FINITE ELEMENT METHODS FOR A BI-WAVE EQUATION MODELING D-WAVE SUPERCONDUCTORS

In this paper we develop two conforming finite element methods for a fourth order bi-waveequation arising as a simplified Ginzburg-Landau-type model for d-wave superconductors inabsence of applied magnetic field.Unlike the biharmonic operator A2,the bi-wave operator□~2 is not an elliptic operator,so the energy space for the bi-wave equation is much largerthan the energy space for the biharmonic equation.This then makes it possible to constructlow order conforming finite elements for the bi-wave equation.However,the existence andconstruction of such finite elements strongly depends on the mesh.In the paper,we firstcharacterize mesh conditions which allow and not allow construction of low order conformingfinite elements for approximating the bi-wave equation.We then construct a cubic and aquartic conforming finite element.It is proved that both elements have the desired approximationproperties,and give optimal order error estimates in the energy norm,suboptimal(and optimal in some cases) order error estimates in the H~1 and L~2 norm.Finally,numericalexperiments are presented to guage the efficiency of the proposed finite element methodsand to validate the theoretical error bounds.