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1 SIAM J SCI COMPUT Vol 34, No 4, pp A48 A7 c Society for Industrial and Applied Mathematics 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 in both the time-harmonic and the time-domain case The optimization process has been performed 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, 65F, 65N DOI 37/84995 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 advantage 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 equations was given in [9], with complete asymptotic results Applications to real-life problems using a discontinuous Galerkin method can be found in [, ] For finite element based nonoverlapping and nonconforming domain decomposition methods for the computation of multiscale electromagnetic radiation and scattering problems, we refer the reader 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 Submitted to the journal s Methods and Algorithms for Scientific Computing section August, ; accepted for publication in revised form May 5, ; published electronically July 7, INRIA Sophia Antipolis-Méditerranée, 69 Sophia Antipolis Cedex, France mohamed el bouajaji@inriafr, stephanelanteri@inriafr Laboratoire J-A Dieudonné, Université de Nice Sophia-Antipolis, Nice, France dolean@ unicefr Section de Mathématiques, Université degenève, CP 64, Genève, Switzerland Martin Gander@unigech A48

2 SCHWARZ METHODS FOR DAMPED MAXWELL S EQUATIONS A49 one of the optimized Schwarz methods in all detail, and asymptotic results are given for all optimized Schwarz algorithms considered in this paper 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 =, where E and H denote the electric and magnetic field, ε is the electric permittivity, and μ is 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= ReJxexpiωt, Ex,t=ReExexpiωt, Hx,t=ReHxexpiωt, where the positive real parameter ω is the pulsation of the harmonic wave The unknown complex-valued vector fields E and H are the solutions of the time-harmonic Maxwell equations iωεe curl H + σe = J, iωμh +curle = 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 =, an impedance condition on Γ a : BnE, H =BnE 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 BnE, 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 also be seen 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 In order to simplify the presentation we consider a decomposition of the computational domain Ω with boundary Ω into two subdomains Ω i, i =,, such that Ω = Ω Ω WesetΓ ij = Ω i Ω j \ Ω, where Ω i is the boundary of Ω i For this decomposition as shown in Figure 3, the classical Schwarz algorithm consists in computing E j,n+, H j,n+ frome j,n, H j,n, for j =,, by the iterative process 3 iωεe,n +curlh,n σe,n = J in Ω, iωμh,n +curle,n = in Ω, Bn E,n, H,n =Bn E,n, H,n on Γ, iωεe,n +curlh,n σe,n = J in Ω, iωμh,n +curle,n = in Ω, Bn E,n, H,n =Bn E,n, H,n on Γ

3 A5 EL BOUAJAJI, DOLEAN, GANDER, AND LANTERI 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 with L = δ and without overlap L =asafunctionof k, ω =π, σ =,andμ = ε = For the classical Schwarz method, we see that at the interfaces artificial boundaries between subdomains, characteristic information is exchanged This type of conditions has been proposed for the first time in [6] for the second order version of the time-harmonic Maxwell equations In the constant coefficient case for the case of two half-spaces Ω =,L R, Ω =, R, an estimate of the convergence factor of this algorithm has been obtained in [9] using Fourier analysis The convergence factor the reduction of the error between two successive iterations for each Fourier mode k =k y,k z is given by ρ cla k, ω, σ, Z, L = k ω +i ωσz i ω e k ω +i ωσzl k ω +i ωσz+i ω Here ω = ω εμ = ω c denotes the adimensionalized frequency or the wave number, c = εμ is the wave propagation speed, and L is the size of the overlap between the subdomains From this result, we see that if σ>, the method converges also without overlap L = But for the high frequencies, the convergence factor is close to one, and therefore the algorithm is not very efficient; see Figure 3

4 SCHWARZ METHODS FOR DAMPED MAXWELL S EQUATIONS A5 The convergence can be improved by modifying the transmission conditions in the classical Schwarz algorithm 3, namely 3 iωεe,n +curl H,n σe,n = J in Ω, iωμh,n +curle,n = 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 +curle,n = in Ω, Bn +S Bn E,n, H,n =Bn +S Bn E,n, H,n on Γ, where S j, j =,, are tangential, possibly pseudodifferential, operators For σ =, different choices of S j, j =,, have been developed in [9] 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> This leads to optimization problems which are very different from the case σ = 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> 4 Convergence analysis for nonzero electric conductivity We present here an analysis of algorithm 3 for the case where the electric conductivity is nonzero, σ > In [9], it was proved that if the operators S l, l =,, have the Fourier symbol [ k 4 FS l = y kz λσz k ] yk z λ + i ωλ + i ω + σz k y k z kz ky, λσz where λ = k ω + i ωσz, then algorithm 3 converges in two iterations It was also shown that these symbols can be written in different forms, FS l = λ+i ωλ + 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 ωλ i ω σz M, where M and M are given by [ ] [ ] k M = y kz λσz k y k z k k y k z kz k y λσz, M = y kz + λσz k y k z k y k z kz k y + 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 nonlocal operator λ by a constant s Note that after this approximation is performed we get local differential operators: for example, ky is the symbol of the partial differential operator yy Corollary 4 For different symbols of the operators S j which approximate the nonlocal operators 4 we get the following convergence factors Algorithm IfS j, j =,, have the symbols 4 σ j = FS j =,

