A conservative relaxation Crank-Nicolson finite element method for the Schrödinger-Poisson equation

In this paper, we propose a novel mass and energy conservative relaxation Crank-Nicolson finite element method for the Schr\"{o}dinger-Poisson equation. Utilizing only a single auxiliary variable, we simultaneously reformulate the distinct nonlinear terms present in both the Schr\"{o}dinger equation and the Poisson equation into their equivalent expressions, constructing an equivalent system to the original Schr\"{o}dinger-Poisson equation. Our proposed scheme, derived from this new system, operates linearly and bypasses the need to solve the nonlinear coupled equation, thus eliminating the requirement for iterative techniques. We in turn rigorously derive 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 emphasize its conservation properties over long-time simulations.