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

Abstract

This paper concerns discontinuous nite element approximations of a fourth order bi-wave equation arising as a simplied Ginzburg-Landau- type model for d-wave superconductors in the absence of an applied magnetic eld. In the rst half of the paper, we construct a variant of the Morley nite 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 (o by a factor j lnhj) 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 eciency of the proposed methods and to validate the theoretical error bounds.