An experimental study of interface relaxation methods for composite elliptic differential equations

Transcription

1 Avalable onlne at Appled Mathematcal Modellng (8) 66 An expermental study of nterface relaxaton methods for composte ellptc dfferental equatons P. Tsompanopoulou *, E. Vavals Unversty of Thessaly, Department of Computer and Communcaton Engneerng, 7 Glavan, GR 8 Volos, Greece Center for Research and Technology Thessaly, Technology Park of Thessaly, GR 85 Volos, Greece Receved January 7; receved n revsed form Aprl 7; accepted 9 Aprl 7 Avalable onlne 6 July 7 Abstract The recently proposed smulaton framework of nterface relaxaton for developng mult-doman mult-physcs smulaton engnes s consdered. An expermental study of the behavor of two representatve nterface relaxaton methods s presented. Three lnear and one non-lnear ellptc two-dmensonal PDE problems are consdered and they are coupled wth both cartesan and general decompostons. The characterstcs and the effectveness of the proposed collaboratve PDE solvng framework n general, and of the two nterface relaxaton methods n partcular are shown. Ó 7 Elsever nc. All rghts reserved. PACS: 5A5; 5A9; 5A Keywords: Doman decomposton methods; Dfferental equatons; nterface relaxaton methods. ntroducton The smulaton and modelng of complex physcal systems usually nvolve many components of dfferent natures and often requre parallel computng strateges. Conventonal modelng strateges that are typcally based on sngle physcal descrptons over a doman seem to be unnatural to some extent and neffectve n many respects exstng smulaton software apples only to smpler geometrcal shapes and physcal stuatons. New approaches for modelng and smulatng complex physcal systems n a more natural way and, at the same tme, n a more convenent manner have been proposed durng the past two decades. There are manly three such approaches for complex problems that nvolve dfferental operators. They are based on () the doman decomposton (substructurng), () the Schwarz splttng and () the nterface relaxaton technques. The nterface relaxaton methods do not consder a mult-doman and mult-physcs (potentally dfferent PDE operators are appled on dfferent subdomans) smulaton engne as a coherent all. They rather treat t as a loosely coupled system of subproblems consstng of much smpler (both as far as the geometry and the * Correspondng author. Present address: Unversty of Thessaly, Department of Computer and Communcaton Engneerng, 7 Glavan, GR 8 Volos, Greece. Tel.: ; fax: E-mal address: yota@nf.uth.gr (P. Tsompanopoulou). 7-9X/$ - see front matter Ó 7 Elsever nc. All rghts reserved. do:.6/j.apm.7..

2 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 6 dfferental operator concerns) PDE problems. Assume that gven some boundary condtons we ether have the analytc soluton or we can easly compute an approxmate soluton of these smpler subproblems. The nterface relaxaton s an teratve method that uses a lbrary of exact or sngle, smple-doman, smple operator PDE solvers and a collecton of smoothng procedures to solve a composte PDE problem as follows: () Guess soluton values (and dervatves f needed) on all subdoman nterfaces. () Solve all sngle PDEs on all the subdomans ndependently usng these estmated values on the nterfaces. () Compare and mprove the values on the nterfaces usng a relaxer (see below). () Return to Step untl satsfactory accuracy s acheved. Collaboratng PDE solvers based on nterface relaxaton have been already proposed [] and have been rather extensvely consdered for ar polluton [6,7], underwater acoustcs [8] and gas turbne engne [9,] related smulatons. A relatvely large class of nterface relaxaton method have been derved [] and theoretcally analyzed (e.g., [,5,,]) for model problems and decompostons. Numercal experments have been also already presented (see for example [,]) but most of them are manly for one-dmensonal problems or smple two-dmensonal ones but wth one-dmensonal decompostons. For ths paper, we have selected from the exstng class of nterface relaxaton methods two typcal representatves that combne most of ther characterstcs. Namely the ROB and the AVE methods have been consdered. The frst one smooths the estmaton of both the soluton and ts normal dervatve on the nterfaces va a sngle per teraton step based on nterface condtons of Robn type, whle the latter relaxes the soluton and ts normal dervatves separately va weghted average schemes n two dstnct steps. The rate of the convergence of both methods s ndependent of the numercal scheme and the dscretzaton detals dependng only on the physcal characterstcs of the PDE problem and the hgh defnton of the nterfaces. For the AVE and ROB methods we have performed an extensve set of numercal experments for general two-dmensonal PDEs wth general two-dmensonal decompostons. The objectve of ths study s to present selected performance data that contrbute to the understandng of the characterstcs and the dosyncraces of nterface relaxaton, further confrm certan exstng theoretcal results and elucdate some applcablty and extensblty ssues. The rest of ths paper s organzed as follows. Secton brefly presents the nterface relaxaton methods consdered n ths study and ther mplementaton on a dstrbuted envronment usng an Agents system. n Secton we defne the PDE problems we consdered and n Sectons we present the numercal results from our expermentaton. Our concludng remarks can be found n Secton 7.. nterface relaxaton methods and the ScAgents framework For the two nterface relaxaton methods n ths study we consder the boundary value problem Lu ¼ f ; x X ¼ [p ¼ X subject to boundary condtons on ox whch, for smplcty, are taken to be homogeneous Drchlet. We denote the restrctons of u, L and f n X by u, L and f respectvely and so we have the set of local PDEs L u ¼ f ; x X ¼ ;...; p: ðþ We assume that the X s do not overlap and that certan condtons are mposed on the nterfaces. These condtons may range from smple contnuty condtons of the soluton and/or ts dervatves to general ones (e.g., conservaton of momentum) mplctly gven as follows:! G ;j u ; ou ; u j ; ou \ j ; J ; J ¼ on C ;j X Xj ; ðþ og ;j og j; where g denotes the outward normal to the nterface dervatve and where J and J are the jump quanttes of the u s and ts dervatves. G,j mght be a functon mappng on the nterface or even a functonal. For smplcty n the presentaton of the two nterface relaxaton methods consdered n ths study we assume no partcular nterface condtons (besdes the ones naturally mposed by the requred contnuty), ðþ