5 A5 EL BOUAJAJI, DOLEAN, GANDER, AND LANTERI then the classical Schwarz algorithm 3 has the convergence factor k ω 43 ρ k, ω, σ, Z, L = +i ωσz i ω e k ω +i ωσzl k ω +i ωσz+i ω Algorithm IfS j, j =,, have the Fourier symbols 44 σ j = FS j = s i ω s + i ω k + sσz M s, s C, then the Schwarz algorithm 3 has the convergence factor k ω 45 ρ k, ω, σ, Z, L, s = +i ωσz s e k ω +i ωσz+s Algorithm 3 IfS j, j =,, have the Fourier symbols 46 σ j = FS j = then 3 has the convergence factor 47 ρ 3 k, ω, σ, Z, L, s = k ω +i ωσzl k ω +i ωσz +i ω + σzs M s, s C, k ω +i ωσz i ω k ω +i ωσz i ω Algorithm 4 IfS j, j =,, have the Fourier symbols 48 σ j = FS j = ρ k, ω, σ, Z, L, s s j i ω s j + i ω k + s j σz M s j, s j C, then the Schwarz algorithm 3 has the convergence factor k 49 ρ 4 k, ω, σ, Z, L, s,s = ω +i ωσz s l e k ω +i ωσz+s l l= Algorithm 5 IfS j, j =,, have the Fourier symbols 4 σ j = FS j = k ω +i ωσzl k ω +i ωσz +i ω + σzs j M sj, s j C, then the Schwarz algorithm 3 has the convergence factor k ω 4 ρ 5 k, ω, σ, Z, L, s,s = +i ωσz i ω ρ 4 k, ω, σ, Z, L, s,s k ω +i ωσz+i ω Proof The convergence factor for Algorithm was already computed in [9] The interesting cases for the computation are Algorithms 4 and 5 since Algorithms and 3 are particular cases of the former, when the parameters are identical for both subdomains In what follows we will illustrate in detail the proof for Algorithm 4, the treatment of the remaining cases being very similar In order to be self-consistent

6 SCHWARZ METHODS FOR DAMPED MAXWELL S EQUATIONS A53 we will recall here the elements of the technique based on Fourier transform already found in [9] Because of the linearity of the equations, it suffices to analyze the convergence to the zero solution when the right-hand side vanishes Performing a Fourier transform of system in the y and z directions, the first and the fourth equation provide an algebraic expression for Ê and Ĥ, which is in agreement with the fact that these are the characteristic variables associated with the null eigenvalue Inserting these expressions into the remaining Fourier transformed equations, we obtain the first order system 4 x Ê Ê 3 Ĥ + Ĥ 3 kykz i ω/z+σ k yk z i ωz ω +k z +i ωσz i ωz ω k z i ωσz i ω/z+σ ω +k y +i ωσz i ω/z+σ k yk z i ω/z+σ ω k y i ωσz i ωz kykz i ωz Ê Ê 3 Ĥ Ĥ 3 = The eigenvalues of the matrix in 4 and their corresponding eigenvectors are 43 k yk z ω k y i ωσz i ω/z+σλ λ TH, = i ω/z+σλ k ω ω + i ωσz, v = +kz +i ωσz i ω/z+σλ, v = kykz i ω/z+σλ, and 44 λ TH 3,4 = k ω + i ωσz, v 3 = kykz i ω/z+σλ ω k z i ωσz i ω/z+σλ, v 4 = k y ω +i ωσz i ω/z+σλ k yk z i ω/z+σλ, where we set λ := k ω + i ωσz Because of the radiation condition, the solutions of system 4 in Ω l, l =,, are given by 45 Ê ; Ê 3; Ĥ ; Ĥ 3 =α v + α v e λx L, Ê ; Ê 3 ; Ĥ ; Ĥ 3 =β v 3 + β v 4 e λx, where the coefficients α j and β j j =, are uniquely determined by the transmission conditions At the nth step of the optimized Schwarz algorithm, the coefficients α =α,α andβ =β,β in 45 satisfy 46 α n = Ā Ā e λl β n, β n = B B e λl α n, where the matrices Āl and B l, l =,, are given by Ā = A + σ A, Ā = A + σ A, B = A + σ A, B = A + σ A, with A l, l =,, defined as

