| Abstract: |
| In this paper, we study 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 an abstract theory of error estimates for a class of nonlinear problems. The nonlinear problem is first formulated as a fixed-point equation of a compact mapping $\mathcal{A}$. By constructing an approximate mapping $\mathcal{A}_h$, we prove that $\mathcal{A}_h$ has a fixed point $u_h$ which is the solution to the nonlinear approximate problem. The error estimate between $u$ and $u_h$ is established. We apply the abstract 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. |
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