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On Two Ways of Stabilizing the Hierarchical Basis Multilevel Methods

1997; Society for Industrial and Applied Mathematics; Volume: 39; Issue: 1 Linguagem: Inglês

10.1137/s0036144595282211

1095-7200

Panayot S. Vassilevski,

Matrix Theory and Algorithms

A survey of two approaches for stabilizing the hierarchical basis (HB) multilevel preconditioners, both additive and multiplicative, is presented. The first approach is based on the algebraic extension of the two-level methods, exploiting recursive calls to coarser discretization levels. These recursive calls can be viewed as inner iterations (at a given discretization level), exploiting the already defined preconditioner at coarser levels in a polynomially-based inner iteration method. The latter gives rise to hybrid-type multilevel cycles. This is the so-called (hybrid) algebraic multilevel iteration (AMLI) method. The second approach is based on a different direct multilevel splitting of the finite element discretization space. This gives rise to the so-called wavelet multilevel decomposition based on $L^2$-projections, which in practice must be approximated. Both approaches---the AMLI one and the one based on approximate wavelet decompositions---lead to optimal relative condition numbers of the multilevel preconditioners.