7 A54 EL BOUAJAJI, DOLEAN, GANDER, AND LANTERI 47 [ k A = y k z kz ω +i ωλ + σzλ + i ω [ A = k y k z kz + ω +i ωλ + σzλ i ω ky ω +i ωλ + σzλ + i ω k y k z ], ky+ ω +i ωλ + σzλ i ω k y k z ] A complete double iteration of the Schwarz algorithm therefore leads to α n+ = Ā Ā B B e λl α n, β n+ = B B Ā Ā e λl β n We first replace σ j from 48 into 46, and we compute the eigenvalues of the iteration matrices Ā Ā B B or equivalently B B Ā Ā : 48 λ = λ s λ + s λ s λ + s, λ = λ λs σz i ω k λs σz + i ω k λs σz i ω k λs σz + i ω k The convergence factor of the algorithm will be given by ρ 4 k, ω, σ, Z, L, s,s = max{ λ, λ }e λl We can see that as soon as the imaginary part of λs j is positive, λ is bigger in modulus than λ This is usually the case, since a simple computation shows that Rλ, Iλ > and the optimized parameter is chosen as s j = p j + i, p j > see[9] and the references therein for details Therefore the convergence factor is ρ 4 k, ω, σ, Z, L, s,s = λ = λ s λ s e λl λ + s λ + s and this ends the proof The different symbols σ j depend on the choice of the parameters s, s,ands 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 /, 49 min s C max k K ρ j k, ω, σ, Z, L, s, j =, 3, min s, s C max k K ρ j k, ω, σ, Z, L, s,s, j =4, 5, 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 5 Optimized transmission conditions In this section, we solve the various min-max problems seen in 49 The fundamental difference with the case σ = is that here we do not need to exclude the resonance frequencies see [9] This changes the nature of the min-max problems We have more local maxima in k, and balancing these maxima by the equioscillation principle is more difficult In a numerical implementation, the range of frequencies is bounded, ie, k K := [k min,k max ], where the minimum frequency k min > is a constant depending on the

8 SCHWARZ METHODS FOR DAMPED MAXWELL S EQUATIONS A55 geometry, and the maximum numerical frequency that can be represented on a mesh is k max = C h,wherec is a constant and h is the mesh size Before solving the min-max problems in 49, we give an asymptotic expression for the maximum of the convergence factor 43 of the classical Schwarz algorithm over k K This allows us to see the behavior of 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 ω 4 σ μ 3 εc 6 8 L h Oh 4, L = CL h, max ρ k, ω, σ, Z, L = k K ω σ μ 3 ε C h 3 + Oh 5, L = 3 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 the solution of the min-max problem 5 min p max ρ k, ω, σ, Z, L, p + i k K 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,, k,p + i = 4ξ ξ p +y ξp 4ξ ξ + p =: Rξ, y, p +y +ξp Since the mapping k ξ k is increasing, the optimization problem 5 can be rewritten as min max ρ ω, Z, σ,, k,p + i =min max Rξ, y, p, p k min k k max p ξ min ξ ξ max where ξ min = ξk min andξ max = ξk max We can prove the following result Theorem 5 In the nonoverlapping case, the solution of the problem is given by 53 p = min p max Rξ, y, p ξ min ξ ξ max { minp,p 3 if y<ξ max ξ min, minp,p if y ξ max ξ min,

9 A56 EL BOUAJAJI, DOLEAN, GANDER, AND LANTERI where p, p, p,andp 3 are constants defined by p = y 4 ξ min y +ξ min + ξ min y, p 3 3 = y 4 ξ min 4 ξ max y +ξ max + ξ max y 3 4 ξ max, p = 8 ξ 4 min +y 8 ξ 4, p = max +y 4 ξ min 4 ξ 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, 54 E := R p ξ, y, p = Rξ, y, p 3 ξ 3 y +64ξ 5 p 4ξ 4 +y 8 ξ 4ξ 4 +8ξ 3 p +8ξ p + y +4yξp 8 ξ4 + y E is negative for p [, 8 ξ4 + y 4 ξ ]andpositiveforp [ 4 ξ, + Moreover, 8 ξ the minimum of the mapping ξ 4 + y 4 ξ is y Thisimpliesthatforp 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 Rξ, y, p by computing the zeros of the derivative of R with respect to ξ, 55 R ξ, y, p = 4 ξ p ξ y 8 ξ 4 +6 y p ξ +y Rξ, y, p4ξ 4 +8ξ 3 p +8ξ p + y +4yξ p The polynomial P ξ = ξ y 8 ξ 4 +6 y p ξ +y hasatmostthree positive roots, and ξ = y is always a root The other roots are given by ξ,3 = 4 y +4p 3 y 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 3yp,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,orξ max From the expression 54 we can derive the following properties of R: a The mapping p Rξ min,y,p is decreasing on [ y,p ] and increasing on [p, + b The mapping p Rξ,y,pisincreasingon[ y, + c The mapping p Rξ max,y,p is decreasing on [ y,p ] and increasing on [p, + To conclude the proof of the theorem, we need the following lemmas

