Optimal $L^2$ error estimates of mass- and energy-conserved FE schemes for a nonlinear Schr\"odinger-type system

TL;DR

This paper develops an implicit Crank-Nicolson finite element scheme for nonlinear Schr"odinger-type systems, proving optimal error estimates and conservation properties, with numerical verification of convergence and stability.

Contribution

It introduces a novel FE scheme that conserves mass and energy and provides rigorous optimal $L^2$ error estimates for nonlinear Schr"odinger-type systems.

Findings

The scheme is well-posed and conserves mass and energy.

Optimal $L^2$ error estimates are established.

Numerical examples confirm convergence and conservation properties.

Abstract

In this paper, we present an implicit Crank-Nicolson finite element (FE) scheme for solving a nonlinear Schr\"odinger-type system, which includes Schr\"odinger-Helmholz system and Schr\"odinger-Poisson system. In our numerical scheme, we employ an implicit Crank-Nicolson method for time discretization and a conforming FE method for spatial discretization. The proposed method is proved to be well-posedness and ensures mass and energy conservation at the discrete level. Furthermore, we prove optimal $L^2$ error estimates for the fully discrete solutions. Finally, some numerical examples are provided to verify the convergence rate and conservation properties.