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

Abstract

In this paper, a linearized Crank-Nicolson Galerkin finite element method (FEM) for generalized Ginzburg-Landau equation (GLE) is considered, in which, the difference method in time and the standard Galerkin FEM are employed. 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(τ 2 + 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.Keywords Nonlinear Ginzburg-Landau equation · Finite element method · Linearized Crank-Nicolson scheme · Ritz projection and interpolated operators · Unconditional superconvergence results IntroductionIn this paper, we are concerned with the numerical solution of the following generalized GLE

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