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1、湘潭大学 硕士学位论文 瀑布型多重网格法中高精度迭代初值的估计 姓名：蒋科兵 申请学位级别：硕士 专业：计算数学 指导教师：黄云清 20081028 . CMG g Bormemann Deufl hard J5 O cu3CMGJ?kN?1U ? aJ L. ECMG 24 T3?y .)k?/? OAO3o O?3 CMG ECMG 5?e ?S ,?3T?CGSv gCq) S |k u u k|6 |k u0 u k| e5 k u0 u k?XJU k?/?k u0 u k 3o?E=fe? S oUJp 3ECMGO5?! g?! ng?! ng? e?SL?mOO? l (pS
2、c:.CMG L.ECMG ?F CG U? I ABSTRACT Cascadic multigrid method (CMG) has been studied much after it was pro- posed by Bornmmann for its simple calculation format.Economical cascadic multi- grid (ECMG) was proposed recently by Shi Zhongci et al.24. Economical cascadic multigrid greatly reduces complexit
3、y especially on coarse grid and remains the same accuracy compared with CMG. The initial values on next grid are usually got by linear interpolation with the solutions on coarser grid in CMG and ECMG methods. And the FEM solutions are got by enough CG-iterations. On considering the inequality as bel
4、ow |k u u k|6 |k u0 u k| we may reduce k u0 u k to get better accuracy. Many scholars only consider to get better accuracy by reduce the . we get better initial values on the next grid by the second interpolations and the third interpolations. Numerical experiments show that the ECMG method with hig
5、h-accurate initial values on coarse grid is of higher effi ciency. Keyword:Cascadic multigrid method(CMG);Economical cascadic multigrid method (ECMG); Conjugate gradient method(CG); Energy norm ; Interpolation 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?
6、 ? U?5?n FcF FcF 1 . Bornemann Deufl hard 3z4Jo ?YDI?Oy 5B.z?dIOSX Richardson !Jacobi !Gauss-Seidel !CG 3oLeZgSvk ? ?y5 7LAOz?Sg AO3og3J.?T U? Bornemann CG 1wf2DS1wf Deufl hard K d P i,j=1 xi(aij u xj) = f,in u= 0,on (2.1) K (2.1) C/ u H1 0() a(u,v) = (f,v),v H1 0() (2.2) a(u,v) = Z d X i,j=1 aij u
7、xj v xi dx,u,v H1() L2= Z fvdx K K (2.1) k?) uh uh H1 0() : a(uh,vh) = hf,vhiL2vh H1 0() (2.3) a(,) L H1 0() V5/f L 2() - X0 X1 Xj Xl H1 0() ink?m (Tj)l j=1 5k?m Xj Xj= u C() : u|T P1(T) T Tj,u|= 0 P1(T) Ln/ T 5Kd (2.3) 31 j ?k? ) ujde5|( uj Xj:a(uj,vj) = hf,vjiL2vj Xj 3 2.1.CMG .OL (i) u? 0 = u0 (i
8、i) u? j = Ij,mju? j1 j = 1,.,l (2.1.1) Ij,mjL31 j ?1 mjg1wS 3.?k?)e?A Sue?vk:L5?S=1 j ? S1 j 1 ?)LJ,fJ,f5 ?fSg mjdeO: mj= mLLl,j = 1,2, ,L 1 m0L2,j = L (2.1.2) 0 ?x L?u x ? e.5?1kA 1 4XJ.ve K. kul u? lka ku ulka = O(nl),nl= dimXl (2.1.3) 2 4UP1wf mj1 j ?d 2.1.2 Sg b35f Tj,mj: Xj Xj uj Ij,mju0 j = Tj
9、,mj(uj u0 j) ?IOS vj Xjve1w5IOSUP1w f 1wf (i) kTj,mjvjka c h1 j m j kvjkL2 (ii) kTj,mjvjka kvjka (2.1.4) :0 21/ c l h l m l kfkH1, = 21/ (2.1.5) n 2 4mj . l X j=1 mjnj c 1 1/2d mlnl, 2d c l mlnl, = 2d (2.1.6) c l ? :0 1 dn 1 n 2 k? v 21/ L0ml= mLLl mL= m0(L L0)2 (2) ? l L0 k I? (2 0)L0 Lml= m 1 2 ?(L (2 0)l)h2 l I
