Domain decomposition methods

Transcription

1 Doman decomposton methods Inspred by Martn J. Gander Laurence Halpern 13 mars Block matrces 1.1 Generaltes Remember frst that two matrces A and B can be multpled f and only f the number of columns of A s equal to the number of rows of B : A s m n and B s n p. Splt A n 4 submatrces as a 11 a 1J a 1J+1 a 1n.... a I1 a IJ a IJ+1 a In A a I+11 a I+1J a I+1J+1 a I+1n.... a m1 a nj a nj+1 a mn The matrx A 11 I,J A 1 I,n J A 1 m I,J a 11 a 1J.. a I1 a IJ a 1J+1 a 1n.. a IJ+1 a In a I+11 a I+1J.. a m1 a mj a I+1J+1 a I+1n A 1 n I,n J.. a mj+1 a mn Consder a matrx B splt as has dmensoni, J, has dmensoni,n J, has dmensonn I, J, B has dmensonm I,n J. B 11 J,K BJ,p K 1 Bn J,K 1 Bn J,p K then the product AB can be done n a natural way as for matrces : A 11 I,J A 1 I,n J, A 1 m I,J A m I,n J Secton de Mathématques. Unversté de Genève. -4 rue du Lèvre, CP 64, CH-111 Genève. SUISSE LAGA et CNRS UMR7539. Unversté Pars 13. Avenue J.B. Clément, Vlletaneuse. FRANCE 1

2 AB A 11 I,J A 1 I,n J A 1 m I,J A m I,n J B 11 J,K B 1 J,p K B 1 n J,K B n J,p K A 11 B 11 +A 1 B 1 A 11 B 1 +A 1 B A 1 B 11 +A B 1 A 1 B 1 +A B To make the product of a matrx A of sze m n by a vector of sze n, t can be useful to decompose A nto I J blocks, and X nto J blocks : A 11 A 1J X 1 A.., X. 1.1 A I1 A IJ X J Block dagonal and trangular matrces have the form D T D 0 0 T 0 0 D T D JJ T JJ The product of lower resp. upper block-trangular matrces s lower resp. upper block trangular matrx. Same for the nverse. An example of block-trdagonal matrx s the equdstant fnte dfferences n D, consttuted of N blocks, each block of sze M M, B C 0 M 0 M A 1 C B C.... h M 0M.... C B C 0 M 0 M C B C I M, B Block relaxaton For a system AX b, splt as n 1.1, t s possble to wrte the same algorthms as before wth A D E F, D beng block-dagonal, E lower block-trangular and F upper block trangular. A A 0 0 A D E A JJ A J1 A JJ 0 For example the Jacob method for a block matrx s A 11 0 X 1 m+1 0 A 1 X 1 0 A X A 1 0 X m b 1 + b

3 and the Gauss-Sedel method s A 11 0 X 1 A 1 A X m+1 0 A X 1 X m b 1 + b They can be rewrtten as systems of two matrx equatons Jacob { A 11 X 1 m+1 A 1 X m +b 1 A X m+1 A 1 X 1 m +b Gauss-Sedel { A 11 X 1 m+1 A 1 X m +b 1 A 1 X 1 m+1 + A X m+1 b Each resoluton needs to nvert the matrces A whch are much smaller matrces!. Schwarz methods We explan n the 1d-case the hstorcal methods of H.A. Schwarz alternate method n [7]. and P.L. Lons parallel method n [6]. Then we dscretze these algorthms, and nterpret those dscrete algorthms as relaxatons algorthms. Then we present the dscrete algorthm, n partcular the addtve Schwarz method of M. Dryja et O. Wdlund [, 3], whch are a major nventon. In the next paragraph, we nterpret the latter as a precondtonng for a lnear system. Ths system nvolves ether the nternal unknowns, or the nterface unknowns. Then we study the condtonng of the precondtoned problem..1 Alternate and parallel Schwarz The problem s u +ηu f n 0,1, u0 g g, u1 g d..1 Ω [0,1] s dvded nto two overlappng subdomans Ω 1 [0,β] and Ω [α,1]. The boundary of Ω 1 n Ω s Γ 1 {β}, and symmetrcally Γ {α}.the overlap s δ β α. In the alternate Schwarz method, a sequence u n 1,un for n 0 s buld by solvng alternatvely the same equaton as n.1, n Ω 1 and Ω, defnng the values on the border by the prevously computed values n the othersubdoman : d u n+1 1 dx +ηu n+1 1 f dans Ω 1, d u n+1 u n g g, u n+1 u n+1 1 β u n dx +ηu n+1 f dans Ω, 1 g d, β, un+1 α u n+1 1 α. The algorthm s ntalzed by g R, wth the conventon u 0 β g, whch means that u1 1 s computed wth u 1 1 β g. In the parallel Schwarz method [5], the computatons n Ω 1 et Ω are made n parallel :. d ũ n+1 1 dx +ηũ n+1 1 f dans Ω 1, d ũ n+1 ũ n g g, ũ n+1 ũ n+1 1 β ũ n β, ũn+1 dx +ηũ n+1 f dans Ω, 1 g d, α ũ n 1 α..3 Then two values g 1 et g are necessary for the ntalzaton. Fgure.1 shows the soluton of.1 n a model case : the dstrbuton of temperature n a bar of length 1, subjected to a source of heat on a part of ts length, wth a fxed temperature at each end. On Fgure. are represented the terates of the two algorthms. 3

