Robust FETI solvers for multiscale elliptic PDEs

Transcription

1 Robust FETI solvers for multscale ellptc PDEs Clemens Pechsten 1 and Robert Schechl 2 Abstract Fnte element tearng and nterconnectng (FETI) methods are effcent parallel doman decomposton solvers for large-scale fnte element equatons. In ths work we nvestgate the robustness of FETI methods n case of hghly heterogeneous (multscale) coeffcents. Our man applcaton are magnetc feld computatons where both large jumps and large varaton n the reluctvty coeffcent may arse. We gve theoretcal condton number bounds whch are confrmed n numercal tests. 1 Introducton Fnte element tearng and nterconnectng (FETI) methods due to Farhat and Roux [2, 14] are parallel solvers for large-scale fnte element (FE) systems arsng from partal dfferental equatons (PDEs). Typcally, the condtonng of such FE system matrces heavly suffers from the total number of degrees of freedom (DOFs). When the number of DOFs grows large, drect solvers are out of queston and effcent precondtoners for teratve solvers are requred. Addtonally, the parallelzaton of numercal algorthms gets ncreasngly mportant to date. FETI methods are known to be parallely scalable and quas-optmal wth respect to the number of DOFs. For a comprehensve presentaton of FETI and related methods we refer to the monograph by Tosell and Wdlund [14]. As an addtonal advantage, one can easly couple fnte and boundary element dscretzatons wthn the same framework, resultng n Clemens Pechsten SFB F013 / Insttute of Computatonal Mathematcs, Johannes Kepler Unversty, Altenberger Str. 69, 4040 Lnz, Austra, e-mal: clemens.pechsten@numa.un-lnz.ac.at, Robert Schechl Department of Mathematcal Scences, Unversty of Bath, Bath BA2 7AY, Unted Kngdom, e- mal: masrs@bath.ac.uk 1

2 2 Clemens Pechsten and Robert Schechl so-called coupled FETI/BETI methods, see [4, 6, 7]. Even exteror domans can be ncorporated to model radaton condtons, see [5, 11] Let us brefly descrbe the FETI method. As a model problem we consder the fnte element dscretzaton of the Posson-type problem (α u) = f (1) n the bounded doman Ω R d, d = 2 or 3, subject to sutable nterface and boundary condtons. In Secton 4 we wll consder a smlar equaton for 2D magnetostatcs. The doman Ω s parttoned nto N non-overlappng subdomans Ω, = 1,...,N, cf. Fg. 1, rght. Introducng separate unknowns u on the subdomans ncludng the DOFs on ther boundares, we obtan the saddle pont problem K 1 0 B K N B N B 1 B N 0 u 1. u N λ f 1 =. f N, (2) 0 where K are the subdoman stffness matrces, and are f the correspondng load vectors. The operators B are sgned Boolean matrces such that each row of the system N B u = 0 =1 has the form u (x h ) u j (x h ) = 0 for a fnte element node x h on the nterface between the subdomans Ω and Ω, thus enforcng the contnuty of the soluton u. The Lagrange multpler λ plays the role of a contnuous flux on the subdoman nterfaces. Introducng a specal projecton P, the dual problem to (2) can be wrtten n the form PF λ = d, (3) ar cols yoke rotor ar gap Fg. 1 Left: Model of an electrc motor. Rght: Possble subdoman parttonng (explosve vew).

3 Robust FETI solvers for multscale ellptc PDEs 3 wth F = N =1 B K B, where the operators K correspond to the soluton of (possbly) regularzed Neumann problems on the subdomans. The FETI method s now a specal projected precondtoned conjugate gradent (PCG) method to solve problem (3). The chosen precondtoner nvolves the soluton of local Drchlet problems, and the projecton P nvolves the soluton of a coarse problem whch corresponds to a sparse lnear system of dmenson O(N). Usually, one chooses the partton n a way that the local subdoman problems can effcently be handled by sparse drect solvers, such as LU-factorzaton wth sutable pvotng. The factorzatons of the local system matrces can be computed n a preprocessng phase and kept n memory durng the whole FETI method. Note that these local, decoupled problems can be parallelzed n a straghtforward manner, e. g., treatng each subdoman on a dfferent processor. Once problem (3) s solved, the actual soluton u can easly be determned from the Lagrange multpler λ. The spectral condton number κ of the precondtoned system can fnally be estmated by κ C (α) N max =1 ( 1 + log(h /h ) ) 2, (4) where the constant C (α) s ndependent of the subdoman dameters H, the mesh parameters h, and the number N of subdomans. If α s (globally) constant, then C (α) 1. As t s well known, the number of PCG teratons needed to acheve a gven accuracy, s essentally determned by κ. In a parallel scheme, the total computatonal complexty of the FETI-PCG method s gven by O ( (D(N) + D(N loc )) log(ε 1 ) κ ), (5) where N loc max N =1 (H /h ) d s the maxmal number of DOFs per subdoman, D( ) s the cost of the drect solver, and ε > 0 s the desred relatve error reducton n the energy norm. However, n many applcatons the orgnal system matrx s ll-condtoned due to heterogeneous coeffcent dstrbutons. As we wll dscuss n Secton 4, n magnetc feld computatons one may have large jumps n the reluctvty coeffcent due to dfferent materals, and smooth but large varaton n the same coeffcent due to nonlnear effects. We are nterested n the queston whether/how the condton number κ of the precondtoned FETI system s affected by ths. If the heterogenetes are resolved by the subdoman partton (. e., α constant on each Ω ), then, usng a specal dagonal scalng, Klawonn and Wdlund [3] proved that C (α) 1. However, n general, usng classcal proof technques, we only get C (α) C max max α(x) =1,...,N x,y Ω α(y), (6) wth C ndependent of α,. e., the bound s proportonal to the maxmum local varaton of α on the subdomans, whch can be rather large. As notced by several au-

4 4 Clemens Pechsten and Robert Schechl thors [4, 13] ths asymptotc bound s n general far too pessmstc, and robustness s observed for many specal knds of coeffcent dstrbutons. The am of the present contrbuton s to gve more theoretcal nsght on the coeffcent-dependency. Due to space lmtatons we summarze our recent work [10] consderng varaton n subdoman nterors n Secton 2, and we gve an outlook to new theoretcal results for the case of varaton near the subdoman nterfaces n Secton 3. Fnally, Secton 4 deals wth the applcaton to magnetostatc problems. 2 Varaton n subdoman nterors In ths secton we gve a sharper estmate than (6) for the case of varaton n the subdoman nterors. On each subdoman Ω wth dameter H and dscretzaton parameter h, we choose a wdth η [h, H /2] and defne the boundary layer Ω,η by the agglomeraton of those fnte elements whch have dstance at most η from the boundary, cf. Fg. 2, left. Under sutable assumptons on the geometrc settng and the subdoman partton, we can prove the bound C (α) C max N ( Hj j=1 η j ) 2 N α(x) max max =1 x,y Ω,η α(y). (7) Ths bound nvolves only the varaton of α n the boundary layer Ω,η and s ndependent of the varaton of α n the subdoman nteror Ω \ Ω,η. For η j H j we reproduce the known estmate (6), n partcular our bound s stll robust wth respect to large jumps across the subdoman nterfaces. However, f α exhbts large (even arbtrary) varaton n the nteror Ω \Ω,η of the subdomans, but vares lttle n the boundary layers, our new bound (7) s n general far better/sharper than (6). Moreover, f n addton the coeffcent s larger n the nteror Ω \ Ω,η than n the boundary layer on each subdoman, then the quadratc factor (H j /η j ) 2 reduces to a lnear factor H j /η j. The detaled proof can be found n our recent paper [10]. In the followng we gve a two-dmensonal numercal example. We partton the unt square nto 25 congruent, square-shaped subdomans. The coeffcent s chosen η nteror Ω condton number case 1 case 2 lnear Ω η, H/eta Fg. 2 Left: Subdoman boundary layer. Rght: Estmated condton numbers κ for varyng wdth parameter η, fxed dscretzaton parameter h (logarthmc scales).

5 Robust FETI solvers for multscale ellptc PDEs 5 to be α = 10 5 (Case 1) and α = 10 5 (Case 2) n the subdoman nterors, and α = 1 on the rest. The dstance between the materal jump and the subdoman nterfaces s denoted by η. We have used a globally unform dscretzaton wth H/h = 512. Fg. 2, rght, shows the estmated condton numbers κ of the precondtoned FETI systems for dfferent values of the wdth parameter η. We see that our asymptotc bound s sharp for Case 1, but stll slghtly pessmstc for Case 2. 3 Interface varaton In ths secton we would lke to gve an outlook on our work for nterface varaton whch wll be exposed n more detal n an upcomng paper. A key tool to the analyss of FETI methods s Poncaré s nequalty, w(x) 2 dx C P H 2 w(x) 2 dx, Ω Ω whch holds for all w H 1 (Ω ) wth vanshng mean value,. e., Ω w(x)dx = 0. The constant C P > 0 depends only on the shape of Ω. A smlar nequalty holds f the average of w over a part of the boundary Ω vanshes. Concernng heterogeneous coeffcents, we would be nterested n an nequalty of the same form but where the ntegrals are weghted wth the coeffcent α(x) and where the constant C P does not depend on α, or at least only very mldly on the heterogenety n α. Such nequaltes are not known n general, but we can show one for a specal case. Assume that each subdoman Ω conssts of two connected subregons Ω (1), Ω (2) where α s mldly varyng,. e., α (k) α(x) α (k) x Ω (k), k = 1, 2, wth moderate ratos α (k) /α (k) ; we can thnk of two quas-homogeneous materals wthn each subdoman. Usng two separate Poncaré nequaltes one can show that Ω α(x) w(x) 2 dx { α (k) } max k=1,2 C(k) P α (k) H 2 Ω α(x) w(x) 2 dx, (8) for all functons w H 1 (Ω ) whch have vanshng mean value over a connected part Λ of the nterface Ω (1) Ω (2),. e., w(x)ds x = 0. The constants C (1) P and C(2) P Λ depend only on the shapes of the subregons Ω (1) and Ω (2) respectvely, and on the relatve shape of Λ. For a varant of FETI called all-floatng FETI method [1, 8, 9], our Poncaré type nequalty (8) eventually allows a proof of the bound C (α) C max N max =1 k=1,2 α (k) α (k), (9)

6 6 Clemens Pechsten and Robert Schechl η η η η j j η k ηk FETI for sland coeffs FETI for sland coeffs condton 10 ref=5 ref=6 ref=7 ref=8 lnear condton 10 ref=5 ref=6 ref=7 ref=8 lnear 1 1e dstance 1 1e dstance Fg. 3 Upper: Sketch of coeffcent slands cuttng through edges and crossponts of the subdoman parttonng. Lower left: Condton numbers for edge slands. Lower rght: Condton numbers for crosspont slands. where the constant C s ndependent of H, h, N, and α, but t depends on the geometry of the subregons Ω (k). Combnng ths dea wth the theory from Secton 2, one can even allow three qualtatvely dfferent subregons per subdoman: two connected subregons of mld varaton n α that cover the boundary layer Ω,η of the subdoman, and a remanng part contaned n the subdoman nteror, where arbtrary varaton of α can be allowed. Under sutable assumptons on the shapes of these subregons t s agan possble to gve explct bounds for C (α) nvolvng (9) and the ratos H /η smlar to (7). For numercal examples we have tested so-called coeffcent slands whch cut through an edge,. e., the nterface of two subdomans, or whch contan a crosspont of four subdomans, cf. Fg. 3, upper. A sutable choce for η, the wtdh of the boundary layer, s also ndcated n that fgure. Note, however, that we have only tested one sland at a tme. In each example we have set the coeffcent α = 10 5 n the sland, and α = 1 elsewhere. The estmated condton numbers for dfferent values of η and dfferent levels of mesh refnement are depcted n Fg. 3, lower. 4 Applcaton to magnetostatc problems In the case of nonlnear magnetostatcs n two dmensons (transverse magnetc mode), we have to solve [ν ( u ) u] = f n Ω, (10)

7 Robust FETI solvers for multscale ellptc PDEs 7 subject to sutable nterface and boundary condtons, where u s the z-component of the magnetostatc vector potental, and ν s the reluctvty. For lnear materals, ν s constant. For other materals, such as ferromagnetc ones, the reluctvty ν depends nonlnearly on the magnetc flux densty B = u, and t s defned by the materal law H = ν ( B )B n Ω, where H denotes the magnetzng force (note that we restrct ourselves to sotropc materals and neglect hysteress). In our numercal computatons we use realstc approxmatons of such materal curves usng the nterproxmaton method proposed n [12]. If we apply Newton s method to (10), the lnearzed system n each Newton step s of smlar form as problem (1), only that we obtan a matrx-valued coeffcent whch depends on the current Newton terate u (k), see, e. g., [4]. For many materal curves, the varaton of the coeffcent depends manly on the varaton of B. However, the flux densty B may vary strongly along subdoman boundares and large values of B appear mostly at sngulartes of the potental u, e. g., near materal corners. Contrary to the usual suggeston to choose subdoman parttons that resolve materal nterfaces n order to obtan robustness (for numercal examples see [4, 5]), our new bounds (7), (9) suggest that t mght be more advantageous to put each peak of B and thus each materal corner nto the center of a subdoman. Fg. 4 shows two such examples. In both cases, the coeffcent varaton s approxmately but our FETI solver performs extremely well (Case 1: condton number 8.5, Case 2: condton number 13.7, compared to 8.3 for a globally constant coeffcent). Our theory for nteror varaton (Secton 2) can perfectly explan the low condton number n Case 1 snce the boundary varaton s small. Inspectng Case 2, we fnd that there are ndeed two regons contaned n the boundary layer wth qualtatvely dfferent coeffcents, see the jump n Fg. 4, lower rght. Thus, Secton 3 partally explans why the condton number s stll qute robust wth respect to the hghly heterogeneous coeffcent. Acknowledgements We would lke to thank Ulrch Langer for hs encouragement. The frst author acknowledges the fnancal support by the Austran Scence Funds (FWF), grant SFB F013. References 1. Zdeněk Dostál, Davd Horák, and Radek Kučera. Total FETI An easer mplementable varant of the FETI method for numercal soluton of ellptc PDE. Commun. Numer. Methods Eng., 22(12): , C. Farhat and F.-X. Roux. A method of fnte element tearng and nterconnectng and ts parallel soluton algorthm. Int. J. Numer. Meth. Engrg., 32(6): , A. Klawonn and O. B. Wdlund. FETI and Neumann-Neumann teratve substructurng methods: Connectons and new results. Comm. Pure Appl. Math., 54(1):57 90, U. Langer and C. Pechsten. Coupled fnte and boundary element tearng and nterconnectng solvers for nonlnear potental problems. ZAMM Z. Angew. Math. Mech., 86(12), U. Langer and C. Pechsten. Coupled FETI/BETI solvers for nonlnear potental problems n (un)bounded domans. In G. Cuprna and D. Ioan, edtors, Scentfc Computng n Electrcal

8 8 Clemens Pechsten and Robert Schechl 1e+07 coeffcent along subdoman boundary 5 1e+06 nu( B ) 00 0 Ω x 1e+07 coeffcent along subdoman boundary 5 1e+06 nu( B ) 00 0 Ω Fg. 4 Upper: Case 1. Lower: Case 2. Left: B -feld, subdoman partton. Rght: Coeffcent ν( B ) plotted along the boundary of subdoman Ω 5. x Engneerng, volume 11 of Mathematcs n Industry: The European Consortum for Mathematcs n Industry. Sprnger, Berln, Hedelberg, Ulrch Langer and Olaf Stenbach. Boundary element tearng and nterconnectng method. Computng, 71(3): , Ulrch Langer and Olaf Stenbach. Coupled boundary and fnte element tearng and nterconnectng methods. In Lecture Notes n Computatonal Scences and Engneerng, volume 40, pages Sprnger, Hedelberg, Günther Of. BETI-Gebetszerlegungsmethoden mt schnellen Randelementverfahren und Anwendungen. PhD thess, Unverstät Stuttgart, Germany, January Günther Of. The all-floatng BETI method: Numercal results. In Doman Decomposton Methods n Scence and Engneerng XVII, volume 60 of Lecture Notes n Computatonal Scence and Engneerng, pages Sprnger, Berln, Hedelberg, C. Pechsten and R. Schechl. Analyss of FETI methods for multscale PDEs. To appear n Numersche Mathematk, 2008; for a preprnt see publcatons/reports/08/rep08-20.pdf. 11. Clemens Pechsten. Boundary element tearng and nterconnectng methods n unbounded domans. To appear n Appled Numercal Mathematcs, Clemens Pechsten and Bert Jüttler. Monotoncty-preservng nterproxmaton of B-H-curves. Journal of Computatonal and Appled Mathematcs, 196(1):45 57, D. Rxen and C. Farhat. A smple and effcent extenson of a class of substructure based precondtoners to heterogeneous structural mechancs problems. Internat. J. Numer. Methods Engrg., 44(4): , A. Tosell and O. B. Wdlund. Doman Decoposton Methods Algorthms and Theory, volume 34 of Sprnger Seres n Computatonal Mathematcs. Sprnger, Berln, Hedelberg, 2005.