One Class of Splitting Iterative Schemes

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1 One Cla of Splitting Iterative Scheme v Ciegi and V. Pakalnytė Vilniu Gedimina Technical Univerity Saulėtekio al. 11, 2054, Vilniu, Lithuania rc@fm.vtu.lt Abtract. Thi paper deal with the tability analyi of a new cla of iterative method for elliptic problem. Thee cheme are baed on a general plitting method, which decompoe a multidimenional parabolic problem into a ytem of one dimenional implicit problem. We ue a pectral tability analyi and invetigate the convergence order of two iterative cheme. Finally, ome reult of numerical experiment are preented. 1 Introduction Iterative method for olving dicrete elliptic problem can be viewed a finite difference cheme for non tationary parabolic problem. The mot important difference i that for elliptic problem we elect the time tep parameter according the requirement of the convergence to the tationary olution and can ignore the approximation error. For plitting method for elliptic problem we refer to [4, 6]. In particular, [5, 6] involve an alternating direction method, [4] preent factorization cheme. The convergence rate can be increaed if optimal nontationary parameter are ued for the definition of each iteration, ee e.g. [6]. Recently the multicomponent verion of alternating direction method were propoed in [1]. Thee cheme are alo ued for olving multidimenional elliptic problem [2]. While in the previou paper the tability of thee plitting cheme wa invetigated by the energy method, we will ue the pectral tability analyi. For ymmetric problem it give neceary and ufficient convergence condition and enable u to find optimal value of iterative parameter. Such analyi wa ued alo in [3] The content of thi paper i organized a follow. In Section 2 we formulate the multicomponent iterative cheme. The convergence of 2D cheme i invetigated in Section 3. In ection 4 we invetigate the tability of 3D iterative cheme, the analyi i done uing numerical experiment. Finally, in Section 5 we tudy the convergence of the p-dimenional multicomponent iterative cheme and prove the energy tability etimate. In Section 6, the Seidel type cheme i formulated and invetigated. P.M.A. Sloot et al. Ed.: ICCS 2002, LNCS 2330, pp , Springer-Verlag Berlin Heidelberg 2002

2 One Cla of Splitting Iterative Scheme Multicomponent iterative cheme Let the trictly elliptic problem be given by i=1 2 u x 2 i = fx, x Q, ux =0 x Q, where Q =0, 1 p. We introduce a uniform grid in Q and approximate the elliptic problem by the following finite difference cheme A α y = f, 1 here A α denote the approximation of a differential operator uing tandard central difference formula. Let, and denote the inner product and the L 2 norm of dicrete function, repectively. Generally, we introduce the following aumption: A α = A α, 0 <m α y A α y, y M α y. It i well known that for the Laplace operator thee aumption are atified with 8 m α, M α 4 h 2, 2 where h i the patial tep ize. In order to olve the ytem of linear equation 1 we define p unknown function y α, α =1, 2,...,p. Then the multicomponent alternating direction MAD cheme i given a ee, [1]: y α ỹ + pa α y α y α + ỹ = 1 p y α, A β yβ = f,, 2,...,p, 3 where y α i the -th iteration of y α. We note that all equation 3 can be olved in parallel. We will invetigate the convergence of the MAD cheme by uing the following error norm: e = ỹ y, r = A αyα f.

3 v. 424 R. Ciegi and V. Pakalnyte 3 Spectral tability analyi of 2D iterative cheme In thi ection we conider a two-dimenional iterative cheme 1 y α ỹ +2A α y α y α + 2 A β yβ = f,, 2. 4 Let denote the error function e α = y α y, α =1, 2,...,p, ẽ = ỹ y. Then the error function atify the following MAD cheme: e α ẽ +2A α e α e α + ẽ = e α, 2 A β eβ =0, α =1, 2, 5 To apply the dicrete von Neumann tability criteria to problem 5, we let e α = N 1 j=1 N 1 k=1 dα,jk injπx 1 inkπx 2, α =1, 2, 6 where N i the number of grid point in one-dimenional grid. It i well known, that injπx α are eigenvector of the operator A α, i.e.: A α injπx α =λ j injπx α, 8 λ 1 λ 2 λ N 1 4 h 2. 7 If we replace e α and e α in 5 by expreion of the form given by equation 6 and ue 7, we get the following matrix equation where d jk i the column vector of pectral coefficient d1,jk d jk =, d 2,jk and Q 2 i the tability matrix of MAD cheme 0.5+λ j 1+2λ j Q 2 = 0.5 λ j 1+2λ k d jk= Q 2 djk, λ k 1+2λ j 0.5+λ k 1+2λ k.

