A structure-preserving relaxation Crank–Nicolson finite element method for the Schrödinger–Poisson equation

In this paper, we propose a mass- and modified energy-conservative relaxation Crank–Nicolson finite element method for the Schrödinger–Poisson (SP) equation. Utilising only a single auxiliary variable, we simultaneously reformulate the distinct nonlinear terms present in both the Schrödinger equation and the Poisson equation into their equivalent expressions, constructing a system equivalent to the original SP equation. Our proposed scheme, derived from this equivalent system, is implemented linearly, avoiding the need for iterative techniques to solve the nonlinear equation. Additionally, it is executed sequentially, eliminating the need to solve a coupled large linear system. We in turn rigorously derive the optimal error estimates for the proposed scheme, demonstrating second-order accuracy in time and $(k+1)$th-order accuracy in space when employing polynomials of degree up to k. Numerical experiments validate the accuracy and effectiveness of our method and emphasise its conservation properties over long-time simulations.