4 Theorem.1 For any η 0, the algorthms of alternate and paralllel Schwarz for problem.1 are convergent. Proof By lnearty, the errors e n u n u are soluton of the same equatons n the subdomans wth f 0 g g 0 and g d 0. They can be solved wth snhx e x e x / for η > 0, modulo a multplcatve constant a n : for η > 0, e n 1 an 1 snh ηx, e n an snh η1 x, for η 0, e n 1 an 1 x, en an 1 x. At frst teraton, a 1 1 s determned by the condton u1 1 β g, thus e1 1 β g uβ : { for η > 0, a 1 1 snh η β g uβ, for η 0, a 1 1 β g uβ. The transmsson condtons e n+1 1 β e n β and en+1 α e n+1 1 α thereafter gve a recurson relaton to determne the coeffcents a n : { for η > 0, a n+1 1 snh η β a n snh η1 β, a n+1 snh η1 α a n+1 1 snh η α, Let for η 0, a n+1 1 β a n 1 β, an+1 1 α a n+1 1 α. ρ 1 snh η1 β snh, ρ η β snh ηα snh η1 α..4 These formulas hold also for η 0 by passng to the lmt. Rewrte the recurson relaton as a n+1 1 ρ 1 a n, a n+1 ρ a n+1 1, or a n+1 ρ 1 ρ a n. The sequences a n 1 and an are geometrc sequences wth rato ρ ρ 1ρ, whch s also called convergence factor of the method. The functon snh s ncreasng, and snce α < β, we have snh η α < snh η β and snh η1 β < snh η1 α. Thus ρ s postve and strctly smaller than 1. The coeffcents a n are now gven by The functons u n satsfy Ω : a n+1 1 ρ n a 1 1, a n+1 ρ ρ n a u n+1 x ux ρu n x ux ρn u 1 x ux. In the doman Ω, the sequence u n converge unformly to u, wth a lnear convergence In the parallel case, there s an smlar relaton between coeffcents ã n de ũ n : ã n+1 1 ρ 1 ã n, ãn+1 ρ ã n 1, or ãn+1 ρã n, and ã n+1 ρ n ã 1. The even and odd terates of ũn converge lnearly wth the same convergence factor ρ. Remark.1 Defnng g g 1, yelds ũ1 n u n 1, then ũn u n. Therefore performng two steps of the parallel algorthm s equvalent to performng one step of the alternate algorthm, as shown n.. Remark. The smaller ρ, the faster the convergence. Ths s realzed for large η, or large overlap δ. 4

