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Zbl 1194.76107
Chen, Yanping; Liu, Huan Wen; Liu, Shang
Analysis of two-grid methods for reaction-diffusion equations by expanded mixed finite element methods.
(English)
[J] Int. J. Numer. Methods Eng. 69, No. 2, 408-422 (2007). ISSN 0029-5981

Summary: We present two efficient methods of two-grid scheme for the approximation of two-dimensional semi-linear 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 nonlinear 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. 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 $H=\cal O (h^{1/4})$ in the first algorithm and $H=\cal O (h^{1/6})$ in second algorithm.
MSC 2000:
*76M10 Finite element methods
76R50 Diffusion
76V05 Interacting phases (fluid mechanics)

Keywords: semi-linear reaction-diffusion equations; expanded mixed finite element; two-grid methods; Newton iteration; correction; error estimates; superconvergence

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