湘潭大学 硕士学位论文 ECMG中具有最优性价比的迭代参数的自适应选取算法 姓名：李朋 申请学位级别：硕士 专业：计算数学 指导教师：黄云清 20080520 .g Bornemann J5$ cu U.? N a L0 m?(L (2 0)l)h2 l ,l L0 m0(L L0)2,l = L 3n m?m0JX . ?3SO XJJ oO op)K J 3L. ECMG : k5d m? m0gAL?OL. S m?, m0E,) m X|uyz? L mXl yL.k5dSgA c:.CMG L.ECMG p dlSGauss-seidel ?FCG U I ABSTRACT Cascadic multigrid method (CMG) has been studied much after it was pro- posed by Bornemann for its simple calculation format. Economical cascadic multi- grid (ECMG) was proposed recently by Shi Zhongci et al.21. Economical cascadic multigrid greatly reduces complexity and remains the same accuracy compared with CMG. The main ingredient of the ECMG method is a new criteria for choosing the smoothing steps on each level as following. ml= mLLl,l L0 m?(L (2 0)l)h2 l ,l L0 m0(L L0)2,l = L The ECMG method is convergent and accurate in theory whatever the parameters m?and m0are. But in actual calculation neither the complexity nor the accuracy of the ECMG method is not good if parameters are selected not well. In order to solve this problem, we make some study about the parameters in this paper. In this paper adaptive choosing iterative parameters algorithm for ECMG is proposed based on ECMG method. We fi rst collect information of the parameters m?, m0and energy normanormand error accuracy by numerical experiments. Secondly, we discover the function about parameters m?, m0and the levels L. At last, we propose the adaptive algorithm for choosing iterative parameters with the best-eff ective in ECMG method. Keyword:Cascadic multigrid method(CMG);Economical cascadic multigrid method (ECMG); Gauss-seidel method; Conjugate gradient method; Energy norm II ?M5( x( ?3e?1 J AOI5SN? ? J ? z8N 3(I ( ?Jd FcF )?k?3! 5 ?3Ik?x?Ef #N ?/ SN? k?1u K! E? ? U?5?n FcF FcF 1 .CMG Bornemann Deufl hard 3z3,4J vko?D CMG ? FCG?FPCG1wf3z?I? L O N?O z3 y dk?z8,9J .gA $ BPX 6! Yserentant ?28e .(J(JL .k?53z18 Shaidurov y 3 H2m CG S1wf. Oz3y 3fK H1+(0 1) e Ke IOSX Jacobi!Gauss-seidel!Richardson nK. ?uKe?F1wf u!nKBraess Dehmen1.A u Stokes aN?19,20,23.AN? K KK ! a! )7 L0 m?(L (2 0)l)h2 l ,l L0 m0(L L0)2,l = L (1.1) 1 L ?L0?l ?c?ml1 l ?1w Sghl?c? m0, m?, 0, 31 4 m0, m?L.O! ) kX K L (1.1) m? L0c?Sg mlk K m0 L0?kK3n m0, m?X L.u K?3SO uyXJ m0, m?L. ,O?$XJ m0, m?L .,p ?O m0, m? / m0, m?O!K/5d c55d k zVg X k8xmax, xmin8 ?u x X eC trans(x) := x xmin xmax xmin X z?=zm 0, 1 L8 X z 1: m0, m?5d 3,? L e z| m0, m?AX|L. O work p cpu $1m time L ! U anorm 3T?e | m0, m?A time, anorm 8 T, N 8 T, N z z?8 T 0, N0 ,| m0, m?eA anorm 0 N 0 time 0 T 0 | m0, m? 5d L= = anorm 0 time 0 dw 5d X m0, m?O ? 2: m0, m?5d ? = 1 5d5d m0, m?k5d N X m0, m?/O 5d 3? ? = 1 L .O )$ k 5d m0, m?L.kX 3L.ECMG: 2 (1.1) m0, m?ZgAL m0, m?k5d L?OL.ECMGS m?, m0OE,)mX| uyz? L mX,l yL .kZ5dSgA e K d P i,j=1 xi(aij u xj) = f,in u= 0,on (1.2) d? Rd d k.f L2() aij f v 1w (1.2)C/ u H1 0 a(u,v) = (f,v),v H1 0 (1.3) a(u,v) = Z d X i,j=1 aij u xj v xi dx,u,v H1 (f,v) = Z fvdx 3A U anorm Lkuka kuka= a(u,u)1/2= ( Z ai,juudxdy)1/2,u H1 0() N?y? ai,j= 1 U Sobolev m H1 0() 1 3?A! 0 . CMG L. ECMG ?EL.k5dS gA ?S ?ygA