Precondtonng technques n Chebyshev collocaton method for ellptc equatons Zh-We Fang Je Shen Ha-We Sun (n 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 llcondtoned. 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 O(l 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 Mathematcs Subject Classfcaton: 35J25, 65F10, 6522, 6535 1 Introducton We consder a two-dmensonal separable ellptc equaton ( a(x u ( b(y u + c(xd(yu(x, y = f(x, y x x y y n Ω = ( 1, 1 2 (1.1 Department of Mathematcs, Unversty of Macau, Macao (fzw913@yeah.net. Department of Mathematcs, Purdue Unversty, West Lafayette, I 47907-1957, USA (shen7@purdue.edu. Ths author was supported n part by SF DMS-1620262 and AFOSR FA9550-16-1-0102. Department of Mathematcs, Unversty of Macau, Macao (hsun@umac.mo. Ths author was supported n part by research grants MYRG2016-00063-FST from Unversty of Macau and 054/2015/A2 from FDCT of Macao. 1
wth homogeneous Drchlet boundary condtons u = 0 on Ω, where the coeffcent functons a(x, b(y, c(x, d(y and f(x, y are contnuous, and 0 < α a(x, b(y β n Ω for some postve constants α and β, and c(xd(y 0. A very effcent accurate method for obtanng approxmate soluton of the above boundaryvalue problem s the Chebyshev collocaton method [3, 4, 7, 9, 11, 12], 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 llcondtoned. Thus t becomes prohbtve to use an 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 solvers convergng to the algebrac soluton wthn a constant number of steps that depends on the requred accuracy, but not on the number of unknowns. However, such a precondtoned method requres to 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 qute 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 dagonal-tmes-toepltz matrces. The man purpose of ths paper s to propose and develop approxmate nverse precond- 2
toners 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 fntedfference operator. Then we construct an approxmate nverse precondtoner to approxmate the nverse of scaled Laplacan-plus-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 usng of the dscrete sne transform (DST, the resultng precondtoner can be effcently mplemented wth O(l 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 operator and the assocated fnte-dfference operator. Let P be the space of polynomals of degree less than or equal to. Let whch are the CGL ponts. x j = cos ( jπ, j = 0, 1,...,, 2.1 The one-dmensonal case Consder the one-dmensonal ellptc problems (a(xu (x + c(xu(x = f(x, x ( 1, 1; u(±1 = 0. (2.1 The Chebyshev-collocaton method for (2.1 s to fnd u X := {v P : v(±1 = 0} such that (au x=xk + c(x k u (x k = f(x k, k = 1, 2,..., 1. (2.2 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 matrx by G = j=0 3
(p j (x k k,j=0,1,...,, we have and u (x k = u (x k = u (x j p j(x k = j=0 j=0 u (x j p j (x k = (G kj u (x j, 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 ( + 1 ( + 1 Chebyshev spectral dfferentaton matrx G be ndexed from 0 to. The entres of ths matrx are (G 00 = 2 2 + 1 6 j=0, (G = 2 2 + 1, 6 (G jj = x j 2(1 x 2 j = 1,..., 1, j, (G kj = γ k ( 1 k+j, k j, k, j = 0,...,, γ j (x k x j where γ 0 = γ = 2 and γ k = 1 for k = 1,..., 1. 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 usng the fast cosne transform [4, 12]. n Chebyshev polynomals and Set D a = dag(a(x 0, a(x 1,..., a(x, Dc = dag(c(x 1, c(x 2,..., c(x 1, A = {( G D a G k,j } 1 k,j 1, and A 1 = A + D c. The Chebyshev-collocaton scheme (2.2 reduces to the followng lnear system A 1 ū = f, where ū = (u (x 1,..., u (x 1 and f = (f(x 1,..., f(x 1. 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 1 (j = 1,...,, h j = (x j+1 x j 1 /2 (j = 1,..., 1, and a k+1/2 = a((x k + x k+1 /2 (k = 0, 1,..., 1. We consder the followng fnte dfference approxmaton: (au x=x a 1/2 u(x 1 h h ( a 1/2 h h + a +1/2 u(x + a +1/2 u(x +1, = 1, 2,..., 1. h h +1 h h +1 4
