Unconditional superconvergence analysis of a linearized Crank-Nicolson Galerkin FEM for generalized Ginzburg-Landau equation

摘要

In this paper, we study an effective numerical method for the generalized Ginzburg-Landau equation (GLE). Based on the linearized Crank-Nicolson difference method in time and the standard Galerkin finite element method with bilinear element in space, the time-discrete and space-time discrete systems are both constructed. We focus on a rigorous analysis and consideration of unconditional superconvergence error estimates of the discrete schemes. Firstly, by virtue of the temporal error results, the regularity for the time-discrete system is presented. Secondly, the classical Ritz projection is used to obtain the spatial error with order O(h(2)) in the sense of L-2-norm. Thanks to the relationship between the Ritz projection and the interpolated projection, the superclose estimate with order O( tau + h(2)) in the sense of H-1-norm is derived. Thirdly, it follows from the interpolated postprocessing technique that the global superconvergence result is deduced. Finally, some numerical results are provided to confirm the theoretical analysis. (C) 2019 Elsevier Ltd. All rights reserved.