27. A Non-Overlapping Optimized Schwarz Method which Converges with Arbitrarily Weak Dependence on h

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1 Fourteent International Conference on Domain Decomposition Metods Editors: Ismael Herrera, David E. Keyes, Olof B. Widlund, Robert Yates c 003 DDM.org 7. A Non-Overlapping Optimized Scwarz Metod wic Converges wit Arbitrarily Weak Dependence on M.J. Gander, G.H. Golub. Introduction. Optimized Scwarz metods ave been introduced in [] to correct te uneven convergence properties of te classical Scwarz metod. In te classical Scwarz metod ig frequency components converge very fast, wereas low frequency components are only converging very slowly and ence slow down te performance of te overall metod. Tis can be corrected by replacing te Diriclet transmission conditions in te classical Scwarz metod by Robin or iger order transmission conditions wic approximate te classical absorbing boundary conditions used to truncate infinite domains for numerical computations on bounded domains. Te new metods are called optimized Scwarz metods because te new transmission conditions are obtained by optimizing teir coefficients for te performance of te metod. Using transmission conditions different from te Diriclet ones is owever not new. P.-L. Lions proposed in [4] to use Robin transmission conditions to obtain a converging nonoverlapping variant of te Scwarz metod, a result not possible wit Diriclet transmission conditions. But it was in te context of a particular problem, namely te Helmoltz equation, were te importance of radiation conditions was first realized in te PD tesis of Deprés [5]. Several publications for te Helmoltz equation followed; in te context of control [], for an overlapping variant in [], and a first approac to optimize te transmission conditions witout overlap in [4]. An interesting variant of a Scwarz metod using perfectly matced layers can be found in [7]. Fully optimized transmission conditions were publised in [9, ] for te non-overlapping variant of te Scwarz metod and a first approac for te overlapping case can be found in []. Very soon it was realized tat approximations to absorbing boundary conditions were very effective for oter types of equations as well. For te convection-diffusion equation, te first paper proposing optimized transmission conditions for a non-overlapping variant of te Scwarz metod is [3]. Around te same time, a discrete version of suc a Scwarz metod was developed at te algebraic level in [6, 5], but it proved to be difficult to optimize te free parameters. Second order optimized transmission conditions for convection-diffusion were explored in [3] for te non-overlapping case and for symmetric positive definite problems in [7] wit te first asymptotic results of te performance of tose metods. Suc transmission conditions are also crucial in te case of evolution problems, as sown in [0], and for systems of equations, for te Euler equations, see [6]. A complete survey for symmetric positive definite problems wit all te asymptotic performances for overlapping and non-overlapping variants, is in preparation [8]. We sow in tis paper tat te transmission conditions in te optimized Scwarz metods can be cosen suc tat te convergence rate of te metod as an arbitrarily weak asymptotic dependence on te mes parameter, even if no overlap is used. Tis result is obtained by coosing a sequence of transmission conditions wic is applied cyclicly in te optimized Scwarz iteration. Closed form expressions for te transmission conditions are derived wic give an asymptotic convergence rate ρ = O /m )form an arbitrary power of.. Te Model Problem. We consider for tis paper te self adjoint coercive model problem Lu) :=η )u = f, in Ω = R.) and we assume tat te solution ux, y) stays bounded at infinity. We can pose an equivalent problem on R decomposed into two overlapping subdomains Ω =,L) R, Ω = McGill University, mgander@mat.mcgill.ca, supported in part by NSERC grant 806. Stanford University, golub@sccm.stanford.edu, supported in part by DOE DE-FC0-0ER4777.

2 8 GANDER, GOLUB y Γ0 0 L Ω x Ω Γ L Figure.: Decomposition for te model problem. 0, ) R, L>0, wit boundaries Γ L at x = L and Γ 0 at x = 0 as sown in Figure., namely η )v = f in Ω, η )w = f in Ω,.) v = w on Γ L, w = v on Γ 0. Ten te restriction of te solution u of te original problem to Ω coincides wit te solution v of te partitioned problem and te restriction of te solution u to Ω coincides wit w of te partitioned problem. If te overlap owever becomes zero, L = 0, ten te subdomain problems do not necessarily coincide wit te solution u of te original problem any more, one as to introduce te additional condition tat te derivatives need to matc, xw = xv on Γ 0 =Γ L. To make te coupling more robust wit respect to small overlap, we introduce te subdomain coupling x + S v)v = x + S v)w on Γ L, x + S w)w = x + S w)v on Γ 0,.3) were S v and S w are for te moment undetermined linear operators acting in te y direction. Note tat for example coosing S v = S w = p for some constant p>0 leads to subdomain solutions v and w wic coincide wit te solution u of te original problem even if te overlap is zero, L = 0, since ten te conditions x +p)v = x +p)w and x p)w = x p)v on Γ 0 imply bot continuity of te subdomain solution and its derivative at x = 0. Te subdomain problems are ten coupled by a Robin transmission condition, an idea introduced in [4]. Te goal of optimized Scwarz metods is to determine good coices for te operators S v and S w to obtain fast domain decomposition metods at a computational cost comparable to te classical Scwarz metod. 3. An Optimized Scwarz Metod. We introduce a Scwarz relaxation to te system coupled wit te new conditions, η )v n = f, in Ω, η )w n = f, in Ω, x + S v)v n = x + S v)w n on Γ L, x + S w)w n = x + S w)v n on Γ 0. 3.) Tis iteration can be analyzed using Fourier analysis, see for example []. Te convergence rate of tis algoritm is ρk) = η + k σ vk) η + η + k + σ k + σ wk) vk) η + k σ e η+k L wk) 3.)

