarz algorithm (3.) has the convergence factor k ω ρ 5 ( k, ω,σ,z,l,s,s ) = +i ωσz i ω k ω +i ωσz +i ω ρ 4( k, ω,σ,z,l,s,s ). (4.) The different symbols σ j depend on the choice of the parameters s, s and s. In order to obtain an efficient algorithm, we will choose σ j, j =, such that ρ l, l =,.., 5 is minimum over a range of frequencies. Therefore the parameters are solutions of the min-max problems min max ρ j ( k, ω,σ,z,l,s),j =,3, s C k K min max ρ j ( k, ω,σ,z,l,s,s ),j = 4,5, s, s C k K (4.) where K denotes the set of relevant numerical frequencies. In the next section, we will analyze these min-max problems for each of the algorithms in Corollary 4...

7 6 5. Optimized transmission conditions. In this section, we solve the various min-max problems seen in (4.). The fundamental difference with the case σ = 0 is that here we do not need to exclude the resonance frequencies (see [8]). This changes the nature of the min-max problems. We have more local maxima in k, and balancing these maxima by the equioscillation principe, is more difficult. In a numerical implementation, the range of frequencies is bounded, i.e k K := [k min, k max ], where the minimum frequency k min > 0 is a constant depending on the geometry, and the maximum numerical frequency that can be represented on a mesh is k max = C h where C is a constant and h is the mesh size. Before solving the min-max problems in (4.), we give an asymptotic expression for the maximum of the convergence factor (4.3) of the classical Schwarz algorithm over k K. This allows us to see the behavior of the algorithm (3.) when h goes to zero. Corollary 5.. The asymptotic convergence factor of the classical Schwarz method (Algorithm ), for small mesh size h is max ρ ( k, ω,σ,z,l) = k K { 4 3 ( 9ω 4 σ µ 3 εc 6 L) 8 h 3 4 +O(h 5 4), L = C L h, ω σ µ 3 ε C 3 h 3 +O(h 5 ), L = 0. (5.) Proof. The proof is obtained by expanding the maximum of ρ over k K for h small. 5.. Optimization of Algorithm. We look for s under the form s = p+iq, with (p,q) R + and p = q, such that p is solution of the min-max problem min p 0 ( max ρ ( k, ω,σ,z,l,p(+i)) k K ). (5.) We choose p = q in order to simplify the computations, this choice being justified for the Helmholtz equation case in [5]. The case where p q is discussed in []. The non-overlapping case. Using the change of variables ( ) ξ( k ) := R k ω +iσ ωz, y := σ ωz, the convergence factor simplifies to ρ ( ω,z,σ,0, k,p(+i)) = 4ξ (ξ p) +(y ξp) 4ξ (ξ +p) =: R(ξ,y,p). +(y +ξp) Since the mapping k ξ( k ) is increasing, the optimization problem (5.) can be re-written as ( ) ( ) min max ρ ( ω,z,σ,0, k,p(+i)) = min max R(ξ,y,p), p 0 k min k k max p 0 ξ min ξ ξ max where ξ min = ξ(k min ) and ξ max = ξ(k max ). We can prove the following result

