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 achieves 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 and Chen and and 20545 and and Yanping Chen and Peng and Luan and and 20546 and and Peng Luan and Zuliang and Lu and and 20547 and 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 achieves 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 - Chen , Yanping AU - Luan , Peng AU - Lu , Zuliang JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 830 EP - 844 PY - 2009 DA - 2009/01 SN - 1 DO - http://doi.org/10.4208/aamm.09-m09S09 UR - https://global-sci.org/intro/article_detail/aamm/8400.html KW - Nonlinear parabolic equations, two-grid scheme, expanded mixed finite element methods, 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 achieves 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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