3 6 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 and that X s decomposed n an one-dmensonal fashon nto the p non-overlappng subdomans X [x,x ], =,...,p wth x < x X for =,...,p... The ROB method The ROB scheme s defned for the above model problem by the followng algorthm:. Defne: g ¼ duðkþ þ dx þ k u ðkþ þ x¼x x¼x g þ ¼ duðkþ dx þ k u ðkþ x¼x x¼x 9 >= ¼ ;...; p : >;. Choose ntal guesses u ðþ for the solutons on each subdoman X, =,,..., p.. Compute the sequence of subdoman solutons u ðkþ ; k ¼ ; ;... by solvng the followng PDE problems: L u ðkþþ ¼ f n X L p up ðkþþ ¼ f p n X p u ðkþþ ¼ x¼x duðkþþ p dx þ k p u ðkþþ p ¼ g p p du ðkþþ x¼xp x¼xp dx þ k u ðkþþ ¼ g x¼x up ðkþþ ¼ x¼x x¼xp 9 L u ðkþþ ¼ f n X duðkþþ dx þ k u ðkþþ ¼ g >= x¼x x¼x ; ¼ ;...; p : du ðkþþ dx þ k u ðkþþ ¼ g >; x¼x x¼x Ths scheme, frst proposed n [], s based on a smple relaxaton technque that nvolves the Robn nterface condtons shown above. The DE problem s solved n each subdoman where the boundary condtons are provded from the prevously computed soluton and ts normal dervatve from the adjacent subdomans. The relaxaton parameter k controls the nfluence of the functon value and/or ts normal dervatve on the smoothng Robn nterface condtons... The two-step average AVE method The AVE [,] method s a two-step teratve scheme descrbed by the followng algorthm:. Choose ntal guesses u ðþ for the soluton on each subdoman X, =,,...,p.. Compute the odd terms of the sequence of subdoman solutons u ðkþþ by solvng the followng PDE problems: g ¼ b du ðkþ þð b dx Þ duðkþ þ ; ¼ ;...; p ; dx x¼x x¼x for ¼ ;...; p L u ðkþþ ¼ f n X L u ðkþþ L ¼ f n X p up ðkþþ ¼ f p n X p u ðkþþ ¼ du ðkþþ du ðkþþ x¼x dx ¼ g p dx ¼ g p p du ðkþþ x¼x x¼xp dx ¼ g du ðkþþ x¼x ¼ g u ðkþþ p ¼ dx x¼xp x¼x

