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 Newton iterations on the fine grid. These two-grid ideas are from Xu's work (SIAM J. Sci. Comput. 1994; 15:231-237; SIAM J. Numer. Anal. 1996; 33:1759-1777) on standard finite element method. We extend the ideas to the mixed finite element method. Moreover, we obtain the error estimates for two algorithms of two-grid method. It is showed that coarse space can be extremely coarse and we achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h 1/4 ) in the first algorithm and H = O(h 1/6 ) in second algorithm.

A two-grid method with expanded mixed element for nonlinear reaction-diffusion equations

“…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.…”

Two grid finite element discretization method for semi‐linear hyperbolic integro‐differential equations

“…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].…”

Analysis of two‐grid discretization scheme for semilinear hyperbolic equations by mixed finite element methods

“…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.…”