Superconvergence Analysis of Bilinear Finite Element for the Nonlinear Schrödinger Equation on the Rectangular Mesh (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we investigate the superconvergence property of a time-dependent nonlinear Schrödinger equation with the bilinear finite element method on the rectangular mesh. We prove the superclose error estimate in $H^1$-norm with order $\mathcal{O}(h^2)$ between the approximated solution and the elliptic projection of the exact solution. Moreover, we obtain the global superconvergence result in $H^1$-norm with order $\mathcal{O}(h^2)$ by the interpolation post-processing operator.