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Parallel multilevel preconditioners. (English) Zbl 0725.65095
Numerical analysis, Proc. 13th Biennial Conf., Dundee/UK 1989, Pitman Res. Notes Math. Ser. 228, 23-39 (1990).
[For the entire collection see Zbl 0689.00014.]
Special constructions of preconditioners \(B_ h\) for finite element approximations of symmetric and positive elliptic operators in the case when successively refined triangulations \(\{T_ k\}\), \(k=1,...,J\) and hierarchical basic functions \(\phi^{\ell}_ k\) are used are suggested. They are such that \(B_ h^{- 1}v=\sum^{J}_{k=1}\sum_{\ell}(v,\phi^{\ell}_ k)_ 0\phi^{\ell}_ k\) and the terms of the double sum can be computed concurrently. Under some assumptions the estimates \(c_ 1J^{-1}B_ h\leq A_ h\leq c_ 2JB_ h\) with \(c_ 1>0\), leading to a weak spectral equivalence of operators, are received. Numerical examples include a grid with 64\(\times 64\times 64\) cubic cells.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
35J25 Boundary value problems for second-order elliptic equations
65Y05 Parallel numerical computation