A robust hierarchical basis preconditioner on general meshes

In this paper, we introduce a multi-level direct sum space decomposition of general, possibly locally refined linear or multi-linear finite element spaces. The resulting additive Schwarz preconditioner is optimal for symmetric second order elliptic problems. Moreover, it turns out to be robust with respect to coefficient jumps over edges in the coarsest mesh, perturbations with positive zeroth order terms, and, after a further decomposition of the spaces, also with respect to anisotropy along the grid lines, Important for an efficient implementation is that stable bases of the subspaces defining our decomposition, consisting of functions having small supports can be easily constructed.