4 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) Compute the even terms of the sequence of subdoman soluton u ðkþ by solvng the followng PDE problems: h ¼ a u ðkþþ þð a Þu ðkþþ þ ; ¼ ;...; p ; x¼x x¼x L u ðkþþ for ¼ ;...; p ¼ f n X u ðkþþ L u ðkþþ ¼ f n X L p up ðkþþ ¼ f p n X p ¼ x¼x u ðkþþ ¼ h u ðkþþ p ¼ h p p u ðkþþ x¼xp ¼ h x¼x x¼x u ðkþþ j x¼x ¼ h up ðkþþ j x¼xp ¼ The relaxaton parameters a and b are to smooth the functon and ts normal dervatve respectvely, and they both take values n (, ). n the frst step (odd terms), the Drchlet problem s solved for each subdoman. The boundary values are computed as a convex combnaton of the prevously computed solutons on adjacent domans. Then a convex combnaton of the normal dervatves of the prevously computed solutons n each subdoman are used to smooth the dervatves on each nterface. Usng these estmates of the normal dervatves, the Neumann problem s solved n the second step (even terms) for all subdomans... The ScAgents expermental framework We have mplemented (manly usng C and Java) a whole class of nterface Relaxaton methods n an agent-based framework for general two-dmensonal decompostons of lnear and non-lnear ellptc PDE problems. Detals about ths mplementaton, whch s known as ScAgents can be found n [8,]. The Sc- Agents explot the nherent parallelsm n the nterface relaxaton methods usng the Agents computng paradgm over a network of heterogeneous workstatons. Specfcally, ScAgents transform the physcal problem (for example, the one assocated wth doman X V n Fg. ) nto a network of local PDE solvers and nterface relaxers (marked as solvers and medators respectvely n Fg. ). Ths network can then easly mapped onto a heterogeneous dstrbuted computng platform whle the coordnaton s takng place accordng to generc teratve workflows lke the ones descrbed above. n all experments we have calculated an ntal guess for the soluton by usng the approprate nterpolant on each nterface segment. For ntal guesses of the normal dervatves we smply mpose the correct sgn (drecton), by settng them equal to the unt outward normal vector. To calculate the requred by the medators dervatves on the nterfaces the assocated buld-n to ELLPACK procedure was used as t s descrbed n []. All experments presented n ths paper were run n parallel usng sngle precson arthmetc on heterogeneous SUN workstatons connected through an ethernet lne. Each subdoman was assgned to a dfferent machne and all the nterfaces were assgned to an addtonal machne. Fg.. The network of solvers and medators for the composte problem assocated wth doman X V shown n Fg..

5 6 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66. The PDE problems under nvestgaton For our expermental analyss we have consdered four domans. They are depcted n Fg. (two on ts left fgure and the other two on ts rght) together wth the assocated boundary condtons, the decompostons of each one of the four domans nto subdomans and the numberng of these subdomans. Specfcally we have consdered [X ]: The doman presented on the left fgure, consstng of the eght rectangular subdomans ðx ; ¼ ;...; 7Þ made dstnct by dotted lnes. [X ]: The doman presented on the left fgure, consstng of the four subdomans ðx ; ¼ ;...; Þ made dstnct by sold lnes. [X ]: The doman presented on the rght fgure, consstng of the four rectangular subdomans ðx ; ¼ ;...; Þ made dstnct by dotted lnes. [X V ]: The doman presented on the rght fgure, consstng of the four subdomans ðx V ; ¼ ;...; Þ made dstnct by sold lnes. Next we defne four composte Partal Dfferental Equatons by defnng, for each one of them, the dfferental operators assocated wth ts subdoman as follows: [PDE] Lnear Helmholtz equatons wth constant coeffcents (on 5 subdomans) and varable coeffcents (on subdomans). Du þ c u ¼ f on X ; ¼ ;...; 7 wth c ¼ expðx þ yþ; c ¼ ; c ¼ 6; c ¼ ; c ¼ ðsnððx þ yþpþþþ; c 5 ¼ 5; c 6 ¼ 5; c 7 ¼ 5 exp x þ y : 8 [PDE] Lnear Helmholtz equatons wth constant coeffcents (on subdomans) and varable coeffcents slghtly more complcated than n PDE (on subdomans). Du þ c u ¼ f on X ; ¼ ; ; ; wth c ¼ expðx þ yþ; c ¼ 6; c ¼ 5 and c ¼ snððx þ yþpþþ 5 exp x þ y þ : 8 [PDE] Lnear Helmholtz equaton wth constant coeffcent (on subdomans) and varable coeffcents (on one subdoman) and a general ellptc operator wth constant coeffcents on one subdoman. Du þ :u þ 6ðx þ y þ Þ ¼; Du þ :u ¼ ; on X and X Du ou ox þ ou þ :u ¼ oy ; on X ; on X : ð6þ [PDE] Lnear Helmholtz equaton wth constant coeffcent (on subdomans) and general non-lnear dfferental operators wth varable coeffcents (on two subdomans). Du þ :u þ u þ 6ðx þ y þ Þ ¼; on X V ; Du þ :u ¼ ; on X V and X V ; ð7þ o u ox þ þ u o u oy þ ou ou 5 ox þ þ :u ¼ ; on XV oy : ðþ ð5þ

