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2017 | OriginalPaper | Chapter

Applying a Patched Mesh Method to Efficiently Solve a Singularly Perturbed Reaction-Diffusion Problem

Authors : José Luis Gracia, Niall Madden, Thái Anh Nhan

Published in: Modeling, Simulation and Optimization of Complex Processes HPSC 2015

Publisher: Springer International Publishing

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Abstract

The solution of linear systems of equations that arise when singularly perturbed partial differential equations are discretized can be difficult: direct solvers scale poorly, but are also known not to be robust with respect to the perturbation parameter, while the design of parameter robust preconditioners is not trivial, primarily due to the specialised layer adapted meshes used for such problems; see MacLachlan and Madden (SIAM J Sci Comput 35:A2225–A2254, 2013). Here we present a multigrid solver strategy that circumvents this problem by using a robust patched mesh method proposed by de Falco and O’Riordan (BAIL 2008—Boundary and Interior Layers vol. 69, pp. 117–127. Springer, Berlin, 2009), as well as permitting parallelization. Numerical results demonstrate the efficiency of the method.

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Andreev, V.B.: On the accuracy of grid approximations of nonsmooth solutions of a singularly perturbed reaction-diffusion equation in the square. Differ. Equ. 42(7), 895–906, 1005 (2006)
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de Falco, C., O’Riordan, E.: A patched mesh method for singularly perturbed reaction-diffusion equations. In: BAIL 2008—Boundary and Interior Layers. Lecture Notes in Computational Science and Engineering, vol. 69, pp. 117–127. Springer, Berlin (2009)
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Nhan, T., Madden, N.: Cholesky factorisation of linear systems coming from finite difference approximations of singularly perturbed problems. In: BAIL 2014—Boundary and Interior Layers. Lecture Notes in Computational Science and Engineering, vol. 108, pp. 209–220. Springer, Berlin (2015)
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Metadata
Title
Applying a Patched Mesh Method to Efficiently Solve a Singularly