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Zbl 0942.65106
Wu, Li; Allen, Myron B.
Two-grid methods for mixed finite-element solution of coupled reaction-diffusion systems.
(English)
[J] Numer. Methods Partial Differ. Equations 15, No.5, 589-604 (1999). ISSN 0749-159X; ISSN 1098-2426/e

Authors' summary: We develop 2-grid schemes for solving nonlinear reaction-diffusion systems: $${\partial{\bold p}\over\partial t}-\nabla\cdot (K\nabla{\bold p})=f(x,{\bold p}),$$ where ${\bold p}= (p,q)$ is an unknown vector-valued function. The schemes use discretizations based on a mixed finite element method. The 2-grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all the Newton-like iterations to grids much coarser than the final one, with no loss in order of accuracy. The iterative algorithms examined here extend a method developed earlier for single reaction-diffusion equations. An application to prepattern formation in mathematical biology illustrates the method's effectiveness.
[H.Marcinkowska (Wrocław)]
MSC 2000:
*65M55 Multigrid methods; domain decomposition (IVP of PDE)
65H10 Systems of nonlinear equations (numerical methods)
35K57 Reaction-diffusion equations
65M60 Finite numerical methods (IVP of PDE)

Keywords: two-grid methods; pattern formation; nonlinear reaction-diffusion systems; mixed finite element method; Newton-like iterations; iterative algorithms

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