Convergence of a Strang splitting finite element discretization for the Schrödinger–Poisson equation^∗

¹ Technische Universität Wien, Institut für Analysis und Scientific Computing, Wiedner Hauptstraße 8-10, 1040 Wien, Austria.
² Leopold-Franzens Universität Innsbruck, Institut für Mathematik, Technikerstraße 13, 6020 Innsbruck, Austria.
³ Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria.
othmar@othmar-koch.org

Received: 18 December 2015
Revised: 25 May 2016
Accepted: 8 September 2016

Abstract

Operator splitting methods combined with finite element spatial discretizations are studied for time-dependent nonlinear Schrödinger equations. In particular, the Schrödinger–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.