Nonconforming quadrilateral finite element analysis for the nonlinear Ginzburg-Landau equation

This paper is devoted to the study of nonlinear Ginzburg-Landau equation (GLE) with the nonconforming modified quasi-Wilson quadrilateral finite element. Based on the special property of this element, that is its consistency error can reach order O ( h 3 ) in the broken H 1-norm when the exact solution belongs to H 4 (Omega), and by use of the interpolated postprocessing technique, the superclose and superconvergence estimates are obtained for the semi- discrete scheme with order O ( h 2 ), the Backward-Euler (B-E) fully-discrete scheme with order O ( h 2 + triangle t ), and the Crank-Nicolson (C-N) fully-discrete scheme with order O ( h 2 + (triangle t)2), triangle t ) 2 ), respectively. In addition, a numerical example is provided to verify the theoretical analysis. Here h is the mesh size and triangle t is the time step.