Mathematics of Computation

Author(s): Yunqing Huang; Zhongci Shi; Tao Tang; Weimin Xue.
Journal: Math. Comp. 73 (2004), 525-539.
MSC (2000): Primary 65F10, 65N30, 65N55
Posted: July 14, 2003
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Abstract: In this paper, a multilevel successive iteration method for solving nonlinear elliptic problems is proposed by combining a multilevel linearization technique and the cascadic multigrid approach. The error analysis and the complexity analysis for the proposed method are carried out based on the two-grid theory and its multilevel extension. A superconvergence result for the multilevel linearization algorithm is established, which, besides being interesting for its own sake, enables us to obtain the error estimates for the multilevel successive iteration method. The optimal complexity is established for nonlinear elliptic problems in 2-D provided that the number of grid levels is fixed.

1.: R. BANK, Analysis of a multilevel iterative method for nonlinear finite element equations. Math. Comp., 39 (1982), 453-465. MR 83j:65105
2.: F. BORNEMANN AND P. DEUFLHARD, The cascadic multigrid method for elliptic problems, Numer. Math., 75 (1996), 135-152. MR 98a:65175
3.: F. BORNEMANN AND P. DEUFLHARD The cascadic multigrid method. Proceedings of the 8th international conference on domain decomposition methods, R.Glowinski, J.Periaux, Z. Shi and O.Widlund eds., Wiley and Sons, 1997, pp. 205-212.
4.: D. BRAESS AND W. DEHMEN, A cascade algorithm for the Stokes equations. Numer. Math., 82 (1999), 179-191. MR 2000c:65114
5.: J.H. BRAMBLE, J.E. PASCIAK AND J. XU, Parallel multilevel preconditioner. Math. Comp., 55 (1990), 1-22. MR 90k:65170
6.: A. BRANDT, Multilevel adaptive solutions to boundary-value problems. Math. Comp., 31 (1977), 333-390. MR 55:4714
7.: C. CHEN AND Y. HUANG, $W^{1,p}$ -Stability of finite element method for elliptic problems. Hunan Annals of Mathematics, 6 (1986), 81-89. (Chinese) MR 92e:65144
8.: C. CHEN AND Y. HUANG, High accuracy theory of finite element methods, Hunan Science and Technology Press, 1995. (Chinese)
9.: P. DEUFLHARD, Cascadic conjugate gradient methods for elliptic partial differential equations. Proceedings of the 7th international conference on domain decomposition methods, eds D. Keyes and J. Xu, AMS, Providence, 1994, pp. 29-42. MR 95i:65006
10.: P. DEUFLHARD, P. LEINEN AND H. YSERENTANT, Concepts of an adaptive hierarchical finite element code. IMPACT 1 (1989), 3-35.
11.: J. FREHSE AND R. RANNACHER, Asymptotic $L_{\infty}$ -error estimate for linear finite element approximation of quasilinear boundary value problems. SIAM J. Numer. Anal., 15 (1978), 418-431. MR 58:19224
12.: W. HACKBUSCH Multi-grid methods and applications. Springer Verlag, Berlin-Heidelberg- New York, 1992. MR 87e:65082
13.: Y. HUANG AND Y. CHEN, A multilevel iterate correction method for solving nonlinear singular problems, Natural Science J of Xiangtan Univ. 16 (1994), 23-26. (Chinese)
14.: H. HUANG AND G. LIU, Successive refinement method for solving elliptic boundary value problems. Mathematica Numerica Sinica, 2 (1978), 41-52, and 3 (1978), 28-35. (Chinese)
15.: R. RANNACHER, On the finite element approximation of general boundary value problems in nonlinear elasticity. Calcolo, 17 (1980), 175-193. MR 82f:73019
16.: R. RANNACHER, On the convergence of the Newton-Raphson method for strongly nonlinear finite element equations. In Nonlinear Computational Mechanics, eds P. Wriggers and W. Wagner, Springer Verlag, 1991.
17.: R. RANNACHER AND R. SCOT, Some optimal error estimates for piecewise linear finite element approximations. Math. Comp., 38 (1982), 437-445. MR 83e:65180
18.: V. SHAIDUROV, Some estimates of the rate of convergence for the cascadic conjugate gradient method. Comp. Math. Appl., 31 (1996), 161-171.
19.: Z. SHI AND X. XU, Cascadic multigrid method for elliptic problems, East-West J. Numer. Math., 7 (1999), 199-222. MR 2000h:66185
20.: Z. SHI AND X. XU, Cascadic multigrid method for a plate bending problem. East-West J. Numer. Math., 6 (1998), 137-153. MR 99g:73109
21.: Z. SHI AND X. XU, Cascadic multigrid method for parabolic problem. J. Comput. Math., 18 (2000), 551-560. MR 2001e:65151
22.: G. VAINIKKO, Galerkin's perturbation method and the general theory of approximation for nonlinear equations. USSR Comput. Math. and Math. Phys., 7 (1967), 1-41. MR 36:1095
23.: K. WITSCH, Projective Newton-Verfahren und Adwendungen auf nicht linear Randwertaufgaben. Numer. Math., 31 (1978/1979), 209-230. MR 80a:65160
24.: J. XU, Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal., 33, (1996), 1759-1777. MR 97i:65169
25.: J. XU, A novel two-grid method for semilinear equations. SIAM J. Sci. Comput., 15 (1994), 231-237. MR 94m:65178
26.: J. XU, Iterative methods by space decomposition and subspace correction, SIAM Review, 34 (1992), 581-613. MR 93k:65029
27.: H. YSERENTANT, On the multilevel splitting of finite element spaces. Numer. Math., 49 (1986), 379-412. MR 88d:65068a; MR 88d:65068b

Retrieve articles in Mathematics of Computation with MSC (2000): 65F10, 65N30, 65N55

Yunqing Huang
Affiliation: Department of Mathematics and Institute for Computational and Applied Mathematics, Xiangtan University, Xiangtan, Hunan 411105, Peoples Republic of China
Email: huangyq@xtu.edu.cn

Zhongci Shi
Affiliation: Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing, 100080, Peoples Republic of China
Email: shi@lsec.cc.ac.cn

Tao Tang
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Email: ttang@math.hkbu.edu.hk

Weimin Xue
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Email: wmxue@math.hkbu.edu.hk

DOI: 10.1090/S0025-5718-03-01566-7
PII: S 0025-5718(03)01566-7
Keywords: Multigrid method, cascadic algorithm, finite element method, nonlinear elliptic problem, error estimate, complexity
Received by editor(s): June 6, 2000
Received by editor(s) in revised form: October 12, 2002
Posted: July 14, 2003
Copyright of article: Copyright 2003, American Mathematical Society

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