Volume 9, Issue 3
A New Two-Grid Method for Expanded Mixed Finite Element Solution of Nonlinear Reaction Diffusion Equations

Adv. Appl. Math. Mech., 9 (2017), pp. 757-774.

Published online: 2018-05

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• Abstract

In the paper, we present an efficient two-grid method for the approximation of two-dimensional nonlinear reaction-diffusion equations using an expanded mixed finite-element method. We transfer the nonlinear reaction diffusion equation into first order nonlinear equations. The solution of the nonlinear system on the fine space is reduced to the solutions of two small (one linear and one non-linear) systems on the coarse space and a linear system on the fine space. Moreover, we obtain the error estimation for the two-grid algorithm. It is showed that coarse space can be extremely coarse and achieve asymptotically optimal approximation as long as the mesh sizes satisfy $h^{k+1}=\mathcal{O}(H^{3k+1})$. A numerical example is also given to illustrate the effectiveness of the algorithm.

• Keywords

Error estimation, mixed finite elements, reaction-diffusion equations, two-grid methods.

65N30, 65N15, 65M12

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• RIS
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@Article{AAMM-9-757, author = {}, title = {A New Two-Grid Method for Expanded Mixed Finite Element Solution of Nonlinear Reaction Diffusion Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {9}, number = {3}, pages = {757--774}, abstract = {

In the paper, we present an efficient two-grid method for the approximation of two-dimensional nonlinear reaction-diffusion equations using an expanded mixed finite-element method. We transfer the nonlinear reaction diffusion equation into first order nonlinear equations. The solution of the nonlinear system on the fine space is reduced to the solutions of two small (one linear and one non-linear) systems on the coarse space and a linear system on the fine space. Moreover, we obtain the error estimation for the two-grid algorithm. It is showed that coarse space can be extremely coarse and achieve asymptotically optimal approximation as long as the mesh sizes satisfy $h^{k+1}=\mathcal{O}(H^{3k+1})$. A numerical example is also given to illustrate the effectiveness of the algorithm.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2015.m1370}, url = {http://global-sci.org/intro/article_detail/aamm/12174.html} }
TY - JOUR T1 - A New Two-Grid Method for Expanded Mixed Finite Element Solution of Nonlinear Reaction Diffusion Equations JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 757 EP - 774 PY - 2018 DA - 2018/05 SN - 9 DO - http://doi.org/10.4208/aamm.2015.m1370 UR - https://global-sci.org/intro/article_detail/aamm/12174.html KW - Error estimation, mixed finite elements, reaction-diffusion equations, two-grid methods. AB -

In the paper, we present an efficient two-grid method for the approximation of two-dimensional nonlinear reaction-diffusion equations using an expanded mixed finite-element method. We transfer the nonlinear reaction diffusion equation into first order nonlinear equations. The solution of the nonlinear system on the fine space is reduced to the solutions of two small (one linear and one non-linear) systems on the coarse space and a linear system on the fine space. Moreover, we obtain the error estimation for the two-grid algorithm. It is showed that coarse space can be extremely coarse and achieve asymptotically optimal approximation as long as the mesh sizes satisfy $h^{k+1}=\mathcal{O}(H^{3k+1})$. A numerical example is also given to illustrate the effectiveness of the algorithm.

Shang Liu & Yanping Chen. (2020). A New Two-Grid Method for Expanded Mixed Finite Element Solution of Nonlinear Reaction Diffusion Equations. Advances in Applied Mathematics and Mechanics. 9 (3). 757-774. doi:10.4208/aamm.2015.m1370
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