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A Hermitian Box-scheme for One-dimensional Elliptic Equations – Application to Problems with High Contrasts in the Ellipticity

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Abstract

We introduce a new box-scheme, called ``hermitian box-scheme'' on the model of the one-dimensional Poisson problem. The scheme combines features of the box-scheme of Keller, [20], [13], with the hermitian approximation of the gradient on a compact stencil, which is characteristic of compact schemes, [9], [21]. The resulting scheme is proved to be 4th order accurate for the primitive unknown u and its gradient p. The proved convergence rate is 1.5 for (u,p) in the discrete L 2 norm. The connection with a non standard mixed finite element method is given. Finally, numerical results are displayed on pertinent 1-D elliptic problems with high contrasts in the ellipticity, showing in practice convergence rates ranging from 1 to 2.5 in the discrete H 1 norm.

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Authors and Affiliations

  1. Laboratoire Mathématiques et Applications de Metz, UMR 7122, Université de Metz, 57045, Metz Cedex, Ile du Saulcy, Metz, France

    J.-P. Croisille

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  1. J.-P. Croisille
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Correspondence to J.-P. Croisille.

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This work has been performed with the support of the GDR MOMAS, (ANDRA, CEA, EDF, BRGM and CNRS): Modélisation pour le stockage des déchets radioactifs. The author thanks especially A. Bourgeat for his encouragements and his interest in this work.

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Croisille, JP. A Hermitian Box-scheme for One-dimensional Elliptic Equations – Application to Problems with High Contrasts in the Ellipticity. Computing 78, 329–353 (2006). https://doi.org/10.1007/s00607-006-0181-3

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  • DOI: https://doi.org/10.1007/s00607-006-0181-3

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