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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
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References:
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[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
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