Finite element error estimates for the nonlinear Schrödinger-Poisson model

In this paper, we study a priori error estimates for the finite element approximation of the nonlinear Schrödinger-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 $\text{Ca}$, 2) $\text{Ca}$ is Fréchet-differentiable at $u$ and $\text{Ci}-{\text{Ca}}^{\mathrm{\prime }}[u]$ has a bounded inverse in a neighborhood of $u$, and 3) there exists an operator ${\text{Ca}}_h$ which converges to $\text{Ca}$ in the neighborhood of $u$. The theory states that ${\text{Ca}}_h$ 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 Schrödinger-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.