An Efficient and Unconditionally Convergent Galerkin Finite Element Method for the Nonlinear Schrödinger Equation in High Dimensions

Abstract

In this paper, we aim to propose and analyze a linearized three-level Galerkin finite element method (FEM) for the nonlinear Schrödinger equation with a general nonlinearity and an external potential. Compared with the existing results in literature, under a weaker assumption on both the exact solution and the nonlinear term,  we give a concise proof to establish the optimal $L^{2}$ error estimate without any grid-ratio restriction.  Besides the standard energy method, the key tools used in our analysis are an induction argument and several Sobolev inequalities. Numerical results are reported to verify our theoretical analysis.