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.

翻译：本文提出一种新型的质量与能量守恒的松弛Crank-Nicolson有限元方法，用于求解薛定谔-泊松方程。仅通过引入单个辅助变量，我们同时将薛定谔方程和泊松方程中不同的非线性项重新表述为等价形式，从而构建了与原薛定谔-泊松方程等价的新系统。基于该新系统提出的格式为线性格式，无需求解非线性耦合方程，因此避免了迭代技术的使用。我们严格推导了该格式的误差估计，证明其在时间上具有二阶精度，在空间上采用最高$k$次多项式时具有$(k+1)$阶精度。数值实验验证了该方法的精度与有效性，并强调了其在长时间模拟中的守恒特性。