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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Discontinuous finite element methods for a bi-wave equation modeling $d$-wave superconductors
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by Xiaobing Feng and Michael Neilan PDF
Math. Comp. 80 (2011), 1303-1333 Request permission

Abstract:

This paper concerns discontinuous finite element approximations of a fourth-order bi-wave equation arising as a simplified Ginzburg-Landau-type model for $d$-wave superconductors in the absence of an applied magnetic field. In the first half of the paper, we construct a variant of the Morley finite element method, which was originally developed for approximating the fourth-order biharmonic equation, for the bi-wave equation. It is proved that, unlike the biharmonic equation, it is necessary to impose a mesh constraint and to include certain penalty terms in the method to guarantee convergence. Nearly optimal order (off by a factor $|\mathrm {ln} h|$) error estimates in the energy norm and in the $H^1$-norm are established for the proposed Morley-type nonconforming method. In the second half of the paper, we develop a symmetric interior penalty discontinuous Galerkin method for the bi-wave equation using general meshes and prove optimal order error estimates in the energy norm. Finally, numerical experiments are provided to gauge the efficiency of the proposed methods and to validate the theoretical error bounds.
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Additional Information
  • Xiaobing Feng
  • Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996
  • MR Author ID: 351561
  • Email: xfeng@math.utk.edu
  • Michael Neilan
  • Affiliation: Center for Computation and Technology and Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70808
  • MR Author ID: 824091
  • Email: neilan@math.lsu.edu
  • Received by editor(s): April 10, 2009
  • Received by editor(s) in revised form: March 24, 2010
  • Published electronically: December 7, 2010
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 80 (2011), 1303-1333
  • MSC (2010): Primary 65N30, 65N12, 65N15
  • DOI: https://doi.org/10.1090/S0025-5718-2010-02436-6
  • MathSciNet review: 2785460