Adv. Appl. Math. Mech., 14 (2022), pp. 652-665.
Published online: 2022-02
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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.
}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0268}, url = {http://global-sci.org/intro/article_detail/aamm/20279.html} }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.