Volume 10, Issue 2
Superconvergence Analysis of Bilinear Finite Element for the Nonlinear Schrödinger Equation on the Rectangular Mesh

Adv. Appl. Math. Mech., 10 (2018), pp. 468-484.

Published online: 2018-10

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• Abstract

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.

65M60, 65M15

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@Article{AAMM-10-468, author = {Tian , ZhikunChen , Yanping and Wang , Jianyun}, title = {Superconvergence Analysis of Bilinear Finite Element for the Nonlinear Schrödinger Equation on the Rectangular Mesh}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {2}, pages = {468--484}, abstract = {

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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0156}, url = {http://global-sci.org/intro/article_detail/aamm/12221.html} }
TY - JOUR T1 - Superconvergence Analysis of Bilinear Finite Element for the Nonlinear Schrödinger Equation on the Rectangular Mesh AU - Tian , Zhikun AU - Chen , Yanping AU - Wang , Jianyun JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 468 EP - 484 PY - 2018 DA - 2018/10 SN - 10 DO - http://doi.org/10.4208/aamm.OA-2017-0156 UR - https://global-sci.org/intro/article_detail/aamm/12221.html KW - Finite element method, nonlinear Schrödinger equation, superconvergence, interpolation post-processing. AB -

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.

Zhikun Tian, Yanping Chen & Jianyun Wang. (2020). Superconvergence Analysis of Bilinear Finite Element for the Nonlinear Schrödinger Equation on the Rectangular Mesh. Advances in Applied Mathematics and Mechanics. 10 (2). 468-484. doi:10.4208/aamm.OA-2017-0156
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