PRECONDITIONING TECHNIQUES IN CHEBYSHEV COLLOCATION METHOD FOR ELLIPTIC EQUATIONS

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1 ITERATIOAL JOURAL OF UMERICAL AALYSIS AD MODELIG Volume 15 umber 1-2 Pages c 2018 Insttute for Scentfc Computng and Informaton PRECODITIOIG TECHIQUES I CHEBYSHEV COLLOCATIO METHOD FOR ELLIPTIC EQUATIOS ZHI-WEI FAG JIE SHE AD HAI-WEI SU Ths paper s dedcated to memory of late Professor Benyu Guo Abstract. When one approxmates ellptc equatons by the spectral collocaton method on the Chebyshev-Gauss-Lobatto CGL grd the resultng coeffcent matrx s dense and ll-condtoned. It s known that a good precondtoner n the sense that the precondtoned system becomes well condtoned can be constructed wth fnte dfference on the CGL grd. However there s a lack of an effcent solver for ths precondtoner n mult-dmenson. A modfed precondtoner based on the approxmate nverse technque s constructed n ths paper. The computatonal cost of each teraton n solvng the precondtoned system s Ol y log where y are the grd szes n each drecton and l s a small nteger. umercal examples are gven to demonstrate the effcency of the proposed precondtoner. Key words. Chebyshev collocaton method ellptc equaton fnte-dfference precondtoner approxmate nverse. 1. Introducton We consder a two-dmensonal separable ellptc equaton 1 ax u by u +cxdyuxy = fxy n Ω = 1 2 x x y y wth homogeneous Drchlet boundary condtons u = 0 on Ω where the coeffcent functons ax by cx dy and fx y are contnuous and 0 < α axby β n Ω for some postve constants α and β and cxdy 0. A very effcent accurate method for obtanng approxmate soluton of the above boundary-value problem s the Chebyshev collocaton method [ ] whch uses the Lagrange nodal bass functons based on the Chebyshev collocaton ponts. However due to the global nature of the Lagrange bass polynomals the assocated lnear systems are dense and ll-condtoned. Thus t becomes prohbtve to use a drect nverson method or an teratve method wthout precondtonng n the mult-dmensonal case so t s mperatve to use an teratve method wth a good precondtoner. Fnte element/fnte dfference precondtoners have been wdely used snce the orgnal work by Orszag [9]. Haldenwang et al. [5] proved that the fnte dfference method based on the Chebyshev collocaton ponts n the one-dmensonal case leads to a good precondtoner. The propertes of the fnte element/fnte dfference precondtoners n the two-dmensonal case were rgorously establshed by Km and Parter [6 7]. Thus the applcaton of the Krylov subspace methods [1] such as generalzed mnmal resdual method GMRES leads to an teratve solver convergng to the algebrac soluton wthn a constant number of steps that depends on the requred accuracy but not on the number of unknowns. Receved by the edtors February and n revsed form Aprl Mathematcs Subject Classfcaton. 35J25 65F

