Time-Space Adaptive Finite Element Method for Nonlinear Schrödinger Equation

Abstract

This paper is devoted to adaptive finite element method for the nonlinear Schrödinger equation. The adaptive method is based on the extrapolation technology and a second order accurate, linear and mass preserving finite element scheme. For error control, we take the difference between the numerical gradient and the recovered gradient obtained by the superconvergent cluster recovery method as the spatial discretization error estimator and the difference of numerical approximations between two consecutive time steps as the temporal discretization error estimator. A time-space adaptive algorithm is developed for numerical approximation of the nonlinear Schrödinger equation. Numerical experiments are presented to illustrate the reliability and efficiency of the proposed error estimators and the corresponding adaptive algorithm.