Then, the precondtoner for A 1 based on the above approxmaton can be wrtten as follows: where (F j := B 1 := F + D c, a 1/2 h h, j = 1, a 1/2 h h + a +1/2 h h +1, j =, a +1/2 h h +1, j = + 1. (2.3 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. 2.2 The two-dmensonal case For the two-dmensonal ellptc equatons (1.1, 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 x 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 b(y replacng a(x, Dc x s the dagonal matrx D c defned n the last sub-secton, Dd y s smlar to D c where d(y s used nstead of c(x, and both I and I y are dentty matrces. The fnte-dfference operator assocated wth the two-dmensonal ellptc operator s defned as follows: B 2 := I x F b y + F a x I y + D c x D d y, (2.4 where F a x s defned by the formula (2.3, F b y s defned analogously wth a(x replaced by b(y. 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 2. 3 Constructon of the precondtoner For the nterest of smplcty, we frst dscuss the basc technques n the one-dmensonal case, and then these technques are utlzed to approxmate B 2. 5
3.1 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 1/2 t = + a +1/2, = 1, 2,..., 1, 2 h h +1 and T = dag(t 1, t 2,..., t 1. (3.1 The frst approxmaton s as follows: where and Then, we construct the frst precondtoner Denote F H T L T, (3.2 ( H = dag, 1 h1 1 h2 1,...,, (3.3 h 1 2 1 0. 1 2.. L =.... R ( 1 ( 1. (3.4.. 1 0 1 2 P 1 = H T L T + D c = H T (L + D c H 1 T 2 T. M = L + D c H 1 T 2. (3.5 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 (3.5. Defne K = L + h c t 2 I, = 1, 2,..., 1, (3.6 where c = c(x. 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 1 e K 1. 6
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 ( 1 P 1 2 = T 1 =1 e e K 1 H 1 T 1. (3.7 We see from above that, to construct P 1 2, we need to compute the nverse of K ( = 1, 2,..., 1. Snce the matrx L can be dagonalzed n O( log operatons by the DST, the product K 1 v for any vector v can be computed n O( log operatons. Let S be the ( 1 ( 1 DST matrx. ote that S s symmetrc, orthogonal and ts (, j-th entry s gven by ( 2 πj sn, 1, j 1. Thus, the nverse of K can be computed by K 1 = S ( Λ + h 1 c t 2 I S, where Λ s a dagonal matrx whose entres are 2 2 cos( jπ, j = 1, 2,..., 1, 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 l(l of values {θ j } l {ξ = π } 1 =1, whch covers (most of the range of values of {ξ } 1 =1. Defne q (θ = 1 λ Λ (θ + w, θ (0, π, where λ Λ (θ = 2 2 cos θ and w = h c. Let t 2 p (θ = φ 1 (θq (θ 1 + φ 2 (θq (θ 2 + + φ l (θq (θ l (3.8 be the pecewse lnear nterpolaton for q (θ based on the l ponts {θ j, q (θ j } l. We apply nterpolaton formula (3.8 to approxmate K 1 K 1 : S Φ j q (θ j S, = 1, 2,..., 1, (3.9 where Φ j = dag (φ j (ξ 1, φ j (ξ 2,..., φ j (ξ 1 are the nterpolaton coeffcent matrces. 7
Fnally,combnng the above consderaton, we defne our fnal precondtoner P 3 by 1 P3 1 =T 1 e e S Φ j q (θ j S H 1 =1 1 =T 1 =T 1 =T 1 =1 ( 1 =1 e e q (θ j S Φ j S H 1 T 1 e e q (θ j W j S Φ j S H 1 T 1, T 1 S Φ j S H 1 T 1 where W j = dag (q 1 (θ j, q 2 (θ j,..., q 1 (θ j are dagonal matrces. ow applyng P 1 3 to any vector requres about O(l 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 1. 