Finite element error estimates for the nonlinear Schr\"{o}dinger-Poisson model

TL;DR

This paper develops a unified theoretical framework for a priori error estimates in finite element approximations of the nonlinear Schrödinger-Poisson model, providing optimal error bounds and verifying convergence through numerical experiments.

Contribution

It introduces a general theory for error estimates of nonlinear problems and applies it to derive optimal error bounds for the Schrödinger-Poisson model's finite element approximation.

Findings

Established a unified error estimate theory for nonlinear problems.

Derived optimal error bounds for the Schrödinger-Poisson finite element approximation.

Numerical experiments confirmed the theoretical convergence rates.

Abstract

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.