2 278 Z. FAG J. SHE AD H. SU However such a precondtoned method requres solvng the precondtoner system.e. solvng the fnte element/fnte dfference system on the spectral collocaton ponts. How to effcently apply the precondtoners s a challengng problem snce the grd formed by spectral collocaton ponts contanng long-thn elements s not shape-regular. We note that Shen et al. [13] developed a fnte element multgrd precondtoner for the second-order ellptc equatons. In ths paper we seek to develop an approxmate precondtoner by explorng the algebrac propertes of the fnte dfference precondtoner. It s obvous that the two-dmensonal fnte dfference precondtoner s a nonsymmetrc block trdagonal matrx. Approxmatng ths matrx to construct a new effcent precondtoner s a natural dea. In [8] g and Pan proposed an approxmate nverse method to modfy crculant-plus-dagonal precondtoners for solvng Toepltz-plus-dagonal systems. Ther dea s to use crculant matrces to approxmate the nverson of Toepltz matrces and then combne the rows of these matrces together. As the resultng precondtoner s already of the nverted form only matrx-vector multplcatons are requred n the precondtonng step. Recently Pan et al. [10] also proposed approxmate nverse precondtoners for dagonaltmes-toepltz matrces. The man purpose of ths paper s to propose and develop approxmate nverse precondtoners for two-dmensonal ellptc operators based on the modfcaton of the fnte-dfference operator dscretzed on the CGL grd. Frst we use a scalng strategy to approxmate the fnte-dfference operator. Then we construct an approxmate nverse precondtoner to approxmate the nverse of scaled Laplacanplus-dagonal matrces and combne them together row-by-row. In order to reduce the nfluence of the varous coeffcents an nterpolaton method wth the egenvalues of Laplacan s utlzed. Specal nterpolaton nodes are chosen to mprove the accuracy of approxmaton. By use of the dscrete sne transform DST the resultngprecondtonercanbeeffcentlymplementedwthol y log operatons where the small nteger l s ndependent of and y. umercal examples are gven to demonstrate the effectveness of the proposed precondtoner. The paper s organzed as follows. In Secton 2 we ntroduce the Chebyshev collocaton method for the ellptc operator and the assocated fnte-dfference operator. In Secton 3 we construct the proposed precondtoners. umercal examples are gven to demonstrate the performance of the proposed precondtoner n Secton 4. In the fnal secton concludng remarks are gven. 2. The Chebyshev-collocaton and the fnte-dfference operator In ths secton we recall the Chebyshev-collocaton method for the ellptc operatorandtheassocatedfnte-dfferenceoperator. LetP bethespaceofpolynomals of degree less than or equal to. Let whch are the CGL ponts. x j = cos jπ j = The one-dmensonal case. Consder the one-dmensonal ellptc problems 2 axu x +cxux = fx x 1; u±1 = 0. The Chebyshev-collocaton method for 2 s to fnd u X := {v P : v±1 = 0} such that 3 au x=xk +cx k u x k = fx k k =

3 PRECODITIOIG TECHIQUES I CHEBYSHEV COLLOCATIO METHOD 279 Let {p j x} j=0 be the Lagrange bass polynomals assocated wth {x j} j=0. Then we can express u x = u x j p j x. Denotng the Chebyshev dfferentaton j=0 matrx by G = p j x k kj=01... we have and u x k = u x j p j x k = G kj u x j u x k = j=0 j=0 u x j p jx k = j=0 G 2 kj u x j. We lst below the formulas for the entres of G for arbtrary cf. [2 14]: Lemma 1. For each 1 let the rows and columns of the Chebyshev spectral dfferentaton matrx G be ndexed from 0 to. The entres of ths matrx are G 00 = j=0 G = G jj = x j 21 x 2 j = 1... j G kj = γ k k+j k j kj = 0... γ j x k x j where γ 0 = γ = 2 and γ k = 1 for k = Explct formulae for the entres of G 2 s also avalable n [3 11]. By usng the dfferentaton matrx G t costs O 2 to compute the dervatve of u at all CGL ponts. However ths process can be accelerated to O log by expressng u n Chebyshev polynomals and usng the fast cosne transform [4 12]. Set D a = dagax 0ax 1...ax Dc = dagcx 1 cx 2...cx A = { G D a G kj } 1 kj and A 1 = A+ D c. The Chebyshev-collocaton scheme 3 reduces to the followng lnear system A 1 ū = f where ū = u x 1...u x and f = fx 1...fx. The matrx A s full and ll-condtoned. As proposed n [9] a good precondtoner for A 1 s to use a fnte-dfference operator on the CGL grd. Denote h j = x j x j j = 1... h j = x j+1 x j /2 j = and a k+1/2 = ax k +x k+1 /2 k = We consder the followng fnte dfference approxmaton: au x=x a /2 h h ux a/2 + a +1/2 ux h h h h +1 + a +1/2 h h +1 ux +1 = Then the precondtoner for A 1 based on the above approxmaton can be wrtten as follows: B 1 := F + D c