6 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) Fg.. Two domans, X and X, on the left and two X and X V on the rght, wth the assocated boundary condtons, ther decompostons and the numberng of these subdomans and ther nterface segments. The rght hand sdes (f s) n both PDE and PDE have been selected so that the true soluton s x + y. Therefore, for these problems we can calculate the errors nvolved n any numercal approxmaton of ther solutons. For each one of these four decompostons we denote the nterface segments by d s ;s, where d denotes the correspondng decomposton, and s, s the ndces of the two subdomans whch are adjacent to ths partcular nterface. For example, by ;, we denote the nterface that lays between the subdomans X and X of the X (see Fg. ). The above gven PDE problems coupled wth the assocated boundary condtons shown n Fg. lead as to a populaton of composte ellptc PDE problems. Ths set of problems s small enough for easy expermentaton but at the same tme t encapsulates many mportant features that mght affect the performance of smulatons based on nterface relaxaton technques. Specfcally, we let us nvestgate the affect of the followng physcal parameters of the PDE problems: Table The theoretcally determned optmal values of the nterface relaxaton parameters used to obtan the data assocated wth the dotteddashed lnes n Fgs. nterface segment ROB relaxaton parameter AVE relaxaton parameters k a b ; ; ; ; ; ; ; ; ; ;

7 66 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 log( u (k+) u (k) ) nterface, nterface, log( u (k+) u (k) ) nterface, nterface, log( u (k+) u (k) ) nterface,5 nterface, log( u (k+) u (k) ) nterface,5 nterface 5, log( u (k+) u (k) ) nterface,7 nterface 5, Fg.. Reducton of the L norm of the dfference of two successve terants on each nterface usng the ROB scheme for PDE. Sold lnes, dotted lnes and crcles denote data usng k =,k = and k = for =,...,8 respectvely. The dash-dotted lnes denote data usng the theoretcally determned optmum values for k s shown n Table.

8 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) log( u (k) u / u ) log( u (k) u / u ) log( u (k) u / u ) log( u (k) u / u ) Subdoman Ω Subdoman Ω Subdoman Ω Subdoman Ω 6 Subdoman Ω Subdoman Ω Subdoman Ω 5 Subdoman Ω Fg.. Reducton of the L norm of the relatve errors n each subdoman usng the ROB scheme for PDE (legends as n Fg. ).

9 68 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 log( u (k+) u (k) ) nterface, nterface, nterface, nterface, log( u (k+) u (k) ) log( u (k+) u (k) ) nterface,5 nterface, log( u (k+) u (k) ) nterface,5 nterface 5, log( u (k+) u (k) ) nterface,7 nterface 5, Fg. 5. Reducton of the L norm of the dfference of two successve terants on each nterface usng the AVE scheme for PDE. Sold lnes, dotted lnes and crcles denote data usng a = b =.5, a = b =.5, and a = b =.75, for =,...,7 respectvely. The dash-dotted lnes denote data usng the theoretcally determned optmum values for the a s and b s shown n Table.

10 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) log( u (k) u / u ) Subdoman Ω Subdoman Ω log( u (k) u / u ) Subdoman Ω Subdoman Ω log( u (k) u / u ) SubdomanΩ Subdoman Ω log( u (k) u / u ) Subdoman Ω 6 Subdoman Ω Fg. 6. Reducton of the L norm of the relatve errors n each subdoman usng the AVE scheme for PDE (legends as n Fg. 5).

11 6 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 log( u (k+) u (k) ) nterface, nterface, log( u (k+) u (k) ) nterface, nterface, log( u (k+) u (k) ) nterface,5 nterface, log( u (k+) u (k) ) nterface,5 nterface 5, log( u (k+) u (k) ) nterface,7 nterface 5, Fg. 7. Reducton of the L norm of the dfference of two successve terants on each nterface usng the ROB scheme wth the optmum values for the k s for PDE. Sold lnes, dotted lnes and dash-dotted denote data usng the 5-pont-star fnte dfference scheme, the collocaton and the fnte element respectvely.

