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Superconvergence in the generalized finite element method

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Abstract

In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in (Babuška et al. in SIAM J Numer Anal 31, 945–981, 1994; Babuška et al. in Int J Numer Meth Eng 40, 727–758, 1997; Melenk and Babuška in Comput Methods Appl Mech Eng 139, 289–314, 1996). In particular, we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non-linear equations; this system does not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non-linear equations. We note that smooth generalized finite element approximation is easy to construct.

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Authors and Affiliations

  1. Institute for Computational Engineering and Sciences, ACE 6.412, University of Texas at Austin, Austin, TX, 78712, USA

    Ivo Babuška

  2. Department of Mathematics, Syracuse University, 215 Carnegie, Syracuse, NY, 13244, USA

    Uday Banerjee

  3. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    John E. Osborn

Authors
  1. Ivo Babuška
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  2. Uday Banerjee
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  3. John E. Osborn
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Corresponding author

Correspondence to Uday Banerjee.

Additional information

I. Babuška’s research was partially supported by NSF Grant # DMS-0341982 and ONR Grant # N00014-99-1-0724.

U. Banerjee’s research was partially supported by NSF Grant # DMS-0341899.

J. E. Osborn’s research was supported by NSF Grant # DMS-0341982.

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Babuška, I., Banerjee, U. & Osborn, J.E. Superconvergence in the generalized finite element method. Numer. Math. 107, 353–395 (2007). https://doi.org/10.1007/s00211-007-0096-8

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  • DOI: https://doi.org/10.1007/s00211-007-0096-8

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