Ewing, R. E.; Lazarov, R. D.; Vassilevski, P. S. Local refinement techniques for elliptic problems on cell-centered grids. III: Algebraic multilevel BEPS preconditioners. (English) Zbl 0726.65137 Numer. Math. 59, No. 5, 431-452 (1991). See the preview in Zbl 0712.65102. Cited in 10 Documents MSC: 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs Keywords:finite differences; cell-centered grids; optimal rate of convergence; local refinement; condition numbers; V-cycle multilevel BEPS preconditioners Citations:Zbl 0712.65102 PDFBibTeX XMLCite \textit{R. E. Ewing} et al., Numer. Math. 59, No. 5, 431--452 (1991; Zbl 0726.65137) Full Text: DOI EuDML References: [1] Axelsson, O., Vassilevski, P.S. (1989) Algebraic multilevel preconditioning methods, 1. Numer. Math.56, 157-177 · Zbl 0673.65069 [2] Axelsson, O., Vassilevski, P.S. (1990) Algebraic multilevel preconditioning methods. II. SIAM J. Numer. Anal.27, 1569-1590 · Zbl 0715.65091 [3] Axelsson, O., Vassilevski, P.S. (1989) A survey of multilevel preconditioned iterative methods. BIT29, 769-793 · Zbl 0694.65009 [4] Bank, R., Dupont, T. (1980) Analysis of a two-level scheme for solving finite element equations. Report CNA-159. Center for Numerical Analysis, Univesity of Texas at Austin [5] Bank, R., Dupont, T., Yserentant, H. (1988) The hierarchical basis multigrid method. Numer. Math.52, 427-458 · Zbl 0645.65074 [6] Bank, R.E., Rose, D.J. (1977) Marching algorithm for elliptic boundary value problems, I: The constant coefficient case. SIAM J. Numer. Anal.14, 792-829 · Zbl 0382.65051 [7] Bank, R.E. (1977) Marching algorithm for elliptic boundary value problems. II: The variable coefficient case. SIAM J. Numer. Anal.14, 950-970 · Zbl 0382.65052 [8] Braess, D., Hackbusch, W. (1983) A new convergence proof for the multigrid method including theV-cycle. SIAM J. Numer. Anal.20, 967-975 · Zbl 0521.65079 [9] Bramble, J.H. (1966) A second order finite difference analog of the first biharmonic boundary value problem. Numer. Math.9, 236-245 · Zbl 0154.41105 [10] Bramble, J.H., Ewing, R.E., Pasciak, J.E., Schatz, A.H. (1988) A preconditioning technique for the efficient solution of problems with local grid refinement. Comput. Meth. Appl. Mech. Eng.67, 149-159 · Zbl 0619.76113 [11] Bramble, J.H., Pasciak, J.E., Wang. J., Xu, J. (1991) Convergence estimates for product iterative methods with applications to domain decomposition and multigrid. Math. Comput. (to appear) · Zbl 0754.65085 [12] Bramble, J.H., Pasciak, J.E., Wang, J., Xu, J. (1991) Multigrid results which do not depend upon elliptic regularity assumptions. Math. Comput. (in press) [13] Dryja, M. (1989) An additive Schwarz algorithm for two- and three-dimensional finite element elliptic problems. In: Chan, T.F., Glowinski, R., Periaux, J., Widlund, O.B. (eds.) Domain Decomposition Methods. SIAM, Philadelphia, pp. 168-172 · Zbl 0681.65075 [14] Dryja, M., Widlund, O.B. (1989) On the optimality of an additive iterative refinement method. Technical Report 442, Department of Computer Science, Courant Institute [15] Dryja, M., Widlund, O.B. (1990) Towards a unified theory of domain decomposition algorithms for elliptic problems. In: Chan, T.F., Glowinski, R., Periaux, J., Widlund, O.B., eds., Domain Decomposition Methods. SIAM, Philadelphia, pp. 3-21 · Zbl 0719.65084 [16] Ewing, R.E., Lazarov, R.D., Vassilevski, P.S. (1991) Local refinement techniques for elliptic problems on cell-centered grids, I: Error Analysis. Math. Comput.56, 437-462 · Zbl 0724.65093 [17] Ewing, R.E., Lazarov, R.D., Vassilevski, P.S. Local refinement techniques for elliptic problems on cell-centered grids, II: Two-grid iterative methods. Math. Comput. (to appear) · Zbl 0840.65124 [18] Hackbusch, W. (1985) Multigrid Methods and Applications. Springer, Berlin Heidelberg New York · Zbl 0595.65106 [19] Hageman, L.A., Young, D.M. (1981) Applied Iterative Methods. Academic Press, New York · Zbl 0459.65014 [20] Kuznetsov, Yu.A. (1989) Algebraic multigrid domain decomposition methods. Soviet J. Numer. Methods Math. Modelling4, 351-380 · Zbl 0825.65091 [21] Kuznetsov, Yu.A. (1989/90) Multilevel domain decomposition methods. Appl. Numer. Math.6, 303-314 · Zbl 0694.65049 [22] Mandel, J., McCormick, S. (1989) Iterative solution of elliptic equations with refinement: The two-level case. In: Chan, T.F., Glowinski, R., Periaux, J., Widlund, O.B., eds., Domain Decomposition Methods. SIAM, Philadelphia, pp. 81-92 · Zbl 0679.65076 [23] Mandel, J., McCormick, S. (1989) Iterative solution of elliptic problems with refinement: The model multilevel case. In: Chan, T.F., Glowinski, P., Periaux, J., Widlund, O.B., eds., Domain Decomposition Methods. SIAM, Philadelphia, pp. 93-102 · Zbl 0679.65077 [24] McCormick, S. (1989) Multilevel adaptive methods for partial differential equations. SIAM, Philadelphia · Zbl 0707.65080 [25] McCormick, S., Thomas, J. (1986) The fast adaptive composite grid (FAC) method for elliptic equations. Math. Comput.46, 439-456 · Zbl 0594.65078 [26] Oganesjan, L.A., Ruhovec, L.A. (1979) Variational difference methods for the solution of elliptic problems. Izd. Acad. Nauk Armjanskoi SSR, Jerevan (Russian) [27] Samarskii, A.A. (1987) Introduction to Theory of Difference Schemes. Nauka, Moskow (Russian) [28] Vassilevski, P.S. (1989) Nearly optimal iterative methods for solving finite element elliptic equations based on the multilevel splitting of the matrix. Report # 1989-09, Institute for Scientific Computation, University of Wyoming, Laramie, Wyoming [29] Vassilevski, P.S. (1990) Hybrid V-cycle algebraic multilevel preconditioners. Preprint # 1990-33, Enhanced Oil Recovery Institute, University of Wyoming, Laramie, Wyoming; and Math Comput. (to appear) · Zbl 0716.65104 [30] Wang, J. Convergence analysis of Schwarz algorithm and multilevel decomposition iterative methods. I: Selfadjoint and positive definite elliptic problems. SIAM J. Numer. Anal. (submitted) · Zbl 0785.65115 [31] Widlund, O.B. (1989) Optimal iterative refinement methods. In: Chan, T.F., Glowinski, R., Periaux, J., Widlund, O.B., eds., Domain Decomposition Methods. SIAM, Philadelphia. pp. 114-125 · Zbl 0682.65066 [32] Yserentant, H. (1986) On the multilevel splitting of finite element spaces. Numer. Math.49, 379-412 · Zbl 0608.65065 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. 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