Approximating the nonlinear Schrödinger equation by a two level linearly implicit finite element method
This provides a theoretical convergence analysis for a numerical scheme for nonlinear Schrödinger equations, which is an incremental contribution to numerical analysis.
The authors propose a two-level linearly implicit finite element method for the nonlinear Schrödinger equation and prove convergence in L2 and H1 norms.
We consider the study of a numerical scheme for an initial- and Dirichlet boundary- value problem for a nonlinear Schrödinger equation. We approximate the solution using a, local (non-uniform) two level scheme in time (see C. Besse [6] and [7]) combined with, an optimal, finite element strategy for the discretization in the spatial variable based on studies outlined as, e.g. in [2] and [10]. For the proposed fully discrete scheme, we show convergence both in $L_2 $ and $H^1$ norms.