12 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 6 The sze of the domans (as movng from doman X to X V ). The type of the decomposton, by comparng results for cartesan (rectangular) decompostons assocated wth rectangular domans X, X to the ones for non-cartesan decompostons assocated wth non-rectangular domans X, X V. The dffculty of the dfferental operator, as movng from lnear Helmholtz the general non-lnear operators (consdered n PDE). Furthermore, we note that one mght consder the decompostons of subdomans X and X as cartesan approxmatons to the ones of subdomans X and X V respectvely and as such they mght be used to calculate good relaxaton parameters for the non-cartesan decompostons (see []). Fnally, t s worth to menton that the above problems can be found n the heart of nuclear reactor smulaton systems.. A lnear PDE wth a cartesan decomposton: PDE For the experments n ths secton we dscretze each of the subdomans usng rectangular meshes wth common dscretzaton parameter h =.. Note that for ths dscretzaton the local to each subdoman grds match on the nterfaces. The local PDE operators were dscretzed usng the ELLPACK s [5] 5-pont-star module, whch s an O(h ) fnte dfference scheme, on all subdomans. Ths resulted nto lnear systems wth N =,,86,,,,5,5 algebrac equatons and unknowns on subdomans,,...,7 respectvely. All these systems were solved usng the ELLPACK s Gauss Elmnaton module for banded matrces. To calculate approprate values for the relaxaton parameters we tred to utlze our one-dmensonal theoretcal results presented n []. For that we collapse approprate rows or columns of rectangles by consderng them as lnes and ther nterface lnes as nterface ponts. For example to obtan the relaxaton parameters on the nterface segments ; ; ;7 and ;5 ; 5;7 we consder the one-dmensonal decomposton [,], [, ] and [, 8], and on the segments ; ; ; and ; the [, ], [,], [, 8] and [8,] whle for the segments ;5 ; 5;6 the [,], [, ] and [,8] and fnally for the segment ;6 we consder the decomposton [, ], [,]. Furthermore, the c s nvolved n the theoretcal expressons of the one-dmensonal analyss of [] were set equal to the average of the coeffcent of u n each subdoman. The new one-dmensonal PDE problems were frst scaled to [, ] by multplyng the c s wth the square of the length of the nterval and then the optmum parameters are computed usng the formulas obtaned n []. The values estmated for PDE by usng the above procedure are gven n Table. To examne the basc convergence propertes of the methods we denote by u (k) the computed at the kth teraton soluton and we present a seres of plots concernng ts accuracy on each subdoman and on each nterface segment by usng the followng two measures: S ¼ log ku ðkþ j C u ðk Þ j C k ; ¼ ; ;...; number of nterfaces; ð8þ E ¼ log ku ðkþ j X u k ; ¼ ; ;...; number of subdomans: ð9þ Table The theoretcally determned optmal values of the nterface relaxaton parameters used to obtan the data assocated wth the dotteddashed lnes n Fgs. 8 and 9 nterface segment ROB relaxaton parameter AVE parameters relaxaton k a b ; ; ; ; ;

13 6 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 log( u (k+) u (k) ) nterface, nterface, log( u (k+) u (k) ) nterface, nterface, log( u (k+) u (k) ) nterface, log( u (k) u / u ) log( u (k) u / u ) Subdoman Ω Subdoman Ω Subdoman Ω Subdoman Ω Fg. 8. Reducton of the L norm of the dfference of two successve terants on each nterface and the L norm of the relatve errors n each subdoman usng the ROB scheme for PDE. Sold lnes, dotted lnes and crcles denote data usng k =, k = and k = for =,..., respectvely. The dash-dotted lnes denote data usng the theoretcally determned optmum values for k s shown n Table.

14 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 6 log( u (k+) u (k) ) nterface, nterface, log( u (k+) u (k) ) nterface, nterface, log( u (k+) u (k) ) nterface, log( u (k) u / u ) Subdoman Ω Subdoman Ω log( u (k) u / u ) Subdoman Ω Subdoman Ω Fg. 9. Reducton of the L norm of the dfference of two successve terants on each nterface and of the L norm of the relatve errors n each subdoman usng the AVE scheme for PDE. The sold lnes, denote data usng a =.5, for =,..., and the dash-dotted lnes denote data usng the theoretcally determned optmum values for the a s and b s as these are shown n Table.