8 7 Theorem 5.. In the non-overlapping case, the solution of the problem ( ) is given by min p 0 max R(ξ,y,p) ξ min ξ ξ max p = { min(p,p 3 ), if y < ξ max ξ min, min(p,p ), if y ξ max ξ min, (5.3) where p, p, p, and p 3 are constants defined by p = y 4 ξ min(y+ξmin +ξmin y) p =, p 4ξ 3 3 = y min 8ξ 4 min +y 8ξ 4 4ξ min, p = max +y 4ξ max. 4 ξ max(y+ξ max +ξmax y) 3 4ξ max, Proof. We start by computing the partial derivative of R with respect to p in order to restrict the range of p in the min-max problem, E is negative for p [0, E := R p (ξ,y,p) = (3ξ 3 y+64ξ 5 ) ( p 4ξ 4 +y ) 8 ξ R(ξ,y,p) (4ξ 4 +8ξ 3 p+8ξ p +y +4yξp). (5.4) 8ξ4 +y 8ξ 4 +y 4ξ ] and positive for p [ 8ξ4 +y 4ξ, + ). Moreover the minimum of the mapping ξ 4ξ is y. This implies that for p y, R decreases with p for all ξ [ξ min, + ). Hence we can restrict the range of p to the interval [ y,+ ). We now look for the eventual extrema of ξ R(ξ,y,p) by computing the zeros of the derivative of R with respect to ξ, R ξ (ξ,y,p) = 4p(ξ y)(8ξ4 +6(y p )ξ +y ). (5.5) R(ξ,y,p)(4ξ 4 +8ξ 3 p+8ξ p +y +4yξp) The polynomial P(ξ) = ( ξ y)( 8ξ 4 +6 ( y p ) ξ +y ) has at most three positive roots, and ξ = y is always a root. The other roots are given by ξ,3 = 4y +4p (3y p )(y p ). We now show that for p 3y/, ξ is a maximum and the other two roots of P cannot be maxima. The second partial derivative of R with respect to ξ at ξ is R ξ (ξ (p 3y)p,y,p) = 4 y p y ( p+ y) 3, which shows that ξ is a local maximum if p < 3y/. Moreover in this case, the other two roots are real and these are minima because ξ ξ ξ 3. In the case p 3y/, ξ is a local minimum and the other two roots are not real. Therefore the maximum of R is either at ξ min, ξ or ξ max. From the expression (5.4) we can derive the following properties of R:

9 8 (a) The mapping p R(ξ min,y,p) is decreasing on [ y,p ] and increasing on [p,+ ). (b) The mapping p R(ξ,y,p) is increasing on [ y,+ ). (c) The mapping p R(ξ max,y,p) is decreasing on [ y,p ] and increasing on [p,+ ). To conclude the proof of the theorem, we need the following lemmas: Lemma 5.3. Let p 3 be the constant defined by ξmin ξ max (y p 3 = +yξmin +4yξ minξ max +yξmax +4ξmin ξ max). 4ξ min ξ max Then the following properties are verified: i) p is the unique solution of the equation R(ξ min,y,p) = R(ξ,y,p) on R + \{0}. ii) p 3 is the unique solution of the equation R(ξ min,y,p) = R(ξ max,y,p) on R + \{0}. iii) p 3 is the unique solution of the equation R(ξ,y,p) = R(ξ max,y,p) on R + \{0}. iv) We have { p, if y < ξ max(p,p ) = max ξ min, p, if y ξ max ξ min. v) We have vi) We have vii) We have max(p,p 3 ) = 3y p,p 3 p 3. max R(ξ,y,p) = ξ min ξ ξ max { p3, if y < ξ max ξ min, p, if y ξ max ξ min. Proof. To prove the first three properties, we compute R(ξ b,y,p) R(ξ m,y,p) R(ξ max,y,p), if y < ξ max ξ min and p [ y,p 3], R(ξ,y,p), if y < ξ max ξ min and p [p 3,+ ), R(ξ min,y,p), if y ξ max ξ min and p [ y,p ], R(ξ,y,p), if y ξ max ξ min and p [p,+ ). = 8p(ξm ξ b)(ξ mξ b y)(8ξ b ξ m p 4ξ b ξ m ξ b y 4yξm ξ b yξ m y ). (4ξb (ξ b+p) +(y+ξ b p) )(4ξm (ξm+p) +(y+ξ mp) ) (5.6) This quantity is zero if ξb ξ m (y p = +yξb +4yξ bξ m +yξm +4ξb ξ m). 4ξ b ξ m By replacing successively the pairs {ξ b,ξ m } by {ξ min,ξ }, {ξ min,ξ max } and {ξ,ξ max } in (5.6) we get the desired result. To prove property iv), we compute p p = (ξ max ξ min )(ξ min +ξ max )(y ξ max ξ min )(y +ξ max ξ min ) 8ξmaxξ min,

