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

Abstract

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.1991 Mathematics Subject Classification. 65J15, 65L05, 65M60 65M12, 65M15..

A novel relaxation scheme for the numerical approximation of Schrödinger-Poisson type systems

“…Usually the Crank-Nicolson, Gaussian beam or time splitting method is used for the temporal integration whilst the finite element, finite difference or spectral method is commonly utilized for the spatial discretization. In [3], the authors conduct an error and stability analysis for an operator splitting finite element discretization of (1.1) whilst an error analysis for the semidiscrete Galerkin finite element scheme is presented in [6]. Utilizing a Crank-Nicolson temporal and finite difference spatial discreitzation of (1.1), a predictor-corrector scheme is studied in [20] and the spherically symmetric case is studied in [12].…”

Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations

“…Despite the fact that splitting schemes are widely used for an efficient time integration of deterministic Schrödinger-type equations, see for instance [10,7,39,26,28,38,4], we are not aware of a numerical analysis of such integrators approximating mild solutions of nonlinear stochastic Schrödinger equations driven by an additive Itô noise. In the present publication we prove ‚ bounds for the exponential moments of the mass of the exact and numerical solutions (Theorem 10); ‚ a kind of longtime stability, a so called trace formula for the mass, of the numerical solutions (Proposition 5); ‚ preservation of symplecticity for the exact and numerical solutions (Proposition 8); ‚ strong convergence estimates (with order) of the splitting scheme, as well as orders of convergence in probability and almost surely (Theorem 14 and Corollary 16).…”