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Two-grid methods for mixed finite-element solution of coupled reaction-diffusion systems. (English) Zbl 0942.65106
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.

65M55Multigrid methods; domain decomposition (IVP of PDE)
65H10Systems of nonlinear equations (numerical methods)
35K57Reaction-diffusion equations
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
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