10 9 which shows that p p if y < ξ max ξ min and p p if y ξ max ξ min. To prove property iv), we compute y p p (ξmax ξ min )(ξ max ξ min y) 3 =. 4ξ max ξ min We can see that p p 3 if y < ξ max ξ min and p p 3 if y ξ max ξ min. We now show vi) by computing p p 3 = (y+4ξ min)(ξ max / y ) 8ξ maxξ min p 3y = y (ξmin / y ) ξ min 0, p 3 p 3 = (4ξ max +y)(ξmin / y ) (ξmax / y ) 0, p 3 3y = y This shows that p i 3y 0 and p i p 3 0, i =,. Finally we treat the last point. After some computations we obtain ξ max 0. E := R(ξ min,y,p) R(ξ max,y,p) 64pξ minξ max(ξ max ξ min)(ξ maxξ min y)(p = p 3 ) (4ξmin (ξmin+p) +(y+ξ minp) )(4ξmax (ξmax+p) +(y+ξ maxp) ), E 3 := R(ξ max,y,p) R(ξ,y,p) = 3pξmax(ξmax / y ) (p p 3) (4ξ (ξ+p) +(y+ξp) )(y+ y p+p ), E 4 := R(ξ min,y,p) R(ξ,y,p) = 3pξmin(ξmin / y ) (p p ) (4ξ (ξ+p) +(y+ξp) )(y+ y p+p ). 8ξ maxξ min 0, From the expression above we can easily see that if y < ξ max ξ min and p [ y,p 3], then E 0 because p 3 p 3 (see proposition (vi)) and E 3 0. if y < ξ max ξ min and p [p 3,+ ), then E 4 0 since p p 3 (see proposition (vi)) and E 3 0. If y ξ max ξ min and p [ y,p ], then E 0 and E 4 0. If y ξ max ξ min and p [p,+ ), then E 3 0 since p p 3 (see proposition (vi)) and E 4 0. This Lemma proves that the solution of the min-max problem (5.) depends mainly on the sign of y ξ max ξ min. We can see on Figure 5. the solution of the min-max problem for different values of y, ξ min and ξ max. The pictures (a), (b), (c) and (d) show the four cases of the theorem (y < ξ max ξ min, p p 3 ), (y ξ max ξ min, p p ), (y < ξ max ξ min, p p 3 ) and (y ξ max ξ min, p p ). Lemma 5.4. If y < ξ max ξ min, the solution of the min-max problem ( ) min max R(ξ,y,p) y p p3 ξ min ξ ξ max is p = min(p,p 3 ) and the solution of the min-max problem ( ) min p p 3 max R(ξ,y,p) ξ min ξ ξ max

11 0 (a) (b) R p p 3 =p * p 3 R(ξ min,y,p) R(ξ,y,p) R(ξ max,y,p) p (c) p R p 3 p 3 p p (d) R(ξ min,y,p) R(ξ,y,p) R(ξ max,y,p) p R p =p * p p 3 p p 3 R(ξ,y,p) min R(ξ,y,p) R(ξ,y,p) max p R p =p * p p 3 p p 3 R(ξ min,y,p) R(ξ,y,p) R(ξ max,y,p) p Fig. 5.. Solutions of the min-max problem for different values of y, ξ min and ξ max. (a) : y =.5, ξ min = 0.5, ξ max = 00, (b) : y = 5.5, ξ min = 0.5, ξ max = 00, (c) : y = 57, ξ min = 3.5, ξ max = 4, (d) : y = 57, ξ min = 3.6, ξ max = 0. is p = p 3. Proof. By using the property (vii) of Lemma 5.3, for p [ y,p 3], we have R(ξ max,y,p) = max ξ min ξ ξ max R(ξ,y,p). Hence the solution of the first min-max problem is the minimum of the mapping p R(ξ max,y,p) on [ y,p 3]. By the property (c) of page 7, the minimum of R(ξ max,y, ) on [ y,p 3] is min(p,p 3 ). By using the property (vii) of Lemma 5.3, for p [p 3,+ ), we have R(ξ,y,p) = max R(ξ,y,p). ξ min ξ ξ max

