A New Error Analysis of Crank-Nicolson Galerkin FEMs for a Generalized Nonlinear Schroedinger Equation

摘要

In this paper, we study linearized Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schroedinger equation. We present the optimal L~2 error estimate without any time-step restrictions, while previous works always require certain conditions on time stepsize. A key to our analysis is an error splitting, in terms of the corresponding time-discrete system, with which the error is split into two parts, the temporal error and the spatial error. Since the spatial error is τ-independent, the numerical solution can be bounded in L~∞-norm by an inverse inequality unconditionally. Then, the optimal L~2 error estimate can be obtained by a routine method. To confirm our theoretical analysis, numerical results in both two and three dimensional spaces are presented.