Analysis of extrapolation cascadic multigrid method (EXCMG). (English) Zbl 1157.65062
Summary: Based on an asymptotic expansion of finite elements, a new extrapolation formula and an extrapolation cascadic multigrid method (EXCMG) are proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on the refined grid. In the case of triple grids, the error of the new initial value is analyzed in detail. A larger scale computation is completed in PC.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 65N15 | Error bounds for boundary value problems involving PDEs |
Keywords:
cascadic multigrid; finite element; new extrapolation; error analysis; Poisson equation; numerical examplesReferences:
| [1] | Brenner S, Scott L. The Mathematical Theory of Finite Element Methods. New York: Springer-Verlag. 1994 · Zbl 0804.65101 |
| [2] | Bornemann F, Deufhard P. The cascadic multigrid method for elliptic problems. Numer Math, 75: 125–152 (1996) · Zbl 0873.65107 · doi:10.1007/s002110050234 |
| [3] | Shi Z C, Xu X J. Cascadic multigrid method for the second order elliptic problem. East-West J Numer Math, 6: 309–318 (1998); 7: 199–209 (1999) · Zbl 0919.65072 |
| [4] | Shi Z C, Xu X J. A new cascadic multigrid. Sci China Ser A-Math, 44(1): 21–30 (2001) · Zbl 1004.65135 · doi:10.1007/BF02872279 |
| [5] | Shi Z C, Xu X J, Huang Y Q. Economical cascadic multigrid methods (ECMG). Sci China Ser A-Math, 50: 1765–1780 (2007) · Zbl 1153.65046 · doi:10.1007/s11425-007-0127-z |
| [6] | Li C L, Chen C M, Xu X J. A cascadic mutligrid method based on superconvergence and extrapolation. Numer Math Sinica, 29(4): 439–448 (2007) · Zbl 1143.65403 |
| [7] | Wang L H, Xu X J. Mathematical Foundations of Finite Element Methods. Beijing: Science Press, 2005 |
| [8] | Chen C M, Xie Z Q, Li C L, et al. Study of a new extrapolation multigrid method. Natur Sci J Hunan Normal Univ, 30(2): 1–5 (2007) · Zbl 1143.65402 |
| [9] | Chen C M, Hu H L, Xie Z Q, Li C L. Extrapolation cascadic multigrid method (EXCMG)-a new algorithm in large scale solution of elliptic problems. Numer Math Sinica (submitted) · Zbl 1212.65487 |
| [10] | Marchuk G, Shaidurov V. Finite Difference and Their Extrapolations. New York: Springer-Verlag, 1983 · Zbl 0511.65076 |
| [11] | Chen C M. Superconvergence results in finite-element analysis. Surveys on Math for Industry, 11: 131–157 (2005) · Zbl 1083.65100 |
| [12] | Chen CM, Huang Y Q. High Accuracy Theory of Finite Elements. Changsha: Hunan Science and Technique Press, 1995 |
| [13] | Chen C M. Introduction to Scientific Computations. Beijing: Science Press, 2007 |
| [14] | Chen C M. Extrapolation of Galerkin solution. J Xiangtan Univ, 4: 1–6 (1980) · doi:10.1631/jzus.2003.0001 |
| [15] | Chen C M, Huang Y Q. Extrapolation of triangular linear elements in general domain. Numer Math J Chinese Univ, 11: 1–16 (1989) |
| [16] | Chen C M, Lin Q. Extrapolation of finite element approximation in a rectangular domain. J Comput Math, 7: 227–233 (1989) · Zbl 0691.65075 |
| [17] | Lin Q, Lu T, Shen S M. Maximum norm estimate, extrapolation and optimal point of stresses for finite element methods on strongly regular triangulation. J Comput Math, 1: 376–383 (1983) · Zbl 0563.65070 |
| [18] | Lin Q, Yan N N. Structure and Analysis of Efficient Finite Elements. Baoding: Hebei University Press, 1996 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
