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Discontinuous finite element methods for a bi-wave equation modeling $ d$-wave superconductors


Authors: Xiaobing Feng and Michael Neilan
Journal: Math. Comp. 80 (2011), 1303-1333
MSC (2010): Primary 65N30, 65N12, 65N15
DOI: https://doi.org/10.1090/S0025-5718-2010-02436-6
Published electronically: December 7, 2010
MathSciNet review: 2785460
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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 $ \vert\mathrm{ln} h\vert$) 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
Email: xfeng@math.utk.edu

Michael Neilan
Affiliation: Center for Computation and Technology and Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70808
Email: neilan@math.lsu.edu

DOI: https://doi.org/10.1090/S0025-5718-2010-02436-6
Keywords: Bi-wave equation, $d$-wave superconductor, Morley-type nonconforming element, discontinuous Galerkin method, error estimate
Received by editor(s): April 10, 2009
Received by editor(s) in revised form: March 24, 2010
Published electronically: December 7, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.