12 Therefore the solution of the second min-max problem is the minimum of the mapping R(ξ,y, ) on [p 3,+ ). By property (b) of page 7, the minimum of R(ξ,y, ) on [p 3,+ ) is p 3. Lemma 5.5. If y ξ max ξ min, the solution of the min-max problem ( ) min max R(ξ,y,p) y p p ξ min ξ ξ max is p = min(p,p ) and the solution of the min-max problem ( ) min max R(ξ,y,p) p p ξ min ξ ξ max is p = p. Proof. By using the property (vii) of Lemma 5.3, for p [ y,p ], we have R(ξ min,y,p) = max R(ξ,y,p). ξ min ξ ξ max The solution of the first min-max problem is the minimum of the mapping p R(ξ min,y,p) on [ y,p ]. Using now (a) of page 7, the minimum of R(ξ min,y, ) on [ y,p ] is min(p,p ). By proposition (vii) of Lemma 5.3, for p [p,+ ), we have R(ξ,y,p) = max ξ min ξ ξ max R(ξ,y,p). Therefore the solution of the second min-max problem is the minimum of the mapping p R(ξ,y,p) on [p,+ ). By property (a) on page 7, the minimum of R(ξ,y, ) on [p 3,+ ) is p. We can now return to the end of the theorem. The min-max problem (5.) can be rewritten in the equivalent form ( min p y (max ξmin ξ ξ max R(ξ,y,p)) = min min (max y p p3 ξ min ξ ξ max R(ξ,y,p)), ) min p p3 (max ξmin ξ ξ max R(ξ,y,p)), which is useful for the case y < ξ max ξ min, because we then get, using Lemma 5.4 ( ) min max R(ξ,y,p) = R(ξ max,y,min(p,p 3 )). y p ξ min ξ ξ max The min-max problem (5.) can also be rewritten as min p y (max ξmin ξ ξ max R(ξ,y,p)) = min ( min (max y p p ξ min ξ ξ max R(ξ,y,p)), ) min p p (max ξmin ξ ξ max R(ξ,y,p)),

13 which is now usefull for the case y ξ max ξ min, because we then get, using Lemma 5.5, ( ) min max R(ξ,y,p) = R(ξ max,y,min(p,p )). y p ξ min ξ ξ max Corollary 5.6. For h sufficiently small in the non-overlapping case, L = 0, the solution of the min-max problem (5.) is given by p = (ωσµ) 4 C and ρ 4(ωσµ) = 3 4 h +O(h). (5.7) 4 h C Proof. If we assume h sufficiently small, then ξ max = ξ(c/h) is large. By this assumption, the solution of the min-max problem (5.) is given by the first equation of (5.3), i.e p = 4 y ξ max ( y +ξmax +ξ max y ) ξ max. By expanding p for h small, using that ξ max = ξ(c/h), we get the desired result. Corollary5.6givestheoptimalparameterp andthecorrespondingconvergencefactor of algorithm in terms of the mesh size h. In some practical situations, where we consider high frequencies, it is interesting to give the optimized parameters in term of ω. For the high frequencies, it has been shown in [] that it is necessary, to avoid the pollution effect, to couple the frequency ω with h by the relation h = C h /ω γ ; where γ depends on the discretization method. In [], it has been shown that γ = 3 for a P finite element method. We have the following corollary: Corollary 5.7. Let h = C h /ω γ. If γ =, then for ω large, the solution of the min-max problem (5.) is given by p 4 = (σµ) (C εµch ) 4 ω 3 4 and ρ 4(σµ) = 3 Ch ω 4 Ch (C εµch 4. (5.8) ) 4 If γ >, for ω large, the solution is p = (σµ) 4 C ω γ+ 4 Ch 4 and ρ 4(σµ) 4 = 3 Ch C ω γ 4. (5.9) Proof. If we assume ω sufficiently large, then ξ max = ξ(ωc/c h ) is large. By this assumption, the solution of the min-max problem (5.) is given by the first equation of (5.3), i.e p = 4 y ξ max ( y +ξmax +ξ max y ) ξ max. By expanding p for ω large, using that ξ max = ξ(ωc/c h ), we get the desired result.