4 280 Z. FAG J. SHE AD H. SU where 4 F j := a /2 h a h /2 + a +1/2 h h a +1/2 h h +1 j = j = h h +1 j = +1. ote that B 1 s a nonsymmetrc trdagonal matrx. Invertng B 1 or solvng a lnear system wth B 1 as the coeffcent matrx requres about O operatons The two-dmensonal case. For the two-dmensonal ellptc equatons 1 the collocaton ponts are the tensor product of unvarate CGL nodes. Assume that and y are the number of the CGL ponts n each drecton respectvely. Then the Chebyshev-collocaton method wll lead to a lnear system wth the matrx A 2 := I x A b y +A a I y + D c D d y where denotes the Kronecker product A a s the matrx A defned n the last sub-secton A b y s smlar to A a wth by replacng ax Dc x s the dagonal matrx D c defned n the last sub-secton D d y s smlar to D c where dy s used nstead of cx and both I x and I y are dentty matrces. The fnte-dfference operator assocated wth the two-dmensonal ellptc operator s defned as follows: 5 B 2 := I x F b y +F a I y + D c D d y where F a s defned by the formula 4 F b y s defned analogously wth ax replaced by by. We remark that B 2 s a non-symmetrc block trdagonal wth trdagonal blocks matrx. Therefore unlke B 1 n the one-dmensonal case t s not an easy task to nvert B 2. In the followng secton we shall construct a precondtoner based on the approxmate nverse strategy for B Constructon of the precondtoner For the nterest of smplcty we frst dscuss the basc technques n the onedmensonal case and then these technques are utlzed to approxmate B Constructon n the one-dmensonal case. We shall construct an effectve precondtoner n the one-dmensonal case through a sequence of approxmatons. Takng the structure of the matrx F nto consderaton we frstly propose a scaled matrx as an approxmaton. Defne 1 a/2 t = + a +1/2 = h h +1 and 6 T = dagt 1 t 2...t. The frst approxmaton s as follows: 7 F H T L T where 8 H = dag 1 h 1 1 h h

5 PRECODITIOIG TECHIQUES I CHEBYSHEV COLLOCATIO METHOD 281 and L = R. 0 2 Then we construct the frst precondtoner Denote P 1 = H T L T + D c = H T L + D c T 2 T. 10 M = L + D c H T 2. Then P 1 = H T MT. We note that the matrces H T are dagonal and can be easly to handle. Therefore we only consder how to nvert the matrx M n 10. Defne 11 K = L + h c t 2 I = where c = cx. Let e be the -th column of the dentty matrx. Accordng to the fact that e M = e K we construct our precondtoner based on the followng approxmaton [8 10] e M e K. Ths means that the -th row of the nverse of M s approxmated by the -th row of the nverse of K. Therefore we propose our second precondtoner P 2 whose nverse s defned by 12 P 2 = T =1 e e K T. We see from above that to construct P2 we need to compute the nverse of K = Snce the matrx L can be dagonalzed n O log operatons by the DST the product K v for any vector v can be computed n O log operatons. Let S be the DST matrx. ote that S s symmetrc orthogonal and ts j-th entry s gven by 2 πj sn 1 j. Thus the nverse of K can be computed by K = S Λ + h S c t 2 I where Λ s a dagonal matrx whose entres are 2 2cos jπ j = the egenvalues of L. Hence mplementng a precondtoner based on P 2 requres O DST per teraton whch s stll too expensve. In order to reduce the computatonal cost we propose to use the nterpolaton method to construct a more effcent precondtoner. We choose a small number

6 282 Z. FAG J. SHE AD H. SU ll of values {θ j } l {ξ = π } =1 whch covers most of the range of values of {ξ } =1. Defne 1 q θ = θ 0π λ Λ θ+w where λ Λ θ = 2 2cosθ and w = h c. Let t 2 13 p θ = φ 1 θq θ 1 +φ 2 θq θ 2 + +φ l θq θ l be the pecewse lnear nterpolaton for q θ based on the l ponts {θ j q θ j } l. We apply nterpolaton formula 13 to approxmate K : 14 K S Φ j q θ j S = where Φ j = dagφ j ξ 1 φ j ξ 2...φ j ξ are the nterpolaton coeffcent matrces. Fnallycombnng the above consderaton we defne our fnal precondtoner P 3 by P3 =T e e S =1 =T =T =T =1 =1 Φ j q θ j S e e q θ j S Φ j S T e e q θ j W j S Φ j S T T S Φ j S T wherew j = dagq 1 θ j q 2 θ j...q θ j aredagonalmatrces. owapplyng P3 to any vector requres about Ol log operatons whch s acceptable for a small number l. Snce the orgnal functon q θ has weak sngulartes near θ = 0 the nterpolaton nodes should be slghtly dense near ξ Constructon n the two-dmensonal case. In the followng we apply smlar technques to construct a sequence of approxmate precondtoner for B 2 defned n 5. Frst usng the approxmaton 7 n x-drecton we defne our frst precondtoner by 15 ˆP 1 =I x F b y +H x T x L x T x I y + D c x D d y =H x T x I y T 2 F b y +L x I y + T 2 Dc x D d y T x I y H x T x I y ˆMT x I y where H x T x and L x are defned as formulas 8 6 and 9 respectvely and ˆM denotes the mddle term n the above decomposton. ote that the frst and