5 . Dscretzed alternate et parallel Schwarz The nterval [0,1] s dvded nto J + 1 subntervals wth length h. The dscretzaton ponts are x j jh for 0 j J + 1. The fnte dfferences schemes assocated to.1 computes u j ux j, wth f j fx j, as follows u j+1 u j +u j h +ηu j f j, 1 j J. These J equatons are complemented wth u 0 g g and u J+1 g d, to obtan the lnear system wth unknowns u u 1,,u J T, n matrx form Au f, A h +η 1 h 1 h h +η h 1 h h +η, f f h g g f. f J f J + 1 h g d The matrx A s wrtten n sparse format sparse n Matlab usng the scrpt spdags : functon AA1deta,a,b,J % A1D one dmensonal fnte dfference approxmaton % AA1deta,a,b,J computes a sparse fnte dfference approxmaton % of the one dmensonal operator eta-delta on the doman % Omegaa,b usng J nteror ponts hb-a/j+1; eonesj,1; Aspdags[-e/h^ eta+/h^*e -e/h^],[-1 0 1],J,J;..6 The scrpt below gves the detals of resoluton n Matlab of the dscrete problem wth non homogeneous boundary condtons The lnear system s solved wth the scrpt "\" of Matlab, based on the Gauss method, optmzed for sparse matrces. Snce A s symmetrc defnte postve, the conjugate gradent wth precondtonng defned n the frst chapter could be used as well. functon usolve1df,eta,a,b,gg,gd % SOLVE1D solves eta-delta n 1d usng fnte dfferences % usolve1df,eta,a,b,gg,gd solves the one dmensonal equaton % eta-deltauf on the doman Omegaa,b wth Drchlet boundary % condtons ugg at xa and ugd at xb usng a fnte % dfference approxmaton wth lengthf nteror grd ponts Jlengthf; AA1deta,a,b,J; hb-a/j+1; f1f1+gg/h^; fendfend+gd/h^; ua\f; u[gg;u;gd]; % construct 1d fnte dfference operator % add boundary condtons nto rhs % add boundary values to soluton In the example, the length of the bar s 1, the source s constant equal to 5 on [ ], vanshes elsewhere. The temperature s fxed at both ends. The Matlab scrpt below Bar.m calls the resoluton scrpt Solve1d for these data, and produces fgure.1. %eta0;j0; x0:1/j+1:1; fzerosj,1; % J number of nteror mesh ponts % fnte dfference mesh, ncludng boundary % source term zero, except for a 5

6 fx>0.4 & x<0.75; % heater n ths poston gg0.1; gd0; % put warm wall on the left, cold on the rght usolve1df,eta,0,1,gg,gd; fgure plotx,u, - ; xlabel x ; ylabel soluton ; soluton Fgure.1 Example of resoluton of equaton.1 by fnte dfferences In order to dscretze the alternate Schwarz algorthm, the pont α s descrbed by α lh and β mh wth m l+d. d represents the overlap, δ dh. The ponts x 1,,x J are nteror to Ω, the ponts x 1,,x b are nteror to Ω 1, twhle ponts x a+1,,x J are nteror to Ω. The dscretzaton of alternate Schwarz algorthm. s un+1 1 j+1 u n+1 1 j +u n+1 1 j h +ηu n+1 1 j f j, 1 j b, u n+1 1 b u n b, un+1 j+1 u n+1 j +u n+1 j h +ηu n+1 j f j, a+1 j J, u n+1 a u n+1 1 a,.7 wth exteror boundary condton u n g g and u n J+1 g d. The scrpt below realzes the algorthm.7 the frst lne Bar;, executes the commands of the above example, whch are mperatvely n the fle Bar.m : eta0;j0; Bar; % to nclude problem parameters ueu;afloorj/; d4; % subdoman decomposton f1f1:a+d-1; ffa+1:j; % subdoman source terms u1[gg; zerosa+d,1]; % zero ntal guess, except boundary value u[zerosj-a+1,1; gd]; x1x1:a+d+1; xxa+1:end; h1/j+1; % fnte dfference meshes fgure1 lne[a,a]*h,[mnue,maxue ], Color, r lne[a,a]+d*h,[mnue,maxue ], Color, r hold on plotx,ue, m ; hold on; pause for 1:00 % Alternatng Schwarz teraton u1solve1df1,eta,x11,x1end,gg,ud+1; usolve1df,eta,x1,xend,u1end-d,gd; x 6

7 plotx1,u1, -,x,u, - ; xlabel x ; ylabel Alternatng Schwarz terates ; hold on; pause end hold off Run n Matlab, t produces the sequence of curves n Fgure.a Alternatng Schwarz terates Parallel Schwarz terates x a Algorthme alternate x b Algorthme parallel Fgure. Example of resoluton of equaton.1 by the Schwarz algorthm dscretzed par fnte dfferences To obtan the parallel Schwarz algorthm, and the results n fgure.b, the loop above has to be modfed nto u1oldu1; u1solve1df1,eta,x11,x1end,gg,ud+1; usolve1df,eta,x1,xend,u1oldend-d,gd; plotx1,u1, -,x,u, - ; xlabel x ; ylabel Parallel Schwarz terates ; Algebrac nterpretaton Algorthm.7 wll now be wrtten as an algebrac algorthm for the vectors u n 1 un 1 1,,u n 1 b T and u n un a+1,,u n J T. The matrx A s splt nto blocs as h +η 1 h 1 h h +η h 1 h h +η 1 h A 1 h h +η 1 h 1 h h +η h 1 h h +η 7..8

