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

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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 2(N)) computing complexity.

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Authors and Affiliations

  1. Department of Mathematics, LMAM, UMR 7122, University Paul Verlaine-Metz, 57012, Metz, France

    Ali Abbas & Jean-Pierre Croisille

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  1. Ali Abbas
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  2. Jean-Pierre Croisille
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Correspondence to Jean-Pierre Croisille.

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Abbas, A., Croisille, JP. A Fourth Order Hermitian Box-Scheme with Fast Solver for the Poisson Problem in a Square. J Sci Comput 49, 239–267 (2011). https://doi.org/10.1007/s10915-010-9458-y

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  • DOI: https://doi.org/10.1007/s10915-010-9458-y

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