10 SCHWARZ METHODS FOR DAMPED MAXWELL S EQUATIONS A57 Lemma 53 Let p 3 be the constant defined by p 3 = ξmin ξ max y +yξ min +4yξ minξ max +yξ max +4ξ min ξ max 4ξ min ξ max Then the following properties are verified: i p is the unique solution of the equation Rξ min,y,p=rξ,y,p on R + \{} ii p 3 is the unique solution of the equation Rξ min,y,p = Rξ max,y,p on R + \{} iii p 3 is the unique solution of the equation Rξ,y,p=Rξ max,y,p on R + \{} iv We have { p if y<ξ max ξ min, maxp,p = p if y ξ max ξ min v We have vi We have vii We have { p 3 if y<ξ max ξ min, maxp,p 3 = p if y ξ max ξ min 3y p,p 3 p 3 max Rξ, y, p ξ min ξ ξ max Rξ max,y,p Rξ,y,p = Rξ min,y,p Rξ,y,p if y<ξ max ξ min and p [ y,p 3], if y<ξ max ξ min and p [p 3, +, if y ξ max ξ min and p [ y,p ], if y ξ max ξ min and p [p, + Proof To prove the first three properties, we compute 56 R ξ b,y,p R ξ m,y,p = 8 p ξ m ξ b ξ m ξ b y 8 ξ b ξ m p 4 ξb ξ m ξ b y 4 yξ m ξ b yξm 4 y ξ b ξ b + p +y +ξ b p 4 ξm ξ m + p +y +ξ m p This quantity is zero if p = ξb ξ m y +yξ b +4yξ bξ m +yξ m +4ξ b ξ m 4ξ b ξ m By successively replacing the pairs {ξ b,ξ m } by {ξ min,ξ }, {ξ min,ξ max },and{ξ,ξ max } in 56 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 8 ξmax, ξ min which shows that p p if y<ξ max ξ min and p p if y ξ max ξ min

11 A58 EL BOUAJAJI, DOLEAN, GANDER, AND LANTERI To prove property v, we compute y p p ξmax ξ min ξ max ξ min y 3 = 4 ξ 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 y +4ξ min ξmax / y p p 3 = 8 ξ max ξ min, 4 ξ max +y ξ min / y p 3 p 3 = p 3y = y ξmin / y 8 ξ max ξ min, ξ min, p 3 3y = y ξmax / y ξ max This shows that p i 3y andp i p 3, i =, Finally we treat the last point After some computations we obtain E := R ξ min,y,p R ξ max,y,p = 64 pξ min ξ max ξ max ξ min ξ max ξ min y p p 3 4 ξ min ξ min + p +y +ξ min p 4 ξ max ξ max + p +y +ξ max p, E 3 := R ξ max,y,p R ξ,y,p 3 pξ max ξmax / y p p 3 = 4 ξ ξ + p +y +ξp, y + y p +p E 4 := R ξ min,y,p R ξ,y,p 3 pξ min ξmin / y p p = 4 ξ ξ + p +y +ξp y + y p +p From the expression above we can easily see that the following hold: If y < ξ max ξ min and p [ y,p 3],thenE because p 3 p 3 see property vi and E 3 If y < ξ max ξ min and p [p 3, +, then E 4 sincep p 3 see property vi and E 3 If y ξ max ξ min and p [ y,p ],thene ande 4 If y ξ max ξ min and p [p, +, then E 3 sincep p 3 see property vi and E 4 Lemma 53 proves that the solution of the min-max problem 5 depends mainly on the sign of y ξ max ξ min We can see in Figure 5 the solution of the min-max problem for different values of y, ξ min,andξ max The pictures a, b, c, and d show the four cases of Theorem 5 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

12 SCHWARZ METHODS FOR DAMPED MAXWELL S EQUATIONS A59 R 8 6 Rξ min,y,p Rξ,y,p Rξ max,y,p 8 6 Rξ min,y,p Rξ,y,p Rξ max,y,p 4 p p 3 =p * p p p R 4 p 3 p 3 p p p R 4 p p =p * p 3 p p 3 Rξ min,y,p Rξ,y,p Rξ max,y,p p R 4 p =p * p 3 Rξ min,y,p Rξ,y,p p Rξ max,y,p p 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 =5, ξ max = b y = 55, ξ min =5, ξ max = c y = 57, ξ min =35, ξ max =4 d y = 57, ξ min =36, ξ max = Lemma 54 If y<ξ max ξ min, the solution of the min-max problem min max Rξ, y, p y p p3 ξ min ξ ξ max is p = minp,p 3,andthesolutionofthemin-maxproblem min p p 3 max ξ min ξ ξ max Rξ, y, p is p = p 3 Proof By using property vii of Lemma 53, for p [ y,p 3], we have Rξ max,y,p= max Rξ, y, p ξ min ξ ξ max Hence the solution of the first min-max problem is the minimum of the mapping p Rξ max,y,pon[ y,p 3] By property c in the proof of Theorem 5, the minimum of Rξ max,y, on[ y,p 3] is minp,p 3 By using property vii of Lemma 53, for p [p 3, +, we have Rξ,y,p= max Rξ, y, p ξ min ξ ξ max

