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Two preconditioners based on the multi-level splitting of finite element spaces. (English) Zbl 0708.65103

The hierarchical preconditioner of the author [Numer. Math. 49, 379–412 (1986; Zbl 0608.65065)] and a preconditioner of J. H. Bramble, J. E. Pasciak and J. Xu (B-P-X) [J. H. Bramble et al., Math. Comput. 55, No. 191, 1–22 (1990; Zbl 0703.65076)] are investigated from a common point of view. Much attention is paid to non-uniformly refined triangulations, and to coefficients of the considered elliptic differential equation which have jumps across the boundaries of triangles of the starting level.
According to the estimates obtained here, the author’s preconditioner is not influenced by such jumps (whereas the B-P-X preconditioner is); the latter preconditioner however – as opposed to the first one – is usable in 3 dimensions, too. For a certain choice of orthogonal operators occurring in the B-P-X preconditioner, a direct connection to the B-P-X preconditioner is established.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods 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
35J25 Boundary value problems for second-order elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling

Software:

PLTMG

References:

[1] Bank, R.E.: PLTMG: A software package for solving elliptic partial differential equations. Philadelphia: SIAM 1990 · Zbl 0717.68001
[2] Bank, R.E., Dupont, T., Yserentant, H.: The hierarchical basis multigrid method. Numer. Math.52, 427-458 (1988) · Zbl 0645.65074 · doi:10.1007/BF01462238
[3] Bank, R.E., Sherman, A.H., Weiser, A.: Refinement algorithms and data structures for regular local mesh refinement. In: Stepleman, R. (eds.) Scientific computing, pp. 3-17. Amsterdam: IMACS/North Holland 1983
[4] Bramble, J.H., Pasciak, J.E., Xu, J.: Parallel multilevel preconditioners. Math. Comput. (to appear) · Zbl 0725.65095
[5] Crouzeix, M., Thom?e, V.: The stability inL p andW 1 p of theL 2-projection onto finiteelement function spaces. Math. Comput.48, 521-532 (1987) · Zbl 0637.41034
[6] Deuflhard, P., Leinen, P., Yserentant, H.: Concepts of an adaptive hierarchical finite element code. IMPACT of Computing in Science and Engineering1, 3-35 (1989) · Zbl 0706.65111 · doi:10.1016/0899-8248(89)90018-9
[7] Hackbusch, W.: Multigrid methods and applications. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0595.65106
[8] Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Stuttgart: Teubner 1986 (English translation in preparation)
[9] Ong, M.E.G.: Hierarchical basis preconditioners for second order elliptic problems in three dimensions. Technical Report No. 89-3, Department of Applied Mathematics, University of Washington, Seattle 1989
[10] Oswald, P.: On estimates for hierarchic basis representations of finite element functions. Technical Report N/89/16, Sektion Mathematik, Friedrich-Schiller Universit?t Jena 1989
[11] Xu, J.: Theory of multilevel methods. Report No. AM48, Department of Mathematics. Pennsylvania State University 1989
[12] Yserentant, H.: On the multi-level splitting of finite element spaces. Numer. Math.49, 379-412 (1986) · Zbl 0608.65065 · doi:10.1007/BF01389538
[13] Yserentant, H.: Hierarchical bases give conjugate gradient type methods a multigrid speed of convergence. Applied Mathematics and Computation19, 347-358 (1986) · Zbl 0614.65114 · doi:10.1016/0096-3003(86)90113-X
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