A two-grid finite element method for the Schrödinger–Poisson system

Abstract A two-grid finite element method is proposed to solve the Schrödinger–Poisson system on a Lipschitz polyhedral domain. To reduce the computational complexity, we solve the Poisson equation of electric potential on a fine mesh ${\cal T}_{h}$ and the Schrödinger eigenvalue problem on a coarse mesh ${\cal T}_{H}$. The electron density $n$ is defined by an infinite series whose terms depend on the eigenvalues and eigenfunctions of the Schrödinger equation. The discrete electron density $n_{H}$ is defined by truncating the series into the sum of $L_{H}$ terms, where $H$ is the mesh size of ${\cal T}_{H}$ and $L_{H}=O(|\!\ln H|^{3/2})$. Optimal $H^{1}$-error estimate is obtained for the discrete electric potential $V_{h}$, provided that the mesh sizes satisfy $h=O(H^{2})$. To solve the discrete problem, we propose two Newton-type methods: the Newton–Raphson method and an inexact Newton method, by seeking an explicit representation of the Fréchet derivative $n^{\prime}_{H}$. We prove that the Newton–Raphson has second-order convergence. The inexact Newton method reduces the computational quantities from $O\big (N_{h}^{3/2}|\!\ln H|^{3/2}\big )$ of the Newton–Raphson method to $O(N_{h}|\!\ln H|^{3})$. Numerical experiments show that the inexact Newton method is very efficient and superior to fixed-point iterations.