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Analysis of additive multilevel methods. (English) Zbl 0963.65124
The paper is devoted to the analysis of the multilevel additive preconditioner introduced by J. H. Bramble, J. E. Pasciak and J. Xu [Math. Comput. 55, No. 191, 1-22 (1990; Zbl 0703.65076)] which is often called the BPX preconditioner. The theory developed is based on a regularity-approximation assumption only. The author shows that both the lower and the upper bounds of the eigenvalues of the iteration operator follow from this assumption. On the contrary in other theories the smoothing interaction hypothesis (also known as the strengthened Cauchy-Schwarz inequality) is used in addition for the upper estimate.
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65Y05 Parallel numerical computation
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling