Volume 6, Issue 2
Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods

Adv. Appl. Math. Mech., 6 (2014), pp. 203-219.

Published online: 2014-06

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• Abstract

In this paper, we study an efficient scheme for nonlinear reaction-diffusion equations discretized by mixed finite element methods. We mainly concern the case when pressure coefficients and source terms are nonlinear. To linearize the nonlinear mixed equations, we use the two-grid algorithm. We first solve the nonlinear equations on the coarse grid, then, on the fine mesh, we solve a linearized problem using Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H =\mathcal{O}(h^{\frac{1}{2}})$. As a result, solving such a large class of nonlinear equations will not be much more difficult than getting solutions of one linearized system.

• Keywords

Two-grid method, reaction-diffusion equations, mixed finite element methods.

65M12, 65M15, 65M60

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@Article{AAMM-6-203, author = {}, title = {Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2014}, volume = {6}, number = {2}, pages = {203--219}, abstract = {

In this paper, we study an efficient scheme for nonlinear reaction-diffusion equations discretized by mixed finite element methods. We mainly concern the case when pressure coefficients and source terms are nonlinear. To linearize the nonlinear mixed equations, we use the two-grid algorithm. We first solve the nonlinear equations on the coarse grid, then, on the fine mesh, we solve a linearized problem using Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H =\mathcal{O}(h^{\frac{1}{2}})$. As a result, solving such a large class of nonlinear equations will not be much more difficult than getting solutions of one linearized system.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.12-m12130}, url = {http://global-sci.org/intro/article_detail/aamm/14.html} }
TY - JOUR T1 - Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 203 EP - 219 PY - 2014 DA - 2014/06 SN - 6 DO - http://doi.org/10.4208/aamm.12-m12130 UR - https://global-sci.org/intro/article_detail/aamm/14.html KW - Two-grid method, reaction-diffusion equations, mixed finite element methods. AB -

In this paper, we study an efficient scheme for nonlinear reaction-diffusion equations discretized by mixed finite element methods. We mainly concern the case when pressure coefficients and source terms are nonlinear. To linearize the nonlinear mixed equations, we use the two-grid algorithm. We first solve the nonlinear equations on the coarse grid, then, on the fine mesh, we solve a linearized problem using Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H =\mathcal{O}(h^{\frac{1}{2}})$. As a result, solving such a large class of nonlinear equations will not be much more difficult than getting solutions of one linearized system.

Luoping Chen & Yanping Chen. (1970). Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods. Advances in Applied Mathematics and Mechanics. 6 (2). 203-219. doi:10.4208/aamm.12-m12130
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