In this paper, we study a priori error estimates for the finite element approximation of the nonlinear Schr\"{o}dinger-Poisson model. The electron density is defined by an infinite series over all eigenvalues of the Hamiltonian operator. To establish the error estimate, we present a unified theory of error estimates for a class of nonlinear problems. The theory is based on three conditions: 1) the original problem has a solution $u$ which is the fixed point of a compact operator $\Ca$, 2) $\Ca$ is Fr\'{e}chet-differentiable at $u$ and $\Ci-\Ca'[u]$ has a bounded inverse in a neighborhood of $u$, and 3) there exists an operator $\Ca_h$ which converges to $\Ca$ in the neighborhood of $u$. The theory states that $\Ca_h$ has a fixed point $u_h$ which solves the approximate problem. It also gives the error estimate between $u$ and $u_h$, without assumptions on the well-posedness of the approximate problem. We apply the unified theory to the finite element approximation of the Schr\"{o}dinger-Poisson model and obtain optimal error estimate between the numerical solution and the exact solution. Numerical experiments are presented to verify the convergence rates of numerical solutions.

翻译：本文研究了非线性薛定谔-泊松模型有限元逼近的先验误差估计。电子密度由哈密顿算符所有特征值的无穷级数定义。为建立误差估计，我们提出了一类非线性问题的统一误差估计理论。该理论基于三个条件：1）原问题存在解$u$，其为紧算子$\Ca$的不动点；2）$\Ca$在$u$处Fr\'{e}chet可微，且$\Ci-\Ca'[u]$在$u$的邻域内具有有界逆；3）存在算子$\Ca_h$，其在$u$的邻域内收敛于$\Ca$。该理论表明$\Ca_h$存在不动点$u_h$，该不动点求解近似问题，并给出了$u$与$u_h$之间的误差估计，无需假设近似问题的适定性。我们将该统一理论应用于薛定谔-泊松模型的有限元逼近，得到了数值解与精确解之间的最优误差估计。通过数值实验验证了数值解的收敛速率。