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In this paper we extend our previous work on the use of domain decomposition (DD) preconditioning for the parallel finite element (FE) solution of three-dimensional elliptic problems [3, 6] and convection-dominated problems [7, 8] to include the use of local mesh refinement. The preconditioner that we use is based upon a hierarchical finite element mesh that is partitioned at the coarsest level. The individual subdomain problems are then solved on meshes that have a single layer of overlap at each level of refinement in the mesh. Results are presented to demonstrate that this preconditioner leads to iteration counts that appear to be independent of the level of refinement in the final mesh, even in the case where this refinement is local in nature: as produced by an adaptive finite element solver for example.

A Parallel Domain Decomposition Algorithm for the Adaptive Finite Element Solution of 3-D Convection-Diffusion Problems