Superconvergence of Finite Element Approximations of the Two-Dimensional Cubic Nonlinear Schrödinger Equation

Abstract

The superconvergence of a two-dimensional time-independent nonlinear Schrödinger equation are analyzed with the rectangular Lagrange type finite element of order $k$. Firstly, the error estimate and superclose property are given in $H^1$-norm with order $\mathcal{O}(h^{k+1})$ between the finite element solution $u_h$ and the interpolation function $u_I$ by use of the elliptic projection operator. Then, the global superconvergence is obtained by the interpolation post-processing technique. In addition, some numerical examples with the order $k = 1$ and $k = 2$ are provided to demonstrate the theoretical analysis.