3.2 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 (2.4. Frst, usng the approxmaton (3.2 n x-drecton, we defne our frst precondtoner by 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 H 1 T 2 F b y + L x I y + H 1 T 2 Dc x x D d y (T x I y (H x T x I y ˆM(T x I y, (3.10 where H x, T x and L x are defned as formulas (3.3, (3.1 and (3.4 respectvely, and ˆM denotes the mddle term n the above decomposton. ote that the frst and the last terms n the decomposton of ˆP1 n (3.10 are dagonal. precondtoner for the mddle term ˆM. Defne ˆK = h t 2 Therefore, we only need to construct a I x F b y + L x I y + h c t 2 I x D d y, = 1, 2,..., 1, where t, h and c are as n (3.6. Let e be the -th column of the dentty matrx I x. ote the fact that (e I y ˆM = (e I y ˆK. 8
As n one-dmensonal case, we use the followng approxmaton (e I y ˆM 1 (e I y ˆK 1, whch means that the (( 1( y 1 + 1-th to ( y 1-th rows of the nverse of ˆM are approxmated by the (( 1( y 1+1-th to ( y 1-th rows of the nverse of ˆK. Therefore, smlarly to P 2 n (3.7 for the one-dmensonal case, we propose the precondtoner ˆP 2 whose nverse s defned by ˆP 1 2 = (T 1 I y We note that ˆK can be factored as ( x 1 (e e I y =1 ˆK 1 (T 1 H 1 I y. ˆK =(S x I y ( h t 2 (S x I y C (S x I y, I x F b y + Λ x I y + h c t 2 I x D d y (S x I y (3.11 where Λ x = dag(λ 1, λ 2,..., λ x 1 s a dagonal matrx whose dagonals are the egenvalues of L x, and C, whch s a block dagonal wth trdagonal blocks matrx, denotes the mddle factor n the factorzaton of ˆK. We remark that the nverse of the frst and the last term n (3.11, 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 1 = (S x I y C 1 (S x I y. 1 evertheless, t s too expensve to calculate ˆP 2 snce we need to compute about nverses of ˆK ( = 1, 2,..., 1. In order to reduce the computatonal workload, as n the onedmensonal case, we propose to explot the nterpolaton method to construct the practcal precondtoner. Denote C = dag(c 1, C 2,..., C (x 1 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 2 cos θ and ξ k = kπ. We nvestgate ts nverse usng the nterpolaton method. Let ( h Q (θ t 2 F b y + h 1 c D d t 2 y + λ Λ (θi y, θ (0, π, be an ( y 1 ( y 1 matrx functon. Then we have C 1 k = Q ( ξ k. 9
We choose a small number l(l of values {θ j } l { ξ k = kπ } x 1 k=1. Let P (θ = φ 1 (θ Q (θ 1 + φ 2 (θ Q (θ 2 + + φ l (θ Q (θ l (3.12 be the pecewse lnear nterpolaton for Q (θ based on the l matrces {θ j, Q (θ j } l. Thus, by nterpolaton formula (3.12 to approxmate C 1, we have C 1 k P ( ξ k = φ j ( ξ k Q (θ j, = 1, 2,..., 1, and C 1 ( dag φ j ( ξ 1, φ j ( ξ 2,..., φ j ( ξ x 1 Q (θ j = Φ j Q (θ j, where the dagonal matrx Φ j s as n (3.9. Fnally, we defne the practcal precondtoner ˆP 3 whose nverse s defned as follows, [ x 1 ( ] ˆP 3 1 =(T 1 I y (e e S I y Φ j Q (θ j (S x I y (T 1 H 1 I y =(T 1 I y =1 [ x 1 =1 ( x 1 =(T 1 I y =1 ( e e Q (Sx ] (θ j Φ j S x I y (T 1 H 1 I y e e Q (θ j (Sx Φ j S x I y =(T 1 I y W j (S x Φ j S x I y ( T 1 H 1 I y, (T 1 H 1 I y (3.13 where W j = dag( Q 1 (θ j, Q 2 (θ j,..., Q x 1(θ j, j = 1, 2,..., l, are block dagonal matrces n whch each block s the nverse of a trdagonal matrx. Therefore, Wj can be nverted wth ˆP 1 only O( y operatons. Applyng 3 to multply any vector requres O(l 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 (3.13. We employ the precondtoned GMRES method to solve the 10