15 6 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 Both Fgs. and clearly show that the estmated, by the above procedure, values of the relaxaton parameters for the ROB scheme are really close to optmum. These values result n convergence that s sgnfcant faster than the one assocated wth k = or. n the AVE case though as seen from Fgs. 5 and 6 the theoretcal determned relaxaton parameters do not show such optmalty. Nevertheless, t s seen that the rate of convergence, depcted by the slopes of the convergence lnes, does not dffer sgnfcantly n the case of the expermentally best relaxaton parameters (found by systematc search). Ths suggests that there exsts only a somehow start-up cost dfference on the convergence rates. By comparng Fgs. and 6 one sees that optmum ROB seems to be slghtly faster than optmum AVE whle both schemes seem to be rather effectve, achevng sgnfcant dgts n less that and 5 dgts n about. We conclude ths secton by nvestgatng the effect of the partcular PDE dscretzaton scheme one mght use to solve the ndvdual PDE subproblems. For ths we plot n Fg. 7 the hstory of convergence for the ROB scheme appled to PDE usng three dfferent dscretzaton modules from ELLPACK. Specfcally, we have performed three experments, one (depcted by dash-dotted lnes n the graphs), usng the fnte element module on all subdomans, another usng the 5-pont-star fnte dfference module (sold lnes) and another usng the collocaton module (dotted lnes). Note that the fnte element and the fnte dfference schemes are of order O(h ) whle the collocaton s of order O(h ). For all methods we used h =.. These lead to lnear systems of sgnfcant dfference on the sze and of rather dfferent mathematcal propertes. For example, the collocaton matrx has 76, 76,, 76, 9, 9, 8, 86 equatons and unknowns, the fnte element has, 6, 6, 6, 67, 67, 9, 6 and the fnte dfference has 99, 8, 88, 99, 8, 7, 5, n subproblems =,...,7 respectvely. t s worth to menton that our expermental data for the AVE method for PDE not presented here exhbt the same behavor as the one found n Fg. 7. t s therefore safe to clam that the convergence behavor of the nterface Relaxaton methods s almost dentcal to all three cases ndcatng the natural expectaton one mght have drawn from the formulaton and the prelmnary analyss already developed (see for example [,]) that the rate of convergence of nterface relaxaton methods does not depend on local dscretzaton parameters and choces. 5. Lnear PDEs wth general decompostons: PDE and PDE We now move nto PDE whch was solved usng the same confguraton as the one descrbed for PDE above wth only the followng two dfferences. Frst we dscretze the doman X usng trangular elements wth h =. on all subdomans. On ths we used the ELLPACK s Fnte Element dscretzaton module whch s also an O(h ) scheme. Secondly to estmate the optmal values for the relaxaton parameters we use the approxmate cartesan decomposton assocated wth X by utlzng our one-dmensonal theoretcal results n the followng way: We frst map the nterface segments of the general decomposton that are not straght lnes parallel to x- or y-axs to ther closest nterface lnes on the approxmate decomposton. Ths map can be done usng geometry or computatonal geometry tools and procedures coupled wth approprate objectve func- Table The values of the relatve error for PDE n the L and L norms of the computed solutons n each subdoman by usng the ELLPACK s Fnte Element module on each sngle subdoman (n the second and thrd columns) and by usng the ROB nterface relaxaton method wth dfferent relaxaton parameters (n the subsequent columns) Sngle doman ROB relaxaton k = k = k = opt L L L L L L L L X.55E.77E.E.6E.79E.6E.E.66E X.85E.7E.5E.7E.8E.7E.E.679E X.9E.E.7E.5E.E.E.99E.E X.6E.9E.9E.E.9E.8E.87E.9E

16 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) log( u (k+) u (k) ) nterface, nterface, log( u (k+) u (k) ) nterface, nterface, log( u (k+) u (k) ) nterface, log( u (k) u / u ) log( u (k) u / u ) Subdoman Ω Subdoman Ω Subdoman Ω Subdoman Ω Fg.. Reducton of the L norm of the dfference of two successve terants on each nterface and of the L norm of the relatve errors n each subdoman usng the ROB scheme and the Fnte Element module on all subdomans for PDE wth k = for =,...,. Sold lnes denote data usng as space dscretzaton parameter h =., the dotted lnes denote data usng h =..

17 66 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 tons. Here we have done t usng a straghtforward naïve way. Specfcally, we use the correspondng PDE operators on each subdoman and for nterface ; the one-dmensonal parttonng [,][,], for nterfaces ; and ; the decomposton [, ][,8], for nterface ; the decomposton [, ][,], whle for nterface ; we just took the average of the values from nterfaces ; and ;. The values determned by the above procedure are shown n Table. Smlarly as for PDE we present n Fgs. 8 and 9 the convergence hstores of the norms of the successve terants and the relatve errors for PDE durng the frst for both the ROB and AVE schemes and dfferent values for the relaxaton parameters k, a and b. We frst note that by comparng Fgs. wth 8 and 9 one sees that both the rate of convergence and the effectveness of the parameters of the nterface Relaxaton methods are very smlar for both the PDE and PDE regardless of the fact that these two problems dffer on many parameters (e.g., dfferent decomposton, dfferent type and number of subdomans). We also note that dfferent local PDE dscretzatons were also used. Furthermore, the common horzontal levelng of the lnes n Fgs. 8 and 9 represent the PDE dscretzaton error whch, n contrast to the PDE case, s due to the geometry of the non-rectangular subdomans for PDE. Both methods converge very fast achevng the dscretzaton error level n about 5. To examne the effect of the ROB (smlar results, not presented here, were obtaned for AVE) nterface Relaxaton procedure on the accuracy of the computed soluton for PDE we gve n Table the two norms of the relatve errors of the computed solutons obtaned by two dfferent ways. Specfcally, n the second and thrd column we lst the errors obtaned by frst mposng Drchlet boundary condtons wth exact (correct) rght hand sdes on all nterfaces and then solvng the ndvdual uncoupled PDE subproblems defned on each subdoman by usng the Fnte Element dscretzaton wth h =. on each one of them. n the subsequent columns we gve the computed values of the same errors obtaned by the experment wth whch we obtaned y axs y axs x axs x axs y axs y axs x axs x axs Fg.. Contour plots of the frst terants (u (k), k =,,,) of PDE computed by the ROB scheme usng the theoretcally determned optmum relaxaton parameters.

18 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) log( u (k+) u (k) ) V nterface, V nterface, log( u (k+) u (k) ) V nterface, V nterface, log( u (k+) u (k) ) V nterface, Fg.. Reducton of the L norm of the dfference of two successve terants on each nterface usng the ROB scheme for PDE (legends as n Fg. ). the data n Fg. 8. As s seen the relaxaton slghtly reduces the accuracy ndcatng that the constant nvolved n the convergence rate of ROB s sgnfcantly small. 5 y 5 6 x Fg.. Space dscretzaton for the PDE usng cartesan grds on subdomans X V and X V and trangular elements on subdomans X V and X V wth dscretzaton respectvely wth h =. n both cases.

19 68 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 To nvestgate the effect of the space dscretzaton parameter h on the rate of convergence we present n Fg. the hstory of the norm of the successve terants and the norms of the relatve errors, for PDE usng the ROB scheme wth k = for =,...,p. The dotted lnes represent data wth h =. and the sold lnes data wth h =.. t s seen that even though we approxmately double the number of elements ths has vrtually no effect on the convergence. To conclude the presentaton of our expermental data for PDE we plot n Fg. the contours of the computed soluton after the frst four. The drastc smoothng effect on the nterfaces s clearly observed. We swtch now to PDE and present n Fg. the reducton of the norm of the successve terants of the ROB scheme for the dfferent values of the k s. The confguraton used to obtan these data was the same as the one for PDE. The optmum values determned from the theoretcal analyss were k =.656, k =.65, k =.6655, k =.987 and k =.8. As s clearly seen these values lead to the fastest convergence. n addton, by comparng the slopes of the lnes n Fgs. and, we see that the rate of convergence for the two problems PDE and PDE that dffer on both the sze of the subdomans and the PDE operators appled on each one of them, s approxmately the same. Fnally, we also see the effectveness of the scheme, n the sense that t offers full accuracy n less than. 6. A non-lnear PDE wth general decomposton: PDE For the experments n ths secton we have dscretzed subdomans X V and X V usng rectangular elements, and subdomans X V and X V usng trangular elements as t s shown n Fg.. n all four subdoman dscretzatons we have used h =.. t s worth to observe that the local dscretzaton ponts do not necessarly match on the nterfaces. Therefore, agents performng local nterpolaton were used (see [] for detals) before relaxng the nterface values. 8 8 T 6 T x 5 y 6 5 x 5 y T y x 8 6 T 6 5 x 5 y Fg.. Three-dmensonal plots of the st, nd, rd and 5th terants (u (k), k =,,,5) of PDE computed by the ROB scheme usng k = for =,...,.

20 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) We have lnearzed the PDE operators n (7) usng the Newton s method. The resultng lnearzed PDEs were dscretzed usng the Ellpack s 5-pont-star module for subdomans X V and X V and the fnte element module for subdomans X V and X V. Ths way subdomans,, and resulted n, 57, and lnear equatons (and unknowns) respectvely. These lnear systems were solved usng the Gauss Elmnaton module for banded matrces. n Fg. we plot n three dmensons the computed by the ROB scheme soluton after,, and 5 and n Fg. 5 the convergence hstory n terms of the dfference of successve terants wth dfferent set of log( u (k+) u (k) ) V nterface, V nterface, log( u (k+) u (k) ) V nterface, V nterface, log( u (k+) u (k) ) V nterface, Fg. 5. Reducton of the L norm of the dfference of two successve terants on each nterface usng the ROB scheme for PDE. Sold lnes, dotted lnes and crcles denote data usng k =,k = and k = for =,..., respectvely. The dash-dotted lnes denote data usng the theoretcally determned optmum values for k s.

21 6 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 values of the relaxaton parameters. The ones represented by the dash-dotted lnes are for the optmum values determned by usng the approxmate cartesan partton (defned by X ) and the assocated procedure used for PDE and descrbed n Secton 5. These values were k =.58, k =.5977, k =.665, k =.57 and k =.7 and as t s seen n Fg. 5 they lead to dvergence. Ths s explaned from the fact that the dfferental operators n the subdomans are far away from the Helmholtz model problem we have analyzed theoretcally. Other than that the scheme seems to converge fast enough, offerng two correct decmal dgts n about 5. Addtonal experments, not presented here, show that, as for the prevous problems, the rate convergence of both schemes for PDE does not sgnfcantly depend on ether the local dscretzaton parameter h nor the local dscretzaton method used locally on each subdoman. 7. Synopss We hope that the presented study numercally further establshes ntuton about the characterstcs and dynamcs of mult-physcs, mult-doman smulaton systems that are based on nterface relaxaton methods and t adds to the further work needed before these methods put nto practcal consderatons. Specfcally, t s our beleve that the numercal experments presented above gve us enough confdence to clam that () Both methods converge rather rapdly achevng numercal convergence n about. () Regardless ther totally dfferent formulaton (e.g., one step vs. two steps, one parameter vs. two parameters) and motvaton and smoothng technques (va Robn vs. Drchlet Neumann condtons) the convergence behavor of both methods s qute smlar. () Although careful tunng nterface relaxaton parameters certanly leads to faster convergence, the rate of convergence does not seem to strongly depend on the values of these parameters. Therefore, just rough estmatons of the optmum values are needed n many cases. Furthermore, technques for utlzng theoretcal results derved for smple model decompostons to tune general decompostons have been proposed and proved ther success. () The much needed ndependence on the local dscretzaton detals s further confrmed and the concept of usng dfferent dscretzatons, possble custom talored, for dfferent subproblems s clearly (re-)proved. (5) Movng from lnear to non-lnear dfferental operators seems easy exhbtng possbly smlar convergence propertes. (6) Software reuse s pushed to ts lmt. Our whole ScAgents mplementaton conssts of more than.5 Mllon lnes of code, the great majorty of whch s consstng of legacy Ellpack modules wearng agent jackets. References [] P.L. Lons, On the Schwarz alternatng method : a varant for nonoverlappng subdomans, n: R. Glownsk, G.H. Golub, G.A. Meurant, J. Peraux (Eds.), Doman Decomposton Methods for Partal Dfferental Equatons, SAM, 99, pp.. [] H.S. McFaddn, J.R. Rce, Collaboratng PDE solvers, Appl. Numer. Math. (99) [] M. Mu, J.R. Rce, Modelng wth collaboratng PDE solvers theory and practce, Comput. Syst. Eng. 6 (995) [] J.R. Rce, An agent-based archtecture for solvng partal dfferental equatons, SAM News (6) (998). [5] M. Mu, Solvng composte problems wth nterface relaxaton, SAM J. Sc. Comput. (999) 96. [6] E. Vavals, A collaboratng framework for ar polluton smulatons, n: NATO-AS Seres, 999, pp [7] E. Vavals, Runtme support for collaboratve ar polluton agents, Syst. Anal. Modell. Smul. () [8] J. Bolon, D.C. Marnescu, J.R. Rce, P. Tsompanopoulou, E. Vavals, Agent based scentfc smulaton and modelng, Concurr.: Pract. Exp. () [9] S. Markus, E. Housts, A. Catln, J. Rce, P. Tsompanopoulou, E. Vavals, D. Gottfred, K. Su, G. Balakrshnan, An agent-based netcentrc framework for multdscplnary problem solvng envronments, nt. J. Comput. Eng. Sc. (). [] E.N. Housts, A.C. Catln, P. Tsompanopoulou, D. Gottfred, G. Balakrshnan, K. Su, J.R. Rce, Gasturbnlab: a multdscplnary problem solvng envronment for gas turbne engne desgn on a network of non-homogeneous machnes, J. Comput. Appl. Math. 9 () () 8.

22 P. Tsompanopoulou, E. Vavals / Appled Mathematcal Modellng (8) 66 6 [] J.R. Rce, P. Tsompanopoulou, E. Vavals, nterface relaxaton methods for ellptc dfferental equatons, Appl. Numer. Methods () 95. [] J.R. Rce, P. Tsompanopoulou, E. Vavals, Fne tunng nterface relaxaton methods for ellptc dfferental equatons, Appl. Numer. Methods () 598. [] P. Tsompanopoulou, E. Vavals, Analyss of a geometry based nterface relaxaton method for ellptc dfferental equatons, submtted for publcaton. [] P. Tsompanopoulou, Collaboratve PDEs: theory and practce, Ph.D. thess, Mathematcs Department, Unversty of Crete, Greece,. [5] J.R. Rce, R.F. Bosvert, Solvng Ellptc Problems Usng ELLPACK, Sprnger-Verlag, New York, NY, 985.