7 PRECODITIOIG TECHIQUES I CHEBYSHEV COLLOCATIO METHOD 283 the last terms n the decomposton of ˆP 1 n 15 are dagonal. Therefore we only need to construct a precondtoner for the mddle term ˆM. Defne ˆK = h t 2 I x F b y +L x I y + h c t 2 I x D d y = where t h and c are as n 11. Let e be the -th column of the dentty matrx I x. ote the fact that e I y ˆM = e I y ˆK. As n one-dmensonal case we use the followng approxmaton e I y ˆM e I y ˆK whch means that the y +1-th to y -th rows of the nverse of ˆM are approxmated by the y +1-th to y -th rows of the nverse of ˆK. Therefore smlarly to P 2 n 12 for the one-dmensonal case we propose the precondtoner ˆP 2 whose nverse s defned by ˆP 2 = T I y x e e I y =1 We note that ˆK can be factored as 16 ˆK =S x I y h t 2 S x I y C S x I y ˆK T I y. I x F b y +Λ x I y + h c t 2 I x D d y S x I y where Λ x = dagλ 1 λ 2...λ x s a dagonal matrx whose dagonals are the egenvaluesofl x andc whchsablockdagonalwthtrdagonalblocksmatrx denotes the mddle factor n the factorzaton of ˆK. We remark that the nverse of the frst and the last term n 16 multplyng any vector can be mplemented by the DST n O y log operatons whle the mddle term C can be nverted n O y operatons;.e. ˆK = S x I y C S x I y. evertheless t s too expensve to calculate ˆP 2 snce we need to compute about nverses of ˆK = In order to reduce the computatonal workload as n the one-dmensonal case we propose to explot the nterpolaton method to construct the practcal precondtoner. Denote C = dagc 1 C 2...C x where C k s an y 1 y 1 trdagonal matrx as follows C k = h t 2 F b y + h c D t 2 d y +λ Λ ξ k I y n whch λ Λ θ = 2 2cosθ and ξ k = kπ. We nvestgate ts nverse usng the nterpolaton method. Let h Q θ t 2 F b y + h c D d t 2 y +λ Λ θi y θ 0π be an y y matrx functon. Then we have C k = Q ξ k.

8 284 Z. FAG J. SHE AD H. SU We choose a small number ll of values {θ j } l { ξ k = kπ } x k=1. Let 17 P θ = φ 1 θ Q θ 1 +φ 2 θ Q θ 2 + +φ l θ Q θ l bethepecewselnearnterpolatonfor Q θbasedonthelmatrces{θ j Q θ j } l. Thus by nterpolaton formula 17 to approxmate C we have and C C k P ξ k = φ j ξ k Q θ j = dag φ j ξ 1 φ j ξ 2...φ j ξ x Q θ j = Φ j Q θ j where the dagonal matrx Φ j s as n 14. Fnally we defne the practcal precondtoner ˆP 3 whose nverse s defned as follows [ x ] ˆP 3 =T I y e e S I y Φ j Q θ j S x I y 18 T =1 [ x =T I y I y =1 =1 e e Q Sx ] θ j Φ j S x I y T I y x =T I y e e Q Sx θ j Φ j S x I y T I y =T I y W j S x Φ j S x I y T I y where W j = dag Q 1 θ j Q 2 θ j... Q xθ j j = 12...l are block dagonal matrces n whch each block s the nverse of a trdagonal matrx. Therefore Wj can be nverted wth only O y operatons. Applyng ˆP 3 to multply any vector requres Ol y log operatons whch s acceptable for a small number l. It s expected that as l the number of nterpolaton nodes ncreases the number of teratons requred for convergence decreases. However the cost of formng and applyng the precondtoner grows proportonally to l. Hence there s a trade-off to determne a sutable number of nterpolaton ponts. 4. umercal experments In ths secton we carry out numercal experments to study the performance of the proposed precondtoner ˆP 3 n 18. We employ the precondtoned GMRES method to solve the collocaton system. In all numercal experments the stoppng crteron s r k 2 r 0 2 < 10 0 where r k s the resdual vector after k teratons and r 0 s the ntal resdual vector. All numercal experments are mplemented usng Matlab on a Dell Optplex 3020 wth the confguraton: IntelR CoreTM CPU GHz and 8.00 GB of memory.