13 A6 EL BOUAJAJI, DOLEAN, GANDER, AND LANTERI Therefore the solution of the second min-max problem is the minimum of the mapping Rξ,y, on[p 3, + By property b in the proof of Theorem 5, the minimum of Rξ,y, on[p 3, + isp 3 Lemma 55 If y ξ max ξ min, the solution of the min-max problem min max Rξ, y, p y p p ξ min ξ ξ max is p = minp,p,andthesolutionofthemin-maxproblem is p = p min p p max Rξ, y, p ξ min ξ ξ max Proof By using property vii of Lemma 53, for p [ y,p ], we have Rξ min,y,p= max Rξ, y, p ξ min ξ ξ max The solution of the first min-max problem is the minimum of the mapping p Rξ min,y,pon[ y,p ] Using now a in the proof of Theorem 5, the minimum of Rξ min,y, on [ y,p ] is minp,p By proposition vii of Lemma 53, for p [p, +, we have Rξ,y,p= max Rξ, y, p ξ min ξ ξ max Therefore the solution of the second min-max problem is the minimum of the mapping p Rξ,y,pon[p, + By property a, the minimum of Rξ,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 max Rξ, y, p =min min y p ξ min ξ ξ max max Rξ, y, p, y p p3 ξ min ξ ξ max min max Rξ, y, p, p p 3 ξ min ξ ξ max which is useful for the case y<ξ max ξ min, because we then get, using Lemma 54, min max Rξ, y, p = Rξ max,y,minp,p 3 y p ξ min ξ ξ max

14 SCHWARZ METHODS FOR DAMPED MAXWELL S EQUATIONS A6 The min-max problem 5 can also be rewritten as min max Rξ, y, p =min min max Rξ, y, p, y p ξ min ξ ξ max y p p ξ min ξ ξ max min max Rξ, y, p, p p ξ min ξ ξ max which is now useful for the case y ξ max ξ min, because we then get, using Lemma 55, min max Rξ, y, p = Rξ max,y,minp,p y p ξ min ξ ξ max Corollary 56 For h sufficiently small in the nonoverlapping case, L =,the solution of the min-max problem 5 is given by 57 p = ωσμ 4 C and ρ = 3 4 ωσμ 4 h + Oh 4 h C Proof If we assume h sufficiently small, then ξ max = ξc/h is large By this assumption, the solution of the min-max problem 5 is given by the first equation of 53, ie, p = 4 y ξ max y +ξmax + ξ max y ξ max By expanding p for h small, using that ξ max = ξc/h, we get the desired result Corollary 56 gives the optimal parameter p and the corresponding convergence factor 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 terms 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 57 Let h = C h /ω γ If γ =, then for ω large, the solution of the min-max problem 5 is given by 58 p = σμ 4 C εμch 4 ω 3 4 and ρ = 3 4 σμ Ch 4 Ch C εμch ω 4 4 If γ >, for ω large, the solution is 59 p = σμ 4 C ω γ+ 4 Ch 4 and ρ = 3 4 σμ 4 Ch C ω γ 4 Proof If we assume ω sufficiently large, then ξ max = ξωc/c h is large By this assumption, the solution of the min-max problem 5 is given by the first equation of 53, ie,

15 A6 EL BOUAJAJI, DOLEAN, GANDER, AND LANTERI p = 4 y ξ max y +ξmax + ξ max y ξ max By expanding p for ω large, using that ξ max = ξωc/c h, we get the desired result 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 obtain tractable best approximation problems and to be able to present a complete set of optimized transmission conditions for all algorithms We start with the case of the previous paragraph, but now consider an overlapping method We set L = h, k min =,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, ie, p is the solution of 5 ρ k, ω, σ, Z, L, p + i = ρ k, ω, σ, Z, L, p + i, 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,andc p, we now solve 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 = C b = ω, ρ k k, ω, σ, Z, L, p + i = C b = C p The equioscillation equation 5 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 Solving the three equations obtained for the constants C b, C b,andc p,andexpressing the results in the original problem parameters ω, ɛ, μ, andσ, we find the asymptotic result 5 p = ωσμ 3 h 3, ρ = 7 6 ωσμ 6 h 3 Lemma 58 Asymptotically, the parameter p defined in 5 is a local minimum of the min-max problem 5 Proof The parameter p is a local minimum if there exists no variation δp such that ρ k, ω, σ, Z,, p + δp + i ρ k, ω, σ, Z,,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,,p + i < fork = 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

