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Fast multigrid solver. (English) Zbl 0824.65016
To solve a system of linear algebraic equations with a positive definite matrix arising from discretization of elliptic boundary value problems, the author suggests a variant of the multigrid algorithm with coarse spaces constructed by the aggregation of unknowns and smoothing. The performance of the algorithm is further improved by a variant of an overcorrection proposed originally by R. Blaheta [J. Comput. Appl. Math. 24, No. 1/2, 227-239 (1988; Zbl 0663.65106)]. The convergence theory for a two-level algorithm is given. Results of numerical experiments indicate that the method presented is efficient.

MSC:
65F10 Iterative numerical methods for linear systems
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65Y15 Packaged methods for numerical algorithms
35J25 Boundary value problems for second-order elliptic equations
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References:
[1] R. Blaheta: Iterative Methods for Numerical Solving of the Boundary Value Problems of Elasticity. Thesis, Ostrava, 1989.
[2] S. Míka, P. Vaněk: Modification of the Two-level Algorithm with Overcorrection. Appl. Math. 37 (1992), no. 1. · Zbl 0753.65028
[3] S. Míka, P. Vaněk: The Acceleration of Two-level Algorithm by Aggregation in Smoothing Process. Appl. Math. 37 (1992), no. 5. · Zbl 0770.65016
[4] P. Vaněk: Acceleration of a Two-level Algorithm by Smoothing Transfer Operators. Appl. Math. 37 (1992), no. 4.. · Zbl 0773.65021
[5] W. Hackbusch: Multi-Grid Methods and Applications. Springer-Verlag, 1985. · Zbl 0595.65106
[6] J. Mandel: Adaptive Iterative Solvers in Finite Elements.
[7] O. Axelsson, V.A. Barker: Finite Element Solution of Boundary Value Problems. Academic Press, 1984. · Zbl 0537.65072
[8] S.F. McCormick: Multi-Grid Methods. SIAM (1987).
[9] P. Leitl: private communication. Nynice, 1993.
[10] J. Mandel: Balancing Domain Decomposition. Communications in Numerical Methods in Engineering 9 (1993). · Zbl 0796.65126 · doi:10.1002/cnm.1640090307
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