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

Auzinger, Winfried; Kassebacher, Thomas; Koch, Othmar; Thalhammer, Mechthild

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1245-1278.

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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