Volume 51, Number 4, July-August 2017
|Page(s)||1245 - 1278|
|Published online||26 June 2017|
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.
Received: 18 December 2015
Revised: 25 May 2016
Accepted: 8 September 2016
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.
Mathematics Subject Classification: 65J15 / 65L05 / 65M60 / 65M12 / 65M15
Key words: Nonlinear Schrödinger equations / operator splitting methods / finite element discretization / stability / local error / convergence
© EDP Sciences, SMAI 2017
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