Superconvergence of Finite Element Approximations of the Two-Dimensional Cubic Nonlinear Schrödinger Equation (Advances in Applied Mathematics and Mechanics, 2022)
The superconvergence of a two-dimensional time-independent nonlinearSchrödinger equation are analyzed with the rectangular Lagrange type finite element of order $k$. Firstly, the error estimate and superclose property are given in $H^1$-normwith 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 isobtained by the interpolation post-processing technique. In addition, some numericalexamples with the order $k = 1$ and $k = 2$ are provided to demonstrate the theoreticalanalysis.