14 The overlapping case. After the complete analysis of the best approximation problem without overlap in the previous paragraph, we focus in the remainder of this paper on asymptotic analysis, in order to be able to present a complete set of optimized transmission conditions for all algorithms. We start with the case of the previousparagraph,butnowconsideranoverlappingmethod. WesetL = h,k min = 0, k max = π h and denote by p the solution of the min-max problem (5.). Solving this min-max problem numerically for different parameter values and different mesh sizes h, we observe that the solution of (5.) equioscillates once, i.e. p is solution of ρ (k, ω,σ,z,l,p (+i)) = ρ (k, ω,σ,z,l,p (+i)), (5.0) where k and k are two interior local maxima of ρ. For small h, we see numerically that these maxima and the optimized parameter p behave like k = C b, k = C b h 3, and p = C p h 3, for some constants Cp, C b, C b. In order to determine the constants C b, C b and C p, we solve now the corresponding equations asymptotically: since k and k are local maxima of the convergence factor, the corresponding derivatives must vanish, and we find asymptotically ρ k (k, ω,σ,z,l,p (+i)) = 0 C b = ω, ρ k (k, ω,σ,z,l,p (+i)) = 0 C b = C p. The equioscillation equation (5.0) must also be satisfied, which gives asymptotically ρ (k, ω,σ,z,l,p (+i)) = ρ (k, ω,σ,z,l,p (+i)) ωσz C p = C b +C p C b. SolvingthethreeequationsobtainedfortheconstantsC b, C b andc p, andexpressing the results in the original problem parameters ω, ǫ, µ and σ, we find the asymptotic result p = (ωσµ) 3 h 3 3, ρ = 7 6 (ωσµ) 6 h 3. (5.) Lemma 5.8. Asymptotically, the parameter p defined in (5.) is a local minimum of the min-max problem (5.). Proof. The parmeter p is a local minimum, if there exists no variation δp such that ρ (k, ω,σ,z,0,(p + δp)( + i)) ρ (k, ω,σ,z,0,p ( + i)), for k = k, k. By the Taylor formula, it suffices to prove that there is no variation δp such that δp ρ p (k, ω,σ,z,0,p ( + i)) < 0, for k = k, k. To prove this, it is necessary to obtain the next higher order terms in the expansions of p, k and k. After a lengthy computation, we find that k ω + C b h 3, k C b h 3 + Cb, p C p h 3 + Cp h 3. With these new three constants determined, we obtain for the derivative of ρ 3 6h 3 ρ p (k, ω,σ,z,h,p (+i)) 4 5 ρ, (ωσµ) 6 p (k, ω,σ,z,h,p (+i)) 5. (ωσµ) 6 Since these leading terms differ in sign, there can not be a variation δp which diminishes the contraction factor obtained for p, and we have asymptotically a local minimum. In Figure 5. we can see the convergence factor in the non-overlapping and overlapping case for the algorithm compared to the classical one. 6h 3

