A Two-Scale Higher-Order Finite Element Discretization for Schrödinger Equation

Abstract

In this paper, a two-scale higher-order finite element discretization scheme is proposed and analyzed for a Schrödinger equation on tensor product domains. With the scheme, the solution of the eigenvalue problem on a fine grid can be reduced to an eigenvalue problem on a much coarser grid together with some eigenvalue problems on partially fine grids. It is shown theoretically and numerically that the proposed two-scale higher-order scheme not only significantly reduces the number of degrees of freedom but also produces very accurate approximations.