3 A ROBUST NON-OVERLAPPING OPTIMIZED SCHWARZ METHOD 83 were k is te Fourier variable in te y direction and σ v and σ w denote te symbols of S v and S w. Te optimal transmission operators S v and S w ave tus te symbols σ v = η + k and σ w = η + k because ten te convergence rate vanises and ence te algoritm converges in steps, independent of te size of te overlap L. Unfortunately tese operators are non-local, tey require te evaluation of a convolution and ence polynomial approximations ave been introduced for various types of partial differential equations, see [3, 3, 4, 7, 0,, 9, ]. Te simplest approximation is to use a constant, wic leads to te Robin transmission conditions S v = S w = p for some constant p>0. Te sign of p is needed for well-posedness, but it also guarantees convergence of te algoritm, since ten te convergence rate becomes ) η + k p ρk, p) = η + k + p e η+k L wic is less tan one for all k<, even if te overlap is zero, i.e. L =0. Tofindtebest Robin parameter, one minimizes te convergence rate over all te frequencies relevant to a given discretization, k min < k <k max, wic leads to te min-max problem ) min p 0 max ρk, p). k min <k<k max Te solution of tis problem, wit or witout overlap, can be found in [8] and te convergence rate depends mildly on te mes parameter ; forl = 0 one finds ρ = O )andfor L = O) te result is ρ = O /3 ). In te following section we will make te convergence rate as weakly dependent on as desired for te case L =0. 4. Arbitrarily Weak Dependence on. Te idea is to use different parameters p j for different steps of te iteration. Suppose we want to use m different values p j, j =,...,m in te Robin transmission condition. We ten cycle troug tese different parameters in te optimized Scwarz algoritm, η )v n = f, in Ω, η )w n = f, in Ω, x + p n mod m+ )v n = x + p n mod m+ )w n on Γ L, x p n mod m+ )w n = x p n mod m+ )v n on Γ 0. 4.) Performing again a Fourier analysis in y wit te parameter k of tis algoritm, we obtain te convergence rate depending on p =p,p,...,p m) ρm, p,η,k)=e η+k L min p 0 m j= ) ) m η + k p j. η + k + p j To optimize te performance of tis new algoritm, te parameters p j, j =,...,m in te vector p ave to be te solution of te min-max problem ) max ρm, p,η,k). k min <k<k max Tis optimization problem as to be solved numerically in general, but for L = 0 and m = l it as an elegant solution in closed form for te ADI metod in [9]. Teorem 4. Wacspress 96)) If m = l ten te optimal coice for te parameters p j, j =,,...,m is given by p j = α 0,j, j =,,...,m 4.)

4 84 GANDER, GOLUB e 0..e k e 07 e 06 e 05.e 3.e.e. Figure 4.: On te left dependence of ρ opt on te frequency k wen,, 4, 8 and 6 optimization parameters are used on a fixed range of frequencies, k max = 00π, and on te rigt dependence of ρ max on as goes to zero for,, 4, 8 and 6 optimization parameters. were te α 0,j are recursively defined using te forward recursion and te backward recursion x 0 = η + k min, x i+ = x iy i y 0 = η + k max, y i+ = x i+y i α l, = x l y l, i =0,,...,l 4.3) α i,j = α i+,j αi+,j xiyi α i,j = α i+,j + α i+,j 4.4) xiyi were i = l,l,...,0 and j =,,... l i for eac i. Te convergence rate obtained wit tese parameters is given by yl ) x l max ρk, m) = k min k k max yl + m. 4.5) x l Proof. Te proof uses te equioscillation property of te optimum similar to te case of te Cebysev polynomials and is due to Wacspress in [9]. An elegant version of te proof can be found in Varga [8]. In Figure 4. we sow ow te optimal coice of an increasing number of parameters p j affects te convergence rate of te optimized Scwarz metod. From Figure 4. on te rigt we see tat te more optimization parameters we use, te weaker te dependence on of te convergence rate becomes. Tis indicates tat we can define a sequence of non-overlapping optimized Scwarz metods wit an arbitrarily weak dependence of te convergence rate on te mes parameter using m different constants in te Robin transmission conditions. To prove tis result, we first need te following Lemma 4. For k max = π/ te recursively defined x i and y i in equation 4.3) ave for small te asymptotic expansion x i = i i η + kmin) y i = π + O i ) i ) i+ π ) i + O ) 5 i ) i =0,, )

