Semi-coarsening AMLI preconditioning of anisotropic trilinear FEM systems

Optimal order algebraic multilevel iteration (AMLI) preconditioners based on recursive application of two-level finite element (FE) methods and polynomial stabilization have been introduced and analyzed in Axelsson and Vassilevski (1989, 1990). The construction follows the natural hierarchical splitting using the fact that the finite element spaces corresponding to two successive mesh refinements are nested. Uniform estimates for the constant γγ in the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality are very important for the derivation of optimal order methods. The value of the upper bound for γ∈(0,1)γ∈(0,1) is a part of the construction of the multilevel extension of the related two-level method.In this paper algebraic two-level and multilevel preconditioning algorithms for second order anisotropic elliptic boundary value problems are constructed. Here we allow big jumps in the coefficients and varying the direction of dominating anisotropy from one element to another in the coarse triangulation. The discretization is done by trilinear conforming finite elements where the semi-coarsening mesh refinement strategy is applied. A new uniform estimates for the related CBS constants are derived. The additive preconditioning strategy for the system with the pivot block in the hierarchical two-level splitting is proposed, where the related sub-problems have two dimensional structure.