A posteriori error control for an energy preserving method for the Schrödinger-Poisson system
In this talk we first introduce a structure preserving, second order in time, relaxation-type scheme for approximating solutions of the Schrödinger-Poisson system and highlight its advantages. We then discuss the a posteriori error analysis for this scheme. In particular, we introduce an appropriate reconstruction and present the main steps of the derivation of optimal order a posteriori error bounds. The main challenges in the analysis arise from the nonlinear nature of the problem and the fact that the potential function, which is also unknown, is linked with the wave function. This means that we should obtain optimal a posteriori error estimates for both the potential and the wave function at the same time. Various numerical experiments verify and complement our findings.