16 SCHWARZ METHODS FOR DAMPED MAXWELL S EQUATIONS A63 ρ Classical Schwarz Algorithm 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 nonoverlapping L = left and overlapping L = h = 3 rightcases k ω + C b h 3, k C b h 3 + Cb, p C p h 3 + Cp h 3 With these three new constants determined, we obtain for the derivative of ρ 3 ρ p k, ω, σ, Z, h, p + i h 3, ωσμ 6 ρ p k, ω, σ, Z, h, p + i 5 6 h 3 ωσμ 6 Since these leading terms differ in sign, there cannot 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 nonoverlapping and overlapping cases for Algorithm compared to the classical Algorithm 5 Optimization of Algorithm 3 Here, we look for s of the form s = p+i, such that p is the solution of the min-max problem 5 min max ρ 3k, ω, σ, Z, L, p + i p k K Again we are only searching for an asymptotic solution of this best approximation problem The nonoverlapping case Setting L =,k min =,k max = C h and denoting by p the solution of 5, we obtain from numerical experiments that the solution of 5 equioscillates once, ie, p is the solution of 53 ρ 3 k max, ω, σ, Z,,p + i = ρ 3 k, ω, σ, Z,,p + i, where k b is an interior local maximum 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,,p + i = C p = C4 b 3 ω σz

17 A64 EL BOUAJAJI, DOLEAN, GANDER, AND LANTERI and the equioscillation equation 53, 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, ɛ, μ, andσ, we find 54 p = C 7 ω 7 μ 4 ɛ 4 σ 7 h 4 7, ρ 3 = ω 7 μ 3 4 ɛ 4 σ 7 h C 3 7 Lemma 59 Asymptotically, the parameter p defined in 54 is a local minimum of the min-max problem 5 Proof The proof follows the same argument as the proof of Lemma 58 We obtain after a lengthy computation that k C b h 7 + Cb h 7, p C p h 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,,p + i 4h C, ρ 3 p k, ω, σ, Z,,p + i σ 3 7 Z 3 7 h 9 7, 4 ω 7 C 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, ie, p is the 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, weget k = C b h, k = C b h 7, p = C p h 5, and solving the corresponding equations asymptotically, we find in the original parameters ω, ɛ, μ, andσ 55 p = 3 5 ε μ 3 ω 5 σ h 5, ρ 3 = ω 5 ε μ 3 σ h 3 In Figure 5 we can see the convergence factor in the nonoverlapping and overlapping cases for Algorithm 3 compared to the classical Algorithm and Algorithm 53 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 solutions of the min-max problem 56 min max ρ 4k, ω, σ, Z, L, p + i,p + i p,p k K As for Algorithm 3, we proceed directly with the asymptotic analysis The nonoverlapping case Letting L =,k min =,k max = C h and denoting by p,p the solution of the min-max problem 56, we find numerically that the solution of 56 equioscillates twice, ie, p,p is the solution of ρ 4 k, ω, σ, Z, L, p + i,p + i = ρ 4k, ω, σ, Z, L, p + i,p + i = ρ 4 k max, ω, σ, Z, L, p + i,p + i,

18 SCHWARZ METHODS FOR DAMPED MAXWELL S EQUATIONS A65 where k and k are two interior local maxima of ρ 4 One can 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 To determine the constants C b, C p,andc q, we solve asymptotically the equioscillation 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 4 Solving this system, we obtain in the original parameters ω, ɛ, μ, andσ the asymptotic result 57 p = ωσμ 8 C 3 4 h 3 4 C 3 p, p = ωσμ 3 8 C 4 h 4, ρ 4 = ωσμ 8 h 4 C 4 The overlapping case Also with overlap, L = h, the solution of of the min-max problem 56 equioscillates twice, and we find that p,p is the 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,andk 3 are three interior local maxima of ρ 4 For small h, weget 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 = 4 C3 q ωσz, C q =Cp After its solution, we obtain in the original parameters ω, ɛ, μ, andσ p = ωσμ 5 h 3 5, p = ωσμ 5 h 5, ρ 4 = 4 ωσμ h 5 In Figure 5 we show again a comparison of the convergence factors 54 Optimization of Algorithm 5 We finally look for s and s of the form s l = p l + i, l =,, such that p l are solutions of the min-max problem 58 min max ρ 5k, ω, σ, Z, L, p + i,p + i p,p k K The nonoverlapping case We set L =,k min =,k max = C h and let p,p be the solution of the min-max problem 56 We observe numerically that this solution equioscillates twice, where p,p solves ρ 5 k, ω, σ, Z, L, p + i,p + i = ρ 5k, ω, σ, Z, L, p + i,p + i = ρ 5 k max, ω, σ, Z, L, p + i,p + i,