4 One Cla of Splitting Iterative Scheme 425 Now we conider the neceary condition for the tability of iterative MAD cheme 4. Since Q 2 i not ymmetric matrix, the dicrete von Neumann tability criteria can not prove that they are alo ufficient for tability of MAD cheme. Theorem 1. All eigenvalue of tability matrix Q 2 atify inequalitie q jk < 1, 1 j, k N 1 unconditionally for any value of parameter and h. Proof. Uing imple computation we get that eigenvalue q of the amplification matrix Q 2 atify the quadratic equation q 2 q + Then the eigenvalue of Q 2 are q 1,2 = 1 1 ± 2 λ j + λ k 1 + 2λ j 1 + 2λ k =0. 1 2λ j 1 2λ k 1 + 2λ j 1 + 2λ k and they are obviouly both le than 1. The theorem i proved. The convergence order of MDAiterative cheme depend on the parameter. We will ue quai - optimal parameter 0 which olve the minimization problem w 0 =min max 1 2λ λ 1 λ λ N 1 1+2λ. Since the function ga = 1 a 1+a i trictly decreaing, we find 0 from the equation After imple computation we get 1 2λ 1 = 2λ N λ 1 1+2λ N 1 0 = 1 2 h λ 1 λ N 1 4π and all eigenvalue of the amplification matrix Q 2 atify the inequality 1 q jk π h. 9 λ 1 /λ N 1 2 Thee etimate are imilar to convergence etimate obtained for the tandard iterative cheme of alternating direction.

5 v. 426 R. Ciegi and V. Pakalnyte 4 Spectral tability analyi of 3D iterative cheme A it wa tated in ection 3, the von Neumann tability criteria give only ufficient tability condition of MAD iterative cheme. In thi ection we will ue the pectral tability analyi for 3D iterative cheme 3. But intead of finding eigenvalue of an amplification matrix Q 3 we ue direct numerical experiment and find the optimal value of the parameter. Such methodology give u a poibility to invetigate the convergence order of MAD cheme in the uual l 2 norm. Let conider the model problem 3 A α y =λ j + λ k + λ l injπx 1 inkπx 2 inlπx 3, 10 which ha the exact olution y = injπx 1 inkπx 2 inlπx 3. Then the olution of 3D MAD cheme can be repreented a y α = dα injπx 1 inkπx 2 inlπx 3, α =1, 2, 3, where d α can be computed explicitly dβ 1 3 d α= 1+3λ α 3 λ β dβ 1 +3λ α dα, α =1, 2, 3. Then the error of the th iteration d i etimated by the following formula d1 + d2 + d3 e = 1 3. Let S,λ j,λ k,λ l be the number of iteration required to achieve the accuracy e ε for given eigenvalue λ α. Then we invetigate the whole interval [m, M], which characterize the tiffne of the problem and compute S = max m λ α M S,λ j,λ k,λ l. Thi problem i olved approximately by computing S,λ j,λ k,λ l for K +1 3 combination of eigenvalue λ α = m + im m/k. Firt, we invetigated the dependence of the number of iteration S on the parameter. It wa proved that there exit the optimal value 0, uch that S 0 S

6 One Cla of Splitting Iterative Scheme 427 Table 1. The number of iteration a a function of ε S and thi value atifie the following condition max m λ α M S 0,λ j,λ k,λ l =S 0,m,m,m. 11 In Table 1 we preent number of iteration S for different value of. Thee experiment were done with m =10,M = The optimal value of the parameter depend lightly on ε. In Table 2 we preent optimal number of iteration S 0 for different value of ε. Table 2. The optimal value of a a function of ε ε 0 S Finally, we invetigated the dependence of the convergence rate of the MAD iterative cheme on m and M. In Table 3 we preent optimal value of and number of iteration S 0 for different pectral interval. We ued ε =10 4 in thee experiment. It follow from reult preented in Table 3 that 0 = c mm, S 0 =O M. m The above concluion agree well with reult of ection 3.

7 v. 428 R. Ciegi and V. Pakalnyte Table 3. The optimal value of a a function of m and M m M 0 S Error etimate for p-dimenional MAD cheme In thi ection we conider p-dimenional iterative cheme 3. Let u introduce the following notation: y y α y α αt = v 2 3 = α,, α>β, ỹ ỹ ỹ t =, v α,β = y α y β, v α,β 2, Q p y= r p 2 2 v 2 3. In the following theorem we etimate the convergence rate of MAD iterative cheme. Theorem 2. Iterative cheme 3 produce a equence converging unconditionally to the olution of problem 2 and the convergence rate i etimated a Q p y 1 q Q p y, 1 q =min 1+mp, 1+, 12 2M p where m and M are the pectral etimate of the operator A: m = min 1 α p m α, M = max 1 α p M α. Proof. Multiplying both ide of 3 by y αt and adding all equalitie we get y α ỹ, y αt + p A αyαt, y αt +p A βyβ f, ỹ t =0. 13

