Analysis of additive multilevel methods. (English) Zbl 0963.65124

Zhang, X.

East-West J. Numer. Math. 8, No. 1, 71-82 (2000).

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.

Reviewer: Peter Matus (Minsk)

MSC:

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

Keywords:

additive multilevel preconditioner; parallel multilevel preconditioner; multigrid