8 Introduce two such decompostons : A1 B A 1 C 1 D 1 D C B A,.9 The sze of A 1 s m m, and that of D s l l, et therefore A has a sze J l J l. The matrces A 1 et D concde when m l+1, that s d 1. The geometrc overlap s n ths case mnmal, and the algebrac overlap s empty. The matrces A 1 et A are the matrces of the operator η over Ω 1 and Ω, dscretzed by fnte dfferences,wth homogeneous Drchlet data on the endponts, they are therefore nvertble. Complete now the matrces B wth zero entres n B 1 [0 m,d B 1 ], B [B 0 J l,d ]. Thus B 1 has sze m 1 J l. To a vector defned on Ω, t assocates a vector defned on Ω 1, extended by 0 outsde Ω. Accordngly, B has sze J l m. To a vector defned on Ω 1 t assocates a vector defned on Ω extended by 0 outsde Ω 1. Wth these notatons, the alternate algorthm.7 takes the form A 1 u n+1 1 f 1 B 1 u n, A u n+1 f B u n+1 1,.10 whch s nothng else but block Gauss-Sedel n+1 A1 0 u1 B A u 0 B1 0 0 u1 u n + f1 f.11 for the augmented system Ãũ f : A1 B1 u1 B A u f1 f..1 When the overlap s mnmal, the augmented matrx concdes wth the matrx A. The dscretzed parallel Schwarz algorthm can also be wrtten n algebrac form A 1 u n+1 1 f 1 B 1 u n, A u n+1 f B u n 1,.13 whch s now a block Jacob method for the augmented system.1,.e. n+1 A1 0 u1 0 A u 0 B1 B 0 u1 u n + f1 f..14 Note that, even though the matrx A s symmetrc, the augmented matrx s not, snce the matrces B T et B 1 are dfferent, except for mnmal overlap.e. d 1 t can be seen as follows : the only non-zero term n B 1 s B 1 b,d, and B T b,d B d,b 0 s d 1. Ths forbdds to use the conjugate gradent for the system A 1 0 A1 B1 u1 0 A B A u A A f1 f,.15 However, Krylov algorthm have been desgned for non symmetrc matrces. Other doman decomposton algothms have been desgn to provde symmetrc augmented matrces, as descrbed n the next paragraph. 8

9 .3 Dscrete Schwarz methods : AS, MS et RAS To understand addtve Schwarz AS, go back to.14. In the case d 1, B B, and the teraton s dentcal to n+1 n u1 u1 A n u1 u u 0 A f A..16 u In ths form, t appears that parallel Schwarz dscretzed wth fnte dfferences.14 s an teratve method for the precondtoned system A 1 0 A1 B 1 u1 0 A B A u A A f1 If the matrx A s symmetrc defnte postve, then so s the precondtoner can be solved by conjugate gradent. Introduce now the restrcton matrces f..17 A 1 0 and.17 0 A R 1 [I b 0 b,j b+1 ], R [0 J a,a I J a ]..18 The precondtoner can be wrtten wth these restrcton matrces as : A A 1 R T A R. Snce n ths case u 1,u u, we deduce a new form of the parallel Schwarz algorthm dscretzed by fnte dfferences wth mnmal overlap, d 1,.e..16 : u n+1 u n + 1 R T A R f Au n..19 Ths algorthm can stll be wrtten for the general overlpa, but t s not so useful, as proved by the followng counter-example. Theorem. If d > 1, algorthm.19 appled to the fnte dfference matrx.6 s not convergent : there exsts an ntal guess u 0 such that the algorthm oscllates betweeen u 0 and u 0. Proof Splt A as n.9. For each teraton, the vector u n s splt nto u n 11,un 1 accordng to the frst decomposton de A, and n u n 1,un accordng to the second decomposton. The rght-hand sde f s decomposed accordngly. Therefore Compute now R 1 Au n [A 1 B 1 ]u n A 1 u n 11 +B 1 u n 1, R Au n [B A ]u n B u n 1 +A u n. R1 TA 1 R 1Au n R1 Tun 11 +RT 1 A 1 B 1u n 1, R TA R Au n R Tun +RT A B u n 1. The equaton.19 can be wrtten as u n+1 u n u n 11 0 A 0 u n 1 B 1u n A B u n 1 A + 1 f A f. To study convergence of the algorthm, choose f 0 nul, and for an ndex j strctly between a et a + d, an ntal guess u 0 e j, the j vector of the canoncal bass n R J. The only non-vanshng 9

10 terms n the rght-hand sde of the prevous equaton are the three frst, and they have the same value u 0, thusu 1 u 0. Itaratng the argument, t appears that the terates oscllate between u 0 and u 0. The scrpt Matlab below BarAS.m realzes the terates of Addtve Schwarz.19 for the same example as before. The terates are drawn fgure.3. The lack of convergence n the overlap appears clearly. eta0;j0; Bar; % to nclude problem parameters ueu;afloorj/; d4; % subdoman decomposton h1/j+1; f1f1+gg/h^; fendfend+gd/h^; % add boundary condtons nto rhs AA1deta,0,1,J; % construct fnte dfference operator R1[speyea+d-1 sparsea+d-1,j-a-d+1]; R[sparseJ-a,a speyej-a]; A1R1*A*R1 ; AR*A*R ; fgure4 lne[a,a]*h,[mnue,maxue ], Color, r lne[a,a]+d*h,[mnue,maxue ], Color, r hold on plotx,ue, b ; hold on; pause uzerosj,1; for 1:0 rf-a*u; uu+r1 *A1\R1*r+R *A\R*r; plotx,[gg;u;gd], -k ; xlabel x ; ylabel Addtve Schwarz statonnary terates ; hold on; pause rnormr; end hold off % keep resdual for plottng later Addtve Schwarz statonnary terates Fgure.3 Attempt to solve.6 by Addtve Schwarz.19 The true addtve Schwarz method or AS s based on the teraton.19, seen as a precondtonneur. It conssts n solvng the lmt sprecondtoned system M AS Au : 1 x R T A R Au 10 1 R T A R f..0

11 If A est symmetrc, M AS RT A R est symmetrc, and the conjugate gradent can be used. The multplcatve Schwarz method or MS see [1] s the sequental verson of addtve Schwarz. For our example, t takes the form u n+1 u n +R1 TA 1 R 1f Au n, u n+1 u n+1 +R TA R f Au n+1..1 For Matlab mplementaton, replace n the loop of the prevous scrpt the computaton of r and u by rf-a*u; uu+r1 *A1\R1*r; rf-a*u; uu+r *A\R*r; whch produces the teratons n Fgure Multplcatve Schwarz terates Fgure.4 Example of multplcatve Schwarz algorthm There s no oscllatng mode n the overlap, the teratve algorthm s convergent. It s n many cases equvalent to dscretzed alternate Schwarz [4]. Snce the precondtoner s not symmetrc, the use of conjugate gradent s however prohbted. Références [1] T. F. Chan and T. P. Mathew, Doman decomposton algorthms, n Acta Numerca 1994, Cambrdge Unversty Press, 1994, pp [] M. Dryja, A capactance matrx method for Drchlet problem on polygon regon, Numer. Math., , pp [3] M. Dryja and O. B. Wdlund, An addtve varant of the Schwarz alternatng method for the case of many subregons, Tech. Rep. 339, also Ultracomputer Note 131, Department of Computer Scence, Courant Insttute, [4] M. J. Gander, Schwarz methods over the course of tme, Electron. Trans. Numer. Anal, , pp [5] P.-L. Lons, On the Schwarz alternatng method I, n Frst Internatonal Symposum on Doman Decomposton Methods for Partal Dfferental Equatons, R. Glownsk, G. H. Golub, G. A. Meurant, and J. Péraux, eds., Phladelpha, PA, 1988, SIAM, pp [6], On the Schwarz alternatng method II : Stochastc nterpretaton and orders propertes, n Doman Decomposton Methods, T. Chan, R. Glownsk, J. Péraux, and O. Wdlund, eds., Phladelpha, PA, 1989, SIAM, pp x

12 [7] H. A. Schwarz, Über enen Grenzübergang durch alternerendes Verfahren, Verteljahrsschrft der Naturforschenden Gesellschaft n Zürch, , pp