Geodesic
Convergence of a Strang splitting finite element discretization for the Schrödinger–Poisson equation
Auzinger, Winfried1 ; Kassebacher, Thomas2 ; Koch, Othmar3 ; Thalhammer, Mechthild2
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.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1245-1278

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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.

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DOI : 10.1051/m2an/2016059
Classification : 65J15, 65L05, 65M60, 65M12, 65M15
Keywords: Nonlinear Schrödinger equations, operator splitting methods, finite element discretization, stability, local error, convergence

Auzinger, Winfried 1 ; Kassebacher, Thomas 2 ; Koch, Othmar 3 ; Thalhammer, Mechthild 2

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. 
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     author = {Auzinger, Winfried and Kassebacher, Thomas and Koch, Othmar and Thalhammer, Mechthild},
     title = {Convergence of a {Strang} splitting finite element discretization for the {Schr\"odinger{\textendash}Poisson} equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1245--1278},
     publisher = {EDP-Sciences},
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Auzinger, Winfried; Kassebacher, Thomas; Koch, Othmar; Thalhammer, Mechthild. Convergence of a Strang splitting finite element discretization for the Schrödinger–Poisson equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1245-1278. doi: 10.1051/m2an/2016059

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