A Two-Grid Finite-Volume Method for the Schrödinger Equation

Abstract

In this paper, some two-grid finite-volume methods are constructed for solving the steady-state Schrödinger equation. The method projects the original coupled problem onto a coarser grid, on which it is less expensive to solve, and then prolongates the approximated coarse solution back to the fine grid, on which it is not much more difficult to solve the decoupled problem. We have shown, both theoretically and numerically,  that our schemes are more efficient and achieve asymptotically optimal accuracy as long as the mesh sizes satisfy $h=\mathcal{O}(H^2)$.