9 PRECODITIOIG TECHIQUES I CHEBYSHEV COLLOCATIO METHOD 285 Table 1. umercal results for ax = by = 1. ˆP3 l = 8 ˆP3 l = 10 ˆP3 l = 12 ˆP3 l = 14 Ave-GMRES = y Iter CPU Iter CPU Iter CPU Iter CPU Iter CPU Table 2. umercal results for ax = e x by = 1. ˆP3 l = 8 ˆP3 l = 10 ˆP3 l = 12 ˆP3 l = 14 Ave-GMRES = y Iter CPU Iter CPU Iter CPU Iter CPU Iter CPU We consder the two-dmensonal ellptc equaton 1 wth the source term fxy = 1. ote that no approxmaton s used n the y-drecton n dervng the precondtoner ˆP 3 whch ndcates that the coeffcent functon by wll not affect the convergence rate of the precondtoner. Therefore we set by = 1 n all the experments. On the other hand the coeffcent functon cxdy s to add nformaton to the man dagonal of the system matrx. cxdy = 0 s chosen to demonstrate the bad condtonal cases. In consequence of the weak sngulartes of the orgnal functon the nterpolaton nodes are selected as θ j = π j l j = 12...l where x denotes the celng of x. We remark the every θ j should be dfferent and we set θ r = θ r +π/ once θ r θ r. For the purpose of comparsons we take the average of each dagonal of the fnte dfference precondtoner resultng n a block trdagonal Toepltz wth trdagonal Toepltz blocks structured precondtoner for the collocaton system whch s denoted by Ave-GMRES. The numercal results are lsted n Table 1-4 where ˆP 3 l = denotes the GMRES method wth the precondtoner ˆP 3 wth l beng the number of nterpolaton nodes Iter denotes the number of teratons requred to solve 1 CPU denotes the CPU tme n seconds for solvng the dscretzed system and - means that the methods do not converge wthn 6000 teratons. In Table 1 the numercal results are reported wth constant coeffcent ax = 1. We see that the precondtoned GMRES methods exhbt excellent performance both n terms of teraton steps and CPU tme and the teraton number only ncreases slghtly as the number of grd ponts ncreases. The number of teratons decreases as expected whle the number of nterpolaton nodes ncreases. Table 2 and 3 lst the numercal results for non-constant coeffcents. The results wthcoeffcentax = e x stestedntable2. Weobservethatl = 8provdesbetter resultsntermsoftheteratonnumberandcputme. InTable3 welsttheresults

10 286 Z. FAG J. SHE AD H. SU Table 3. umercal results for ax n large varaton case and by = 1 wth l = 8. ax e 6x 10 3 cosx cos8πx+10 = y Iter CPU Iter CPU Iter CPU Table 4. umercal results for axy = bxy = e x+y +1. ˆP3 l = 5 ˆP3 l = 6 ˆP3 l = 7 = y Iter CPU Iter CPU Iter CPU wth ax beng functons wth large varatons. We observe that the precondtoners s not senstve for problems wth coeffcents havng large varatons. In the last example we examne the effectveness of the precondtoner to nonseparable ellptc equatons: x axy u x y bxy u = fxy n Ω = 1 2. y For problems wth non-separable coeffcents we buld the precondtoner by usng averages of the coeffcents as āx = ax ydy and by n the smlar way. Then our precondtoners could be appled to the relatve separable coeffcent systems. umercal results are gven for ax y = bxy = e x+y +1 n Table 4. We observe that the teraton numbers are larger than the separable case but are stll acceptable. We remark that the current algorthm s based on the approxmaton n x drecton. However problem 1 s symmetrc wth respect to x and y. The roles of x and y can be swtched to get an alternatve algorthm. When y < the alternatve one would be less computatonal expensve. 5. Concludng remarks The man contrbuton of ths paper s to develop a precondtoner based on the approxmate nverses n Chebyshev collocaton method for two-dmensonal ellptc equatons. The complexty of the matrx-vector multplcaton of ˆP 3 s of Ol y log. It sshownnumercallythattheprecondtonedgmresmethod for solvng these precondtoned collocaton systems converges very quckly. We only consdered two-dmensonal case n ths paper. But snce one drecton s approxmated n the two-dmensonal case our strategy for constructng precondtoners can be easly extended to three-dmensonal cases. Indeed n the

11 PRECODITIOIG TECHIQUES I CHEBYSHEV COLLOCATIO METHOD 287 three-dmensonal case we can frst apply the approxmaton 7 to x and y drectons of the fnte dfference operator resultng n a scaled tensor product. Usng the row-by-row approxmaton we can obtan the second precondtoner whch s of nverted form. Fnally by defnng a two-dmensonal tensor functon the nterpolaton method can be utlzed to construct a practcal precondtoner whch requres only Ol y z log y operatons. evertheless even the numercal results show the effcency and fast convergence of the proposed method the convergence of our algorthm has not been studed theoretcally but wll be tackled n our future work. Acknowledgments Ths research was supported n part by SF DMS and AFOSR FA and n part by research grants MYRG FST from Unversty of Macau and 054/2015/A2 from FDCT of Macao. References [1] M. Benz Precondtonng technques for large lnear systems: A survey J. Comput. Phys pp [2] D. Gottleb M. Y. Hussan and S. A. Orszag Introducton: Theory and Applcatons of Spectral Methods n R. G. Vogt D. Gottleb and M. Y. Hussan eds. Spectral Methods for Partal Dfferental Equatons SIAM Phladelpha [3] D. Gottleb and L. Lustman The Dufort-Frankel Chebyshev method for parabolc ntal boundary value problems Comput. Fluds pp [4] D. Gottleb and S. A. Orszag umercal Analyss of Spectral Methods: Theory and Applcatons SIAM [5] P. Haldenwang G. Labrosse S. Abboud and M. DeVlle Chebyshev 3-D spectral and 2-D pseudospectral solvers for the Helmholtz equaton J. Comput. Phys pp [6] S. Km and S. Parter Precondtonng Chebyshev spectral collocaton method for ellptc partal dfferental equatons SIAM J. umer. Anal pp [7] S. Km and S. Parter Precondtonng Chebyshev spectral collocaton by fnte-dfferences operators SIAM J. umer. Anal pp [8] M. g and J. Pan Approxmate nverse crculant-plus-dagonal precondtoners for Toepltzplus-dagonal matrces SIAM J. Sc. Comput pp [9] S. A. Orszag Spectral methods for problems n complex geometres J. Comput. Phys pp [10] J. Pan R. Ke M. g and H. Sun Precondtonng technques for dagonal-tmes-toepltz matrces n fractonal dffuson equatons SIAM J. Sc. Comput pp. A2698 A2719. [11] R. Peyret Introducton to Spectral Methods Von Karman Insttute Rhode-St.-Genèse Belgum [12] J. Shen T. Tang and L. Wang Spectral Methods: Algorthms Analyss and Applcatons Sprnger Seres n Computatonal Mathematcs Sprnger [13] J. Shen F. Wang and J. Xu A fnte element multgrd precondtoner for Chebyshevcollocaton methods Appl. umer. Math pp [14] J. A. C. Wedeman and S. C. Reddy A MATLAB Dfferentaton Matrx Sute ACM Trans. Math. Software Department of Mathematcs Unversty of Macau Macao E-mal: fzw913@yeah.net Department of Mathematcs Purdue Unversty West Lafayette I USA E-mal: shen7@purdue.edu Department of Mathematcs Unversty of Macau Macao E-mal: hsun@umac.mo

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