19 A66 EL BOUAJAJI, DOLEAN, GANDER, AND LANTERI 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 3, p = C q h 4 3 Proceeding as in the other cases, we find for the constants the system Cb 4 =6C q ωσz, C b = C p C q, C q = C, C p = 8 ω σzc 3, which allows us to find the asymptotic result in the original parameters to be C p 3 p = 8 3 ω 3 ε 6 μ 3 6 σ 3 C 3, p h = 3 ω 6 3 σ 3 3 ε 3 6 μ 9 6 C 4 3, h 4 3 ρ 5 = 34 3 ω 3 σ 3 ε 6 μ 3 6 h C 3 3 The overlapping case We finally treat the overlapping case L = h, where the solution of 58 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, Table 5 Asymptotic convergence factor and optimal choice of the parameters in the transmission conditions 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 ωσμ/3 p = 3 7 ω 4 εμ 3 σ h h 3 p = 5 ω 4 εμ 3 σ h ωσμ h 5 p = ωσμ 5, p = ωσμ ω 4 εμ 3 σ 3 h h 3 5 p = ω4 εμ 3 σ h 5 8 Without overlap, L =, p = ω σ μ 3 ε C 3 h 3 none 3 4 ωσμ 4 h C p = ωσμ 4 C 4 h h 5 ω 4 εμ 3 σ h ω 4 εμ 3 σ 4 h C ωσμ 8 h 4 C ω 4 εμ 3 σ 6 h C 3 3 p = ωσμ 8 C 3 4 p = 4 7 ω 4 εμ 3 σ 4 C 4 7 h h 4 7, p = ωσμ3/8 C /4 p = 3 8 ω 4 εμ 3 σ 6 C h, p = 3 h 4 3 ω 4 εμ 3 σ 6 3 C h 3 4

20 SCHWARZ METHODS FOR DAMPED MAXWELL S EQUATIONS A67 and k, k,andk 3 are three interior local maxima of ρ 5 Proceeding as before, we arrive after a long asymptotic analysis at p = ω 4 ε 6 μ 3 6 σ h 5 8 ωε, p = 8 μ 3 8 σ 3 4, ρ h 5 = 4 8 ω 8 ε 3 μ 3 3 σ 6 h 3 6 In Figure 5 we show the convergence factor in the nonoverlapping and overlapping cases for Algorithm 5 compared to the classical Algorithm, 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 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 equations are approximated by a discontinuous Galerkin method DG P p ; see [7] for further details In short, given a mesh T h of Ω j such that Ω j = τ k T h,thedg 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} In all that follows we note that the computational cost for one iteration of all algorithms is very similar since optimized algorithms introduce only a few modified entries in the discretized problems at the nodes of the interface, and these modified entries have no influence on the cost of the direct solvers we use in the interior 6 Performance for a two subdomain decomposition We first test the propagation of a plane wave in a homogeneous and conductor medium using a transverse magnetic formulation To perform these tests, which confirm the theoretical asymptotic results, we used the optimized Schwarz methods as solvers without a Krylov method The domain is Ω =,, and the parameters are constant in Ω, with ε = μ =, σ = 5, andω = π We impose on the boundary an incident field W inc = Hx inc,hinc y,einc z = ky μω, kx μω, e ik x with k = k x,k y = ω ε i σ ω,, x =x, y The domain Ω is decomposed into two subdomains Ω = Overlapping case Non overlapping case Iterations Alg Oh 3/4 Alg Oh /3 Alg 3 Oh 3/ Alg 4 Oh /5 Alg 5 Oh 3/6 Iterations Alg Oh 3/4 Alg Oh /3 Alg 3 Oh 3/ Alg 4 Oh /5 Alg 5 Oh 3/6 3 h 3 h Fig 6 Number of iterations against the mesh size h, to attain a relative residual reduction of 8

21 A68 EL BOUAJAJI, DOLEAN, GANDER, AND LANTERI Iterations DGTH P Oω / DGTH P Oω 3/8 DGTH P3 Oω /3 DGTH P4 Oω 5/6 ω Fig 6 Number of iterations against the wave number ω for four different DG-element orders, and theoretically predicted asymptotics, /+L, and Ω =/,, ; 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 For this test, we use a DG P method, ie, 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 nonoverlapping and overlapping cases We have also represented in this figure the theoretical asymptotic results denoted by Oh 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 [8] for Cauchy Riemann equations We next show a numerical experiment in order to illustrate the weak dependence of our optimized Schwarz method on the wave number ω, and to illustrate Corollary 57 We choose the same setting as in the previous experiment, but now set h := C h /ω γ, with γ =p +/p, where p is the polynomial degree of the DG-element we are using This choice of the mesh size avoids the pollution effect; see [] We show in Figure 6 for σ = the iteration number our optimized Schwarz method from Corollary 57 needs when we increase the wave number ω These numerical experiments illustrate well the weak dependence of the convergence behavior on the wave number ω, as predicted by our analysis 6 A more realistic application We now present a second, more realistic, test which consists in the simulation of electromagnetic wave propagation in a heterogeneous subsurface medium For this test case, the optimized Schwarz methods are used as preconditioners for a Krylov method here BiCGStab This kind of simulation is very important in imaging; see [3] for details The configuration of the subsurface is shown in Figure 63 and is composed of media which are characterized by various parameter values ε and σ: ε =5, 5, 35, 45 and σ = 6, 5, 4, 3 The computational domain is decomposed into several subdomains without overlap; an example with eight subdomains for a mesh with 4943 elements is shown in Figure 64 We test here the performance of the algorithms for a decomposition into two, four, eight, and 6 subdomains, for two mesh resolutions: 4943 and 977 elements In Table 6, we show the number of iterations needed for convergence, ie,

