Superconvergence analysis of BDF-Galerkin FEM for nonlinear Schrödinger equation

Abstract

A nonlinear iteration scheme for nonlinear Schrödinger equation with 2-step backward differential formula (BDF) finite element method (FEM) is proposed. Energy stability is testified for the constructed scheme, which leads to the boundedness of $\|{U_{h}^{n}}\|_{0}$ and $\sqrt {\tau }\|\nabla {U_{h}^{n}}\|_{0}$. Auxiliary equation known as a time-discrete system is constructed to get rid of the restriction of τ. The solutions of the time-discrete equation in H¹-norm is deduced. On the one hand, $\sqrt {\tau }\|\nabla U^{n}\|_{0}$ reduces to the temporal error in H²-norm. On the other hand, with the help of the boundedness about $\sqrt {\tau }\|\nabla {U_{h}^{n}}\|_{0}$, the unconditional optimal estimate for spatial error is derived. Without any restriction of the time step, $\|{U_{h}^{n}}\|_{0,\infty }$ is bounded through surmounting the difficulty of nonlinear term. By taking difference between two time levels n and n − 1 of the error equation, an unconditional superconvergence estimate is derived. At last, global superconvergence result is achieved through the known interpolated postprocessing technique. Here, τ is the time step, and Uⁿ and ${U_{h}^{n}}$ denote the solutions of the time-discrete system and the finite element approximation equation, respectively. Furthermore, by introducing modified mass and energy functions, the numerical scheme is proved to preserve the total mass and energy in the discrete senses. Finally, numerical results are given to support the theoretical analysis.