A Fourth Order Hermitian Box-Scheme with Fast Solver for the Poisson Problem in a Square

Authors:
Ali Abbas;Jean-Pierre Croisille
Affiliations:
Department of Mathematics, LMAM, UMR 7122, University Paul Verlaine-Metz, Metz, France 57012;Department of Mathematics, LMAM, UMR 7122, University Paul Verlaine-Metz, Metz, France 57012
Venue:
Journal of Scientific Computing
Year:
2011

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Abstract

A new fourth order box-scheme for the Poisson problem in a square with Dirichlet boundary conditions is introduced, extending the approach in Croisille (Computing 78:329---353, 2006). The design is based on a "hermitian box" approach, combining the approximation of the gradient by the fourth order hermitian derivative, with a conservative discrete formulation on boxes of length 2h. The goal is twofold: first to show that fourth order accuracy is obtained both for the unknown and the gradient; second, to describe a fast direct algorithm, based on the Sherman-Morrison formula and the Fast Sine Transform. Several numerical results in a square are given, indicating an asymptotic O(N 2log驴2(N)) computing complexity.