Superconvergence analysis of a two grid finite element method for Ginzburg–Landau equation

摘要

In this paper, a two-grid method (TGM) is presented for the complex Ginzburg–Landau equation, in which the original nonlinear system is analyzed on the coarse grid based on the Newton iteration method for the backward Euler fully-discrete scheme, and then a simple linearized problem developed through Taylor’s expansion on the fine grid is solved. By use of the character of the element and some special techniques, the superclose estimation in the H1-norm is deduced for the TGM scheme. Furthermore, the global superconvergent result is derived through interpolation postprocessing skill. Numerical results illustrate that the proposed TGM is very effective and its computing cost is much less than that of the traditional Galerkin finite element method (FEM) without loss of accuracy.