Low order nonconforming finite element method for time-dependent nonlinear Schrödinger equation

Abstract

The main aim of this paper is to apply a low order nonconforming EQ rot 1 finite element to solve the nonlinear Schrödinger equation. Firstly, the superclose property in the broken H 1-norm for a backward Euler fully-discrete scheme is studied, and the global superconvergence results are deduced with the help of the special characters of this element and the interpolation postprocessing technique. Secondly, in order to reduce computing cost, a two-grid method is developed and the corresponding superconvergence error estimates are obtained. Finally, a numerical experiment is carried out to confirm the theoretical analysis.