Volume 1, Issue 6
Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods

Adv. Appl. Math. Mech., 1 (2009), pp. 830-844.

Published online: 2009-01

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• Abstract

In this paper, we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method. We use two Newton iterations on the fine grid in our methods. Firstly, we solve an original nonlinear problem on the coarse nonlinear grid, then we use Newton iterations on the fine grid twice. The two-grid idea is from Xu$'$s work [SIAM J. Numer. Anal., 33 (1996), pp. 1759--1777] on standard finite method. We also obtain the error estimates for the algorithms of the two-grid method. It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy $h=\mathcal{O}(H^{(4k+1)/(k+1)})$.

• Keywords

Nonlinear parabolic equations two-grid scheme expanded mixed finite element methods Gronwall's Lemma

65N30 65N15 65M12

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@Article{AAMM-1-830, author = {Yanping Chen, Peng Luan and Zuliang Lu}, title = {Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2009}, volume = {1}, number = {6}, pages = {830--844}, abstract = {

In this paper, we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method. We use two Newton iterations on the fine grid in our methods. Firstly, we solve an original nonlinear problem on the coarse nonlinear grid, then we use Newton iterations on the fine grid twice. The two-grid idea is from Xu$'$s work [SIAM J. Numer. Anal., 33 (1996), pp. 1759--1777] on standard finite method. We also obtain the error estimates for the algorithms of the two-grid method. It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy $h=\mathcal{O}(H^{(4k+1)/(k+1)})$.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.09-m09S09}, url = {http://global-sci.org/intro/article_detail/aamm/8400.html} }
TY - JOUR T1 - Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods AU - Yanping Chen, Peng Luan & Zuliang Lu JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 830 EP - 844 PY - 2009 DA - 2009/01 SN - 1 DO - http://dor.org/10.4208/aamm.09-m09S09 UR - https://global-sci.org/intro/aamm/8400.html KW - Nonlinear parabolic equations KW - two-grid scheme KW - expanded mixed finite element methods KW - Gronwall's Lemma AB -

In this paper, we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method. We use two Newton iterations on the fine grid in our methods. Firstly, we solve an original nonlinear problem on the coarse nonlinear grid, then we use Newton iterations on the fine grid twice. The two-grid idea is from Xu$'$s work [SIAM J. Numer. Anal., 33 (1996), pp. 1759--1777] on standard finite method. We also obtain the error estimates for the algorithms of the two-grid method. It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy $h=\mathcal{O}(H^{(4k+1)/(k+1)})$.

Yanping Chen, Peng Luan & Zuliang Lu. (1970). Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods. Advances in Applied Mathematics and Mechanics. 1 (6). 830-844. doi:10.4208/aamm.09-m09S09
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