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Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients

Published online by Cambridge University Press:  28 May 2015

Eveline Rosseel ,
Nico Scheerlinck  and
Stefan Vandewalle
Eveline Rosseel*
Affiliation:
Computer Science Department, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium
Nico Scheerlinck*
Affiliation:
Computer Science Department, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium
Stefan Vandewalle*
Affiliation:
Computer Science Department, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium
*
Corresponding author.Email address:eveline.rosseel@cs.kuleuven.be
Corresponding author.Email address:nico.scheerlinck@cs.kuleuven.be
Corresponding author.Email address:stefan.vandewalle@cs.kuleuven.be
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Abstract

An algebraic Newton-multigrid method is proposed in order to efficiently solve systems of nonlinear reaction-diffusion problems with stochastic coefficients. These problems model the conversion of starch into sugars in growing apples. The stochastic system is first converted into a large coupled system of deterministic equations by applying a stochastic Galerkin finite element discretization. This method leads to high-order accurate stochastic solutions. A stable and high-order time discretization is obtained by applying a fully implicit Runge-Kutta method. After Newton linearization, a point-based algebraic multigrid solution method is applied. In order to decrease the computational cost, alternative multigrid preconditioners are presented. Numerical results demonstrate the convergence properties, robustness and efficiency of the proposed multigrid methods.

Keywords

35K5735Q9265M5565N35Multigridstochastic Galerkin finite element methodreaction-diffusion problemsimplicit Runge-Kutta method and PDEs with random coefficients

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 1 , February 2012 , pp. 62 - 84
Copyright
Copyright © Global Science Press Limited 2012

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