On finite element approximation of the Schrödinger–Poisson model

In this paper, we study 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 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 [Formula: see text]. By constructing an approximate mapping [Formula: see text], we prove that [Formula: see text] has a fixed point [Formula: see text] which is the solution to the nonlinear approximate problem. The error estimate between [Formula: see text] and [Formula: see text] is established. We apply the abstract 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.