collocaton system. In all numercal experments, the stoppng crteron s r k 2 r 0 2 < 10 10, 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: Intel(R Core(TM CPU 5-4590 3.30 GHz and 8.00 GB of memory. We consder the two-dmensonal ellptc equaton (1.1 wth the source term f(x, y = 1. ote that no approxmaton s used n the y-drecton n dervng the precondtoner ˆP 3 whch ndcates that the coeffcent functon b(y wll not affect the convergence rate of the precondtoner. Therefore we set b(y = 1 n all the experments. On the other hand, the coeffcent functon c(xd(y s to add nformaton to the man dagonal of the system matrx. c(xd(y = 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 ( 1 j 1 l 1, j = 1, 2,..., l, where x denotes the celng of x. We remark the every θ j should be dfferent, and we set θ r = θ r 1 + π/ once θ r θ r 1. 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 = 8, 10, 12, 14 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 (1.1, CPU denotes the CPU tme n seconds for solvng the dscretzed system, and - means that the methods do not converge wthn 6000 teratons. Table 1: umercal results for a(x = b(y = 1. = y ˆP3 (l = 8 ˆP3 (l = 10 ˆP3 (l = 12 ˆP3 (l = 14 Ave-GMRES Iter CPU Iter CPU Iter CPU Iter CPU Iter CPU 64 16 0.34 15 0.26 15 0.29 15 0.33 300 2.66 128 17 0.90 16 0.96 16 1.09 15 1.15 685 15.89 256 18 3.66 17 4.02 16 4.38 16 4.96 1532 123.65 512 19 16.20 18 18.33 17 20.20 17 22.95 3494 1621.91 1024 20 63.95 19 73.39 18 81.29 17 88.26 - - In Table 1, the numercal results are reported wth constant coeffcent a(x = 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 11
of grd ponts ncreases. The number of teratons decreases as expected whle the number of nterpolaton nodes ncreases. Table 2: umercal results for a(x = e x, b(y = 1. = y ˆP3 (l = 8 ˆP3 (l = 10 ˆP3 (l = 12 ˆP3 (l = 14 Ave-GMRES Iter CPU Iter CPU Iter CPU Iter CPU Iter CPU 64 16 0.33 16 0.29 16 0.32 16 0.35 549 4.98 128 18 0.94 17 1.04 17 1.17 16 1.23 1362 33.40 256 20 4.06 18 4.31 17 4.70 17 5.28 3627 304.40 512 21 18.15 19 19.60 18 21.57 18 24.50 - - 1024 22 70.91 20 78.06 19 86.49 18 93.99 - - Table 2 and 3 lst the numercal results for non-constant coeffcents. The results wth coeffcent a(x = e x s tested n Table 2. We observe that l = 8 provde better results n terms of the teraton number and CPU tme. In Table 3, we lst the results wth a(x beng functons wth large varatons. We observe that the precondtoners s not senstve for problems wth coeffcents havng large varatons. Table 3: umercal results for a(x n large varaton case and b(y = 1 wth l = 8. a(x e 6x 10 3 cos(x cos(8πx + 10 = y Iter CPU Iter CPU Iter CPU 64 21 0.36 16 0.29 18 0.33 128 22 1.15 16 0.85 19 0.99 256 23 4.63 17 3.44 20 4.03 512 24 20.58 19 16.36 20 17.23 1024 25 80.18 19 61.19 22 70.57 In the last example, we examne the effectveness of the precondtoner to non-separable ellptc equatons: x ( a(x, y u ( b(x, y u = f(x, y n Ω = ( 1, 1 2. x y y For problems wth non-separable coeffcents, we buld the precondtoner by usng averages of the coeffcents as ā(x = 1 2 1 1 12 a(x, ydy,
and b(y n the smlar way. Then our precondtoners could be appled to the relatve separable coeffcent systems. umercal results are gven for a(x, y = b(x, y = e (x+y +1 n Table 4. We observe that the teraton numbers are larger than the separable case, but are stll acceptable. Table 4: umercal results for a(x, y = b(x, y = e (x+y + 1. = y ˆP3 (l = 5 ˆP3 (l = 6 ˆP3 (l = 7 Iter CPU Iter CPU Iter CPU 64 39 0.44 40 0.47 40 0.49 128 42 1.37 42 1.44 42 1.52 256 44 5.42 41 5.43 42 6.01 512 45 26.39 43 27.24 43 29.47 We remark that the current algorthm s based on the approxmaton n x drecton. However, problem (1.1 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 O(l y log. It s shown numercally 1 that the precondtoned GMRES method 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 three-dmensonal case, we can frst apply the approxmaton (3.2 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 O(l 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. 13
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