Approximating the nonlinear Schr\"odinger equation by a two level linearly implicit finite element method

TL;DR

This paper introduces a new two-level linearly implicit finite element method for solving the nonlinear Schrödinger equation, demonstrating convergence in both $L_2$ and $H^1$ norms.

Contribution

It combines a local two-level time scheme with an optimal finite element spatial discretization, providing a convergent numerical method for nonlinear Schrödinger equations.

Findings

Proves convergence in $L_2$ norm.

Proves convergence in $H^1$ norm.

Validates the method through theoretical analysis.

Abstract

We consider the study of a numerical scheme for an initial- and Dirichlet boundary- value problem for a nonlinear Schr\"odinger 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.