22 SCHWARZ METHODS FOR DAMPED MAXWELL S EQUATIONS A69 Fig 63 Configuration of a subsurface - -4 y x Fig 64 Mesh and decomposition in eight subdomains Table 6 Iteration number comparison for the wave propagation problem in a subsurface with two meshes, the original one shown in Figure 64 and a refined one where each triangle was cut into four 4943 elements 977 elements No of subdomains/processors Algorithm Algorithm Algorithm Algorithm Algorithm

23 A7 EL BOUAJAJI, DOLEAN, GANDER, AND LANTERI to attain a relative residual of 8, depending on the number of subdomains and the resolution These results show that the optimized algorithms converge much faster than the classical algorithm ie, Algorithm We can also see that the hierarchy of the optimized algorithms, in terms of number of iterations, is respected, as predicted by the theoretical results in Corollary 4, and that refining the mesh leads only to a very moderate increase in the iteration numbers required for the solution when using optimized algorithms Even increasing the number of subdomains only leads to a moderate increase in the iteration numbers in these experiments 7 Conclusions We have developed several domain decomposition algorithms based on optimized transmission conditions for Maxwell s equations with nonzero electric conductivity Two algorithms are obtained by approximating the transparent operator by zeroth order transmission conditions, while two further algorithms are based on second order transmission conditions We have shown that the convergence factor of each algorithm can be written as Oh αi, i =,,5 Our results are well confirmed by the first numerical test with a decomposition into two subdomains, where we also obtained numerical convergence factors in the form Oh βi, i =,,4, with α i β i The first test also shows that optimized Schwarz algorithms converge much faster than the classical one This is also confirmed by the second more realistic test which also shows that these optimized algorithms can be effective for more complex problems and for arbitrary decompositions REFERENCES [] A Alonso Rodriguez and L Gerardo-Giorda, New nonoverlapping domain decomposition methods for the harmonic Maxwell system, SIAM J Sci Comput, 8 6, pp [] I M Babuška and S A Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM J Numer Anal, , pp [3] P Chevalier and F Nataf, An OO Optimized Order method for the Helmholtz and Maxwell equations, in th International Conference on Domain Decomposition Methods in Science and in Engineering, AMS, Providence, RI, 997, pp 4 47 [4] P Collino, G Delbue, P Joly, and A Piacentini, A new interface condition in the nonoverlapping domain decomposition for the Maxwell equations, Comput Methods Appl Mech Engrg, , pp 95 7 [5] B Després, Décomposition de domaine et problème de Helmholtz, CR Acad Sci Paris, Sér I Math, 3 99, pp [6] B Després, P Joly, and J Roberts, A domain decomposition method for the harmonic Maxwell equations, in Iterative Methods in Linear Algebra, North-Holland, Amsterdam, 99, pp [7] V Dolean, M El Bouajaji, M J Gander, S Lanteri, and R Perrussel, Domain decomposition methods for electromagnetic wave propagation problems in heterogeneous media and complex domains, in Domain Decomposition Methods in Science and Engineering XIX, Lect Notes Comput Sci Eng 78, Springer, Heidelberg,, pp 5 6 [8] V Dolean and M J Gander, Can the discretization modify the performance of Schwarz methods?, in Domain Decomposition Methods in Science and Engineering XIX, Lect Notes Comput Sci Eng 78, Springer, Heidelberg,, pp 7 4 [9] V Dolean, M J Gander, and L Gerardo-Giorda, Optimized Schwarz methods for Maxwells equations, SIAM J Sci Comput, 3 9, pp 93 3 [] V Dolean, S Lanteri, and R Perrussel, A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods, J Comput Phys, 7 8, pp 44 7 [] V Dolean, S Lanteri, and R Perrussel, Optimized Schwarz algorithms for solving timeharmonic Maxwell s equations discretized by a discontinuous Galerkin method, IEEE Trans Magn, 44 8, pp [] M El Bouajaji, V Dolean, M J Gander, and S Lanteri, Comparison of a one and two parameter family of transmission conditions for Maxwell s equations with damping, in

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