Convergence of a Strang splitting finite element discretization for the Schrödinger-Poisson equation
Provides rigorous error bounds for a widely used numerical method for nonlinear Schrödinger equations, benefiting computational scientists solving such PDEs.
The paper proves second-order temporal and polynomial spatial convergence for a Strang splitting finite element method applied to the Schrödinger-Poisson equation, confirmed by numerical examples.
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.