Finite element error estimates for the nonlinear Schrödinger-Poisson model
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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 , 2) is Fréchet-differentiable at u and -'[u] has a bounded inverse in a neighborhood of u, and 3) there exists an operator _h which converges to in the neighborhood of u. The theory states that _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.
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