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

Sabine Le Borne and Jeffrey Ovall

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Submission date: 06. Jun. 2011 (revised version: June 2011)
Pages: 17
published in: Numerical linear algebra with applications, 20 (2013) 5, p. 743-760 
DOI number (of the published article): 10.1002/nla.1841
Bibtex
MSC-Numbers: 65N22, 65N55, 65N30, 65F08, 65F05
Keywords and phrases: higher-order finite elements, hierarchical bases, hierarchical matrices, block Gauss-Seidel
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Abstract:

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 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 support these claims.