Unconditionally convergence and superconvergence error analysis of a mass- and energy-conserved finite element method for the Schrödinger–Poisson equation

This paper aims to investigate the unconditionally optimal and superconvergent error estimates of a mass- and energy-conserved finite element method for the Schrödinger–Poisson equation. Firstly, a priori error bound of the numerical solutions in \(H^1\) $H^1$ -norm is obtained by the conserved property. Secondly, the unconditionally optimal error estimates in \(L^2\) $L^2$ -norm are derived without any timestep restriction in terms of the bound of the numerical solution. Thirdly, the unconditionally superclose error estimates in \(H^1\) $H^1$ -norm are got by treating the coupled nonlinear term rigorously and skillfully. Furthermore, the unconditionally superconvergent error estimates in \(H^1\) $H^1$ -norm are acquired by the interpolation post-processing approach. Finally, some numerical results are provided to verify the theoretical analysis.