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Volume 14, Issue 3
Superconvergence of Finite Element Approximations of the Two-Dimensional Cubic Nonlinear Schrödinger Equation

Jianyun Wang & Zhikun Tian

Adv. Appl. Math. Mech., 14 (2022), pp. 652-665.

Published online: 2022-02

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  • 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.

  • Keywords

Superconvergence, nonlinear Schrödinger equation, finite element method, elliptic projection.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-14-652, author = {Wang , Jianyun and Tian , Zhikun}, title = {Superconvergence of Finite Element Approximations of the Two-Dimensional Cubic Nonlinear Schrödinger Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {3}, pages = {652--665}, 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0268}, url = {http://global-sci.org/intro/article_detail/aamm/20279.html} }
TY - JOUR T1 - Superconvergence of Finite Element Approximations of the Two-Dimensional Cubic Nonlinear Schrödinger Equation AU - Wang , Jianyun AU - Tian , Zhikun JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 652 EP - 665 PY - 2022 DA - 2022/02 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2020-0268 UR - https://global-sci.org/intro/article_detail/aamm/20279.html KW - Superconvergence, nonlinear Schrödinger equation, finite element method, elliptic projection. AB -

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.

Jianyun Wang & Zhikun Tian. (2022). Superconvergence of Finite Element Approximations of the Two-Dimensional Cubic Nonlinear Schrödinger Equation. Advances in Applied Mathematics and Mechanics. 14 (3). 652-665. doi:10.4208/aamm.OA-2020-0268
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