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Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods

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Abstract

A model second-order elliptic equation on a general convex polyhedral domain in three dimensions is considered. The aim of this paper is twofold: First sharp Hölder estimates for the corresponding Green’s function are obtained. As an applications of these estimates to finite element methods, we show the best approximation property of the error in \({W^1_{\infty}}\) . In contrast to previously known results, \({W_p^{2}}\) regularity for p > 3, which does not hold for general convex polyhedral domains, is not required. Furthermore, the new Green’s function estimates allow us to obtain localized error estimates at a point.

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, 02906, USA

    J. Guzmán

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    D. Leykekhman

  3. Institut für Mathematik, Universität Rostock, 18051, Rostock, Germany

    J. Rossmann

  4. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    A. H. Schatz

Authors
  1. J. Guzmán
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  2. D. Leykekhman
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  3. J. Rossmann
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  4. A. H. Schatz
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Corresponding author

Correspondence to D. Leykekhman.

Additional information

J. Guzmán was supported by NSF grand DMS-0503050, D. Leykekhman was supported in part by NSF grands DMS-0240058 and DMS-0811167, and A. H. Schatz by NSF grand DMS-0612599.

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Guzmán, J., Leykekhman, D., Rossmann, J. et al. Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods. Numer. Math. 112, 221–243 (2009). https://doi.org/10.1007/s00211-009-0213-y

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  • DOI: https://doi.org/10.1007/s00211-009-0213-y

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