A Two-grid Method with Expanded Mixed Element for Nonlinear Reaction-diffusion Equations

Wei Liu^{1,2}, Hong-xing Rui^{1}, Hui Guo^{3}

1. School of Mathematics, Shandong University, Jinan 250100, China;
2. School of Mathematics and Information, Ludong University, Yantai 264025, China;
3. School of Mathematics and Computational Science, China University of Petroleum, Dongying 257061, China

Abstract Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O(Δt + h^{k+1} + H^{2k+2-d/2}) (k ≥ 1), where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.

Abstract：
Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O(Δt + h^{k+1} + H^{2k+2-d/2}) (k ≥ 1), where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.

Supported by the National Natural Science Foundation of China Grant (No. 10771124); the Research Fund for Doctoral Program of High Education by State Education Ministry of China (No. 20060422006); the Program for Innovative Research Team in Ludong University; the Discipline Construction Fund of Ludong University.

Cite this article:

.A Two-grid Method with Expanded Mixed Element for Nonlinear Reaction-diffusion Equations[J] Acta Mathematicae Applicatae Sinica, English Serie, 2011,V27(3): 495-502

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