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Rapid error reduction for block Gauss-Seidel based on p-hierarchical bases
Sabine Le Borne and Jeffrey Ovall
We consider a two-level block Gauss-Seidel iteration for solving systems arising from finite element 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 $H^1$-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 support these claims.