Superconvergence of a new energy dissipation finite element scheme for nonlinear Schrödinger equation with wave operator

Abstract

This paper is devoted to develop a new energy dissipation scheme by finite element method (FEM) for a nonlinear Schrödinger equation with wave operator (NLSW) and investigate its superconvergence behavior through introducing a new second order discrete approximation with respect to time. The new scheme not only guarantees the mass and energy stable but also is efficient in achieving the numerical solution at the last point. Also, an application to other general nonlinear equations with wave operator is discussed. On the one hand, the boundedness of the solution about the numerical equation is deduced and as a result, the existence and uniqueness of the solution are obtained. On the other hand, the superconvergence error estimate is derived and some numerical results are given to confirm the theoretical predictions.