Superconvergence analysis of bi-<Emphasis Type="Italic">k</Emphasis>-degree rectangular elements for two-dimensional time-dependent Schrödinger equation

Superconvergence analysis of bi-k-degree rectangular elements for two-dimensional time-dependent Schrödinger equation

Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schrödinger equation with the finite element method. The error estimate and superconvergence property with order O(h^k+1) in the H¹ norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(h^k+1 + τ²) in the H¹ norm can be obtained in the Crank-Nicolson fully discrete scheme.