2006
DOI: 10.1002/nme.1775
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Analysis of two‐grid methods for reaction‐diffusion equations by expanded mixed finite element methods

Abstract: SUMMARYWe present two efficient methods of two-grid scheme for the approximation of two-dimensional semilinear reaction-diffusion equations using an expanded mixed finite element method. To linearize the discretized equations, we use two Newton iterations on the fine grid in our methods. Firstly, we solve an original non-linear problem on the coarse grid. Then we use twice Newton iterations on the fine grid in our first method, and while in second method we make a correction on the coarse grid between two Newt… Show more

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Cited by 73 publications
(30 citation statements)
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“…In recent years, reaction-diffusion equations have received a great deal of attention, motivated by their widespread occurrence in models of both hydrologic and bio-geochemical phenomena [9,11] . The two-grid methods for semi-linear reaction-diffusion equations have been considered by Dawson-Wheeler [6] , Wu-Allen [13] and Chen [2,3] . In this paper, we consider an expanded mixed element scheme for strongly nonlinear reaction-diffusion equations in which the coefficient K is nonlinear and the source term f relies on both p and ∇p.…”
Section: Introductionmentioning
confidence: 99%
“…In recent years, reaction-diffusion equations have received a great deal of attention, motivated by their widespread occurrence in models of both hydrologic and bio-geochemical phenomena [9,11] . The two-grid methods for semi-linear reaction-diffusion equations have been considered by Dawson-Wheeler [6] , Wu-Allen [13] and Chen [2,3] . In this paper, we consider an expanded mixed element scheme for strongly nonlinear reaction-diffusion equations in which the coefficient K is nonlinear and the source term f relies on both p and ∇p.…”
Section: Introductionmentioning
confidence: 99%
“…The main idea of this method is using a coarse-grid space to produce a rough approximation of the solution for nonlinear problems, and then use it as the initial guess for one Newton-like iteration on the fine grid. Two-grid discretization method has been widely used for different kinds of problems, such as elliptic Equations [9,10], parabolic equations [12][13][14][15][16][17], eigenvalue problems [18][19][20] stochastic partial differential equations [21] and fractional differential equations [22,23]. The two-grid discretization idea is also used for nonlinear coupled equations, such as the complicated miscible displacement problems [24][25][26] and fluid flow in porous media [27].…”
Section: Introductionmentioning
confidence: 99%
“…Mixed finite element methods pioneered by Babuska 8 and Brezzi 9 is known to be an efficient approach in the analysis of engineering and scientific computation. For example, see previous works [3][4][5][6][7][10][11][12][13][14][15][16][17][18][19] and the references cited therein. One main advantage of the mixed method lies in that the method can simultaneously approximate both the displacement and the stress or the pressure and the flux.…”
Section: Introductionmentioning
confidence: 99%