15 4 5.. Optimization of algorithm 3. Here, welookforsoftheforms = p(+i), such that p is solution of the min-max problem ( ) min max ρ 3(k, ω,σ,z,l,p(+i)). (5.) p 0 k K Again we are only searching for an asymptotic solution of this best approximation problem. The non-overlapping case. Setting L = 0, k min = 0, k max = C h and denoting by p the solution of (5.), we obtain from numerical experiments that the solution of (5.) equioscillates once, i.e. p is solution of ρ 3 (k max, ω,σ,z,0,p (+i)) = ρ 3 (k, ω,σ,z,0,p (+i)), (5.3) where k b is an interior local maximun of ρ 3. For small h we find the asymptotic behavior k = C b h 7, p = C p h 4 7, and to determine C b and C p, we solve asymptotically ρ 3 k (k, ω,σ,z,0,p (+i)) = 0 C p = C4 b 3 ω σz, and the equioscillation equation (5.3), which leads to C b = (6C ω 4 σ Z ) 7. Solving the two equations for the two constants, and using again the original parameters ω, ǫ, µ and σ, we find p = C 7 ω 7 µ 4 ǫ 4 σ 7 h 4 7, ρ 3 = 4 67ω 4 7µ 4ǫ 3 4σ 7h 3 7. (5.4) 3 C 3 7 Lemma 5.9. Asymptotically, the parameter p defined in (5.4) is a local minimum of the min-max problem (5.). Proof. The proof follows the same argument as the proof of Lemma 5.8. We obtain after a lengthy computation that k C b h 7 +Cb h 7, p C p h 4 7 +Cp h 7. Determining the constants in the higher order terms as well, we can show that ρ 3 p (k max, ω,σ,z,0,p (+i)) 4h C, ρ 3 p (k, ω,σ,z,0,p (+i)) σ 7Z 3 7h 3 9 7, 4 ω 7C 9 7 which shows by the opposite sign that p is indeed a local minimum. The overlapping case. Also with L = h the solution of the min-max problem (5.) equioscillates once, i.e. p is solution of ρ 3 (k, ω,σ,z,l,p (+i)) = ρ 3 (k, ω,σ,z,l,p (+i)), where k and k are two interior local maxima of ρ 3. For small h, we get k = C b h 0, k = C b h 7 0, p = C p h 5, and solving the corresponding equations asymptotically, we find in the original parameters ω, ǫ, µ and σ p = 3 35ε 0µ 0ω 3 5σ 5 5 h 5, ρ 3 = ω 5 ε 0 µ 3 0 σ 0 h 3 0. (5.5) In Figure 5. we can see the convergence factor in the non-overlapping and overlapping case for algorithm 3 compared to the classical one and algorithm.

16 5.3. Optimization of algorithm 4. Here, we look for s and s of the form s l = p l (+i), l =, such that the p l are solution of the min-max problem ( ) min max ρ 4(k, ω,σ,z,l,p (+i),p (+i)). (5.6) p,p 0 k K As for algorithm 3, we proceed directly with the asymptotic analysis. The non-overlapping case. Letting L = 0, k min = 0, k max = C h and denoting by (p,p ) the solution of the min-max problem (5.6), we find numerically that the solution of (5.6) equioscillates twice, i.e. (p,p ) is solution of ρ 4 (k, ω,σ,z,l,p (+i),p (+i)) = ρ 4 (k, ω,σ,z,l,p (+i),p (+i)) = ρ 4 (k max, ω,σ,z,l,p (+i),p (+i)), where k and k are two interior local maxima of ρ 4. One can to show that k = ω, and for small h we get k = C b h, p = C p h 3 4, p = C q h 3 4. TodeterminetheconstantsC b, C p andc q, wesolveasymptoticallytheequioscillation conditions and the equation stating that k is a maximum, which leads to the system C b = C p C q, C q = C, C p = ( ωσz) 3 8 C Solving this system, we obtain in the original parameters ω, ǫ, µ and σ the asymptotic result p = (ωσµ) 8C 3 4 h 3 4 C 3 p, p = (ωσµ)3 8C 4 h 4 4., ρ 4 = (ωσµ) 8h 4 C 4 5. (5.7) The overlapping case. Alsowithoverlap, L = h, thesolutionofofthemin-max problem (5.6) equioscillates twice, and we find that (p,p ) is solution of ρ 4 (k, ω,σ,z,l,p (+i),p (+i)) = ρ 4 (k, ω,σ,z,l,p (+i),p (+i)) = ρ 4 (k 3, ω,σ,z,l,p (+i),p (+i)), where k, k and k 3 are three interior local maxima of ρ 4. For small h, we get k = C b, k = C b h 5, k3 = C b h 4 5, p = C p h 3 5, and solving the corresponding equations asymptotically, we find after a lengthy calculation for the constants the system C b = ω, C b = C p C q, C b3 = C p, C p = 4C3 q ωσz, C q = C p. After its solution, we obtain in the original parameters ω, ǫ, µ and σ p = (ωσµ) 5 h 3 5, p = (ωσµ) 5 h 5, ρ 4 = 4 (ωσµ) 0 h 5. In Figure 5. we show again a comparison of the convergence factors.

