A new family of expanded mixed finite element methods for reaction–diffusion equations

2021 ◽  
Author(s):  
Bingjie Ji ◽  
Jiansong Zhang ◽  
Yun Yu ◽  
Jun Liu ◽  
Hui Guo
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Mixed Finite Element ◽  
Reaction Diffusion ◽  
Mixed Finite Element Methods ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
New Family ◽  
Expanded Mixed Finite Element

Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods

2014 ◽  
Vol 6 (2) ◽  
pp. 203-219 ◽  
Author(s):  
Luoping Chen ◽  
Yanping Chen
Keyword(s):  
Finite Element ◽  
Nonlinear Equations ◽  
Finite Element Methods ◽  
Mixed Finite Element ◽  
Reaction Diffusion ◽  
Mixed Finite Element Methods ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Fine Mesh ◽  
Nonlinear Reaction

AbstractIn 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 satisfyH=O(h1/2). As a result, solving such a large class of nonlinear equations will not be much more difficult than getting solutions of one linearized system.

A New Two-Grid Method for Expanded Mixed Finite Element Solution of Nonlinear Reaction Diffusion Equations

2017 ◽  
Vol 9 (3) ◽  
pp. 757-774 ◽  
Author(s):  
Shang Liu ◽  
Yanping Chen
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Grid Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Nonlinear Reaction ◽  
Expanded Mixed Finite Element ◽  
Coarse Space

AbstractIn the paper, we present an efficient two-grid method for the approximation of two-dimensional nonlinear reaction-diffusion equations using a 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. An numerical example is also given to illustrate the effectiveness of the algorithm.

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