Rapid error reduction for block Gauss-Seidel based on p-hierarchical basis

Abstract

We consider a two-level block Gauss-Seidel iteration for solving systems arising from finite element discretizations employing higher-order elements. A p-hierarchical basis is used to induce this block structure. Using superconvergence results normally employed in the analysis of gradient recovery schemes, we argue that a massive reduction of the H1-error occurs in the first iterate, so that the discrete solution is adequately resolved in very few iterates-sometimes a single iteration is sufficient. Numerical experiments on uniform and adapted meshes support these claims.