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

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. Copyright © 2012 John Wiley & Sons, Ltd.