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Two‐grid methods for mixed finite‐element solution of coupled reaction‐diffusion systems
Author(s) -
Wu Li,
Allen Myron B.
Publication year - 1999
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/(sici)1098-2426(199909)15:5<589::aid-num6>3.0.co;2-w
Subject(s) - grid , finite element method , mathematics , nonlinear system , reaction–diffusion system , iterative method , partial differential equation , diffusion , newton's method , function (biology) , mathematical optimization , mathematical analysis , geometry , physics , thermodynamics , quantum mechanics , evolutionary biology , biology
We develop 2‐grid schemes for solving nonlinear reaction‐diffusion systems:$$ {{\partial {\bf p}}\over{\partial t}} - \nabla\cdot( K\nabla {\bf p}) = {\bf f}({\bf x, p}),$$where 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. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 589–604, 1999