On Two Ways of Stabilizing the Hierarchical Basis Multilevel Methods

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 L2-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.