Convergence of a Strang splitting finite element discretization for the Schr\"odinger-Poisson equation

TL;DR

This paper analyzes the convergence and stability of a Strang splitting finite element method for solving the nonlinear Schr"odinger-Poisson equation, demonstrating second-order temporal and polynomial spatial convergence.

Contribution

It provides a rigorous stability and error analysis for the second-order Strang splitting combined with finite element discretization for the Schr"odinger-Poisson equation.

Findings

Proves second-order convergence in time for regular solutions.

Establishes polynomial convergence in space.

Numerical results confirm theoretical convergence rates.

Abstract

Operator splitting methods combined with finite element spatial discretizations are studied for time-dependent nonlinear Schr\"odinger equations. In particular, the Schr\"odinger-Poisson equation under homogeneous Dirichlet boundary conditions on a finite domain is considered. A rigorous stability and error analysis is carried out for the second-order Strang splitting method and conforming polynomial finite element discretizations. For sufficiently regular solutions the classical orders of convergence are retained, that is, second-order convergence in time and polynomial convergence in space is proven. The established convergence result is confirmed and complemented by numerical illustrations.