8 One Cla of Splitting Iterative Scheme 429 The firt term of 13 can be rewritten a I 1 = 1 p = p ỹ t p y α y β, y αt + 1 p α,, α>β v α,β, v t α,β = p ỹ t p v t p 2 v 2 3 v 2 3 By adding equation 3 we get that ỹ t = A α y α f, y β y β, y αt. 14 hence uing the third term of 13 we can prove that p ỹ t 2 + p A βyβ f, ỹ t p r 2 r The econd term of 13 i etimated imilarly. Thu the convergence rate etimate 12 follow trivially. The theorem i proved. A a corollary of Theorem 2 we can find the optimal value of parameter 0 =1/p 2mM, which i obtained from the equation 1+pm =1+ 6 Seidel-type iterative cheme 1 2M p. In thi ection we invetigate the convergence rate of Seidel-type iterative cheme: y α y α + α A β y β + β=α+1 y 1 = y 1, A β yβ = f,, 2,...,p, 16 yα y α =0.5 + y α Spectral tability analyi of 2D cheme To apply the dicrete von Neumann tability criteria to problem 16, we write the global error a a erie: e α = N 1 j=1 N 1 k=1 dα,jk injπx 1 inkπx 2, α =1, 2,

9 v. 430 R. Ciegi and V. Pakalnyte Subtituting thi expanion into 16, we obtain the equation for coefficient d jk= Q 2 djk. 17 where d jk i the column vector of pectral coefficient and Q 2 i the tability matrix of cheme 16 1 λ k 1+λ j 1+λ j Q 2 = 0.51 λ j 0.5+λ j + 2 λ j λ k. 1 + λ j 1 + λ k 1 + λ j 1 + λ k Now we conider the neceary condition for the tability of cheme 16. The eigenvalue of the amplification matrix Q 2 atify the quadratic equation q λ j 1+ q = λ j 1 + λ k 1+λ j Theorem 3. All eigenvalue of tability matrix Q 2 atify inequalitie q jk < 1, 1 j, k N 1 unconditionally for any value of parameter and h. Proof. Application of the Hurwitz criterion give that q jk 1i atified if and only if 0.5 < 1, 1+λ j λ j λ j 1 + λ k < λ j Simple computation prove that both inequalitie are atified unconditionally. The theorem i proved. 6.2 Spectral tability analyi of 3D iterative cheme Let conider the model problem 10. The olution of 3D cheme 16 can be repreented a y α = dα injπx 1 inkπx 2 inlπx 3, α =1, 2, 3, where d α,, 2, 3, are computed explicitly 1 d α= dα + α 1 dα 1 λ β d β 1+λ α 2 e = max dα 1 α 3 3 β=α+1 λ β dβ + We etimate the error of the th iteration dα by the following formula. 3 λ β.

10 One Cla of Splitting Iterative Scheme 431 Table 4. The optimal value of a a function of m and M m M 0 S We invetigated numerically the dependence of the convergence rate of the iterative cheme 16 on m and M. In Table 4 we preent optimal value of and number of iteration S 0 for different pectral interval. We ued ε =10 4 in thee experiment. It follow from reult preented in Table 4 that 0 = c mm, S 0 =O M m The convergence rate of the Seidel type cheme 16 i the ame a of cheme 3.. Reference 1. Abrahin, V.N.: On the tability of multicomponent additive direction method. Differentialnyje uravnenyja in Ruian. 2. Abrahin, V.N., Zhadaeva, N.G.: On the convergence rate of economical iterative method for tationary problem of mathematical phyic. Differentialnyje uravnenyja in Ruian. 3. Aleinikova, T., Čiegi, R.: Invetigation of new iterative method for olving multidimenional elliptic equation. Differentialnyje uravnenyja in Ruian. 4. Marchuk, G.I. Splitting method. Nauka, Mocow, in Ruian. 5. Peaceman, D., Rachford, H.: The numerical olution of parabolic and elliptic differential equation. SIAM, Samarkii, A.A., Nikolajev, E.S. Method for olving difference equation. Nauka, Mocow, 1978.in Ruian.

SOME RESULTS ON INFINITE POWER TOWERS

NNTDM 16 2010) 3, 18-24 SOME RESULTS ON INFINITE POWER TOWERS Mladen Vailev - Miana 5, V. Hugo Str., Sofia 1124, Bulgaria E-mail:miana@abv.bg Abtract To my friend Kratyu Gumnerov In the paper the infinite