17 Optimization of the algorithm 5. We finally look for s and s of the form s l = p l (+i), l =, such that p l are solution of the min-max problem ( ) min max ρ 5(k, ω,σ,z,l,p (+i),p (+i)). (5.8) p,p 0 k K The non-overlapping case. We set L = 0, k min = 0, k max = C h and let (p,p ) be the solution of the min-max problem (5.6). We observe numerically that this solution equioscillates twice, where (p,p ) solves ρ 5 (k, ω,σ,z,l,p (+i),p (+i)) = ρ 5 (k, ω,σ,z,l,p (+i),p (+i)) = ρ 5 (k max, ω,σ,z,l,p (+i),p (+i)), and k and k are two interior local maxima of ρ 5. Asymptotically, we obtain the behavior k = C b h 3, k = C b h 7 3, p = C p h 0 3, p = C q h 4 3. Proceeding as in the other cases, we find for the constants the system C 4 b = 6C q ωσz, C b = C p C q, C q = C p 3 C, C p = (8 ω σzc 0 ) 3, which allows us to find the asymptotic result in the original parameters to be p = 3ω 8 3ε 6µ 6σ 3 3C 0 3, p 3ω 3σ h 3 0 = 3 3ε6µ 3 6C h , ρ 3ω 3σ 3ε 6µ 6h = C The overlapping case. We finally treat the overlapping case L = h, where the solution of (5.8) equioscillates twice, ρ 5 (k, ω,σ,z,l,p (+i),p (+i)) = ρ 5 (k, ω,σ,z,l,p (+i),p (+i)) = ρ 5 (k 3, ω,σ,z,l,p (+i),p (+i)), and k, k and k 3 are three interior local maxima of ρ 5. Proceeding as before, we arrive after a long asymptotic analysis at ωε 8µ 8σ 3 4 p = ω 4ε 6µ 3 6σ h 5 8, p = h 4, ρ 5 = 3 8 ω 8ε 3µ 3σ 3 6h In Figure 5. we show the convergence factor in the non-overlapping and overlapping cases for the algorithm 5 compared to the classical one, algorithm, algorithm 3 and algorithm 4. Table 5. summarizes the asymptotic results that have been obtained for all algorithms. 6. Numerical results. We present here some numerical experiments in order to illustrate the performance of the optimized algorithms developed in the previous sections. The domain Ω is partitioned into several subdomains Ω j. The Maxwell s equations are approximated by a discontinuous Galerkin method (DG P p ), see [] for further details. In short, given a mesh T h of Ω j such that Ω j = τ k T h, the (DG P p ) method consists in searching an approximate solution of each field component in the set V h = { V L (Ω) / τ k T h,v τk P p [τ k ] }, P p [τ k ] = {polynomial function on τ k of degree p}.

18 7 ρ Classical Schwarz Algorithm 0. Algorithm 3 Algorithm 4 Algorithm k ρ Classical Schwarz Algorithm Algorithm 3 Algorithm 4 Algorithm k Fig. 5.. Convergence factor comparisons of different algorithms for ω = π, σ = and µ = ε =, for both non-overlapping (L = 0) (left figure) and overlapping (L = h = 30 ) (right figure) cases with overlap, L = h Algorithm ρ parameters ( 4 3 9ω 4 σ µ 3 ε ) 8 h 3 4 none 7 6 (ωσµ) /6 h 3 p = (ωσµ)/ (ω 4 εµ 3 σ ) 0h h 3 p = 5(ω 4 εµ 3 σ ) h (ωσµ) 0 h 5 p = (ωσµ) 5, p = (ωσµ) (ω 4 εµ 3 σ ) 3h h 3 5 p = (ω4 εµ 3 σ ) h 5 8 without overlap, L = 0,p = ω σ µ 3 ε C h 3 none 3 h 3 4(ωσµ) 4 C p = (ωσµ) (ω 4 εµ 3 σ ) 4h C (ωσµ) 8h C 4 3(ω 4 εµ 3 σ ) 6h C p = (ωσµ) 8C 3 4 h 3 4 p = 8 3(ω 4 εµ 3 σ ) 6C h h5 (ω 4 εµ 3 σ ) 8 C p = 4 7(ω 4 εµ 3 σ ) 4C h h h 4, p = (ωσµ)3/8 C /4,p = h4 3(ω 4 εµ 3 σ ) 6C h Table 5. Asymptotic convergence factor and optimal choice of the parameters in the transmission conditions. 6.. Performance for a two subdomain decomposition. We first test the propagation of a plane wave in a homogeneous and conductor medium using a TM formulation. The domain is Ω = (0,), and the parameters are constant in Ω, with ε = µ =, σ = 5 and ω = π. We impose on the boundary an incident field W inc = (Hx inc,hy inc,ez inc ) = ( ky µω, kx,)e ik x µω with k = (k x,k y ) = (ω ε i σ ω,0), x = (x,y). The domain Ω is decomposed into two subdomains Ω = (0,/+L) (0,) and Ω = (/,) (0,); L is the size of the overlap and is equal to the mesh size h for the methods with overlap and is equal to zero for the methods without overlap.

19 8 Overlapping case Non overlapping case Iterations 0 Alg. O(h 3/4 ) Alg. O(h /3 ) Alg. 3 O(h 3/0 ) Alg. 4 O(h /5 ) Alg. 5 O(h 3/6 ) Iterations Alg. O(h 3/4 ) Alg. O(h /3 ) Alg. 3 O(h 3/0 ) Alg. 4 O(h /5 ) Alg. 5 O(h 3/6 ) h h Fig. 6.. Number of iterations against the mesh size h, to attain a relative residual reduction of 0 8 Number of subdomains and processors Algorithm Algorithm Algorithm Algorithm Algorithm Table 6. Iteration number comparison for the wave propagation problem in a subsurface For this test, we use a DG P method, i.e. a uniform polynomial approximation of order two. The performance of the algorithms is shown in Figure 6.. In this figure, we show the number of iterations as a function of the mesh size h for both the non-overlapping and overlapping cases. We have also represented in this figure the theoretical asymptotic results denoted by O(h α ). The numerical results are in good agreement with the theoretical asymptotic results which are summarized in Table 5., except for algorithm (the classical algorithm) without overlap where we get better results. The fact that the discretization can improve the performance of Schwarz algorithms has also been observed and explained in [7] for Cauchy-Riemann equations. 6.. A more realistic application. We present now a second more realistic test which consists in the simulation of electromagnetic wave propagation in a heterogenous subsurface medium. This kind of simulation is very important in imaging, see [3] for details. The configuration of the subsurface is shown in Figure 6. and is constituted of media which are characterized by various parameter values ε and σ, ε =.5,.5, 3.5, 4.5 and σ = 0 6, 0 5, The computational domain is decomposed into several subdomains without overlap (a decomposition into eight subdomains is shown for example in Figure 6.3). We test here the performance of the algorithms for a decomposition into two, four, eight and sixteen subdmains. In Table 6., we have summarized the number of iterations needed for convergence, i.e to attain a relative residual of 0 8, depending on the number of subdomains. These

20 9 Fig. 6.. Configuration of a subsurface y x Fig Mesh and decomposition in 8 subdomains results show that the optimized algorithms converge much faster than the classical algorithm (i.e. algorithm ). We can also see that the hierarchy, in terms of number of iterations, of the optimized algorithms is respected as predicted by the theoretical results in Corollary 4..

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