5 A ROBUST NON-OVERLAPPING OPTIMIZED SCHWARZ METHOD 85 Proof. Te proof is done by induction. For i =0,weave π ) x 0 = η + k min, y0 = π η + = + O). Now we assume tat 4.6) olds for i and compute for i +, using te recursive definition 4.3) first for x i+ x i+ = x iy i = i i η+kmin ) = i i η + kmin ) = i+) i η + kmin) = i+) i η + kmin) and ten for y i+ ) ) ) i +O 5 ) i π ) +O i ) i ) i+ π ) i+ π ) i+ π i+ i+ π i + O ) 5 i ) +O i ) ) i+ + O 5 ) i+ ) y i+ = x i+y i = i i η+k min ) i+ π ) i +O ) 5 i )+ i π +O ) i ) π = + O i+ ) i ) wic completes te induction. Te asymptotic convergence rates for small mes parameter are given in te following Teorem 4. Te non-overlapping optimized Scwarz metod 4.) wit m = l optimally cosen parameters p j, j =,,...,m in te Robin transmission conditions according to 4.) as for small mes parameter te asymptotic convergence rate ρ opt = m η + k min ) 4m mπ m m + O m ). 4.7) Proof. We first need te asymptotic expansions of te square roots of x l and y l given in Lemma 4., xl = l l η + k min) yl = l+ π l ). ) l π + O ) ) l+ + O ) 9 l+ ), Inserting tese expansions into te expression for te optimized convergence rate 4.5) of Teorem 4. we obtain ) yl ρ opt = ) x l m yl + x l = l η+k min ) l+ π ) l+ m +O l ) + l η+k min ) l+ π l+ +O l ) ) m = l η+k = l η+kmin ) l+ π l+ l+ +O l ) mπ min ) l+ l+ l+ +O l ) and te result follows by noting tat m = l. Hence increasing m we can acieve an as weak dependence of te convergence rate on te mes parameter as we like. Te numerical experiments in te next section sow tat tis result also olds for te discretized algoritm.

6 86 GANDER, GOLUB 5. Numerical Experiments. We perform all our computations on a bounded domain for te model problem Lu) :=η )u = f, in Ω = [0, ] wit omogeneous Diriclet boundary conditions. We decompose te domain into two nonoverlapping subdomains Ω =[0, ] [0, ] and Ω =[, ] [0, ] and apply te optimized non-overlapping Scwarz metod for various values of te parameter m. We simulate directly te error equations, i.e. f = 0, and we sow te results for η =. To solve te subdomain problems, we use te standard five point finite difference discretization wit uniform mes spacing in bot te x and y directions. We start te iteration wit a random initial guess so tat it contains all te frequencies on te given mes and we iterate until te relative residual is smaller tan e 6. Table 5. sows te number of iterations required as one refines te mes parameter. Tere are two important tings to notice: first one can Scwarz as a solver Scwarz as a preconditioner m = m = m =4 m = m = m =4 / / / / / Table 5.: Dependence on and m of te number of iterations wen te optimized Scwarz metod is used as a solver or as a preconditioner for a Krylov metod. see tat te dependence of te number of iterations gets weaker as m becomes larger, as predicted by te analysis. Second for small m, using Krylov acceleration leads to significant improvement in te performance, wereas for bigger m, te improvement is almost negligible, Scwarz by itself is already suc a good solver tat Krylov acceleration is not needed. Tis is a property also observed for multi-grid metods applied to tis problem. To see te dependence of te convergence rate on more clearly, we plotted in Figure 5. te number of iterations togeter wit te asymptotic rates expected from our analysis. One can see tat te asymptotic analysis predicts very well te numerically observed results. One even gains almost te additional square-root from te Krylov metod wen Scwarz is used as a preconditioner. Finally we empasize tat te number of iterations given in Table 5. is te number of times we cycled troug all parameter values. In te current implementation terefore te cost of one iteration wit m = 4 is four times te cost of one iteration wit m =. But note tat not eac iteration of te Scwarz metod needs te same resolution now, since it only needs to be effective in te frequency range around te corresponding p j. Te values of p j for m =4wit =/400 are for example p =4.78, p =5.85, p 3 =60.6 and p 4 = Hence te solve wit p in te transmission condition can be done on a very coarse grid, te one wit p on quite a coarse grid, te one wit p 3 on an intermediate grid and only te solve wit p 4 needs to be on a fine grid. In addition we do not need to solve exactly; it is only required to reduce te error in te corresponding frequency range, using a relaxation iteration, wic leads to an algoritm wit natural inner and outer iterations. Doing tis, te cost for arbitrary m will be only a constant times te cost for m = and ence te number of iterations we gave become te relevant ones to compare. Furtermore in tat case, te factor /m in te asymptotic convergence rate 4.7) disappears because now te relevant quantities to compare are ρ m opt and ence one can obtain a convergence rate

7 A ROBUST NON-OVERLAPPING OPTIMIZED SCHWARZ METHOD parameter 0 parameter number of iterations 0 0 / parameters /4 4 parameters /8 number of iterations 0 /4 parameters /8 4 parameters / Figure 5.: Asymptotic beavior of te optimized Scwarz metod, on te left used as an iterative solver and on te rigt as a preconditioner. independent of by coosing te number m like te logaritm of / as is refined. Suc an algoritm ten as te key properties of multigrid, but is naturally parallel like te Scwarz algoritm. REFERENCES [] J. D. Benamou. A domain decomposition metod for te optimal control of system governed by te Helmoltz equation. In G. Coen, editor, Tird international conference on matematical and numerical wave propagation penomena. SIAM, 995. [] X.-C. Cai, M. A. Casarin, F. W. Elliott Jr., and O. B. Widlund. Overlapping Scwarz algoritms for solving Helmoltz s equation. In Domain decomposition metods, 0 Boulder, CO, 997), pages Amer. Mat. Soc., Providence, RI, 998. [3] P. Carton, F. Nataf, and F. Rogier. Métode de décomposition de domaine pour l equation d advection-diffusion. C. R. Acad. Sci., 339):63 66, 99. [4] P. Cevalier and F. Nataf. Symmetrized metod wit optimized second-order conditions for te Helmoltz equation. In Domain decomposition metods, 0 Boulder, CO, 997), pages Amer. Mat. Soc., Providence, RI, 998. [5] B. Deprés. Métodes de décomposition de domains pour les problèms de propagation d ondes en régime armonique. PD tesis, Université Paris IX Daupine, 99. [6] V. Dolean and S. Lanteri. A domain decomposition approac to finite volume solutions of te Euler equations on triangular meses. Tecnical Report 375, INRIA, oct 999. [7] B. Engquist and H.-K. Zao. Absorbing boundary conditions for domain decomposition. Appl. Numer. Mat., 74):34 365, 998. [8] M. J. Gander. Optimized Scwarz metods for symmetric positive definite problems. in preparation, 000. [9] M. J. Gander. Optimized Scwarz metods for Helmoltz problems. In Tirteent international conference on domain decomposition, pages 45 5, 00. [0] M. J. Gander, L. Halpern, and F. Nataf. Optimal convergence for overlapping and nonoverlapping Scwarz waveform relaxation. In C.-H. Lai, P. Bjørstad, M. Cross, and O. Widlund, editors, Elevent international Conference of Domain Decomposition Metods. ddm.org, 999. [] M. J. Gander, L. Halpern, and F. Nataf. Optimized Scwarz metods. In T. Can, T. Kako, H. Kawarada, and O. Pironneau, editors, Twelft International Conference on Domain Decomposition Metods, Ciba, Japan, pages 5 8, Bergen, 00. Domain Decomposition Press.

8 88 GANDER, GOLUB [] M. J. Gander, F. Magoulès, and F. Nataf. Optimized Scwarz metods witout overlap for te Helmoltz equation. SIAM J. Sci. Comput., 00. to appear. [3] C. Japet. Conditions aux limites artificielles et décomposition de domaine: Métode oo optimisé d ordre ). application àlarésolution de problèmes en mécanique des fluides. Tecnical Report 373, CMAP Ecole Polytecnique), 997. [4] P.-L. Lions. On te Scwarz alternating metod. III: a variant for nonoverlapping subdomains. In T. F. Can, R. Glowinski, J. Périaux, and O. Widlund, editors, Tird International Symposium on Domain Decomposition Metods for Partial Differential Equations, eld in Houston, Texas, Marc 0-, 989, Piladelpia, PA, 990. SIAM. [5] H. San and W. P. Tang. An overdetermined Scwarz alternating metod. SIAM J. Sci. Comput., 7: , 996. [6] W. P. Tang. Generalized Scwarz splittings. SIAM J. Sci. Stat. Comput., 3: , 99. [7] A. Toselli. Some results on overlapping Scwarz metods for te Helmoltz equation employing perfectly matced layers. Tecnical Report 765, Courant Institute, New York, June 998. [8] R. Varga. Matrix Iterative Analysis. Prentice Hall, first edition, 96. [9] E. L. Wacspress. Optimum alternating-direction-implicit iteration parameters for a model problem. J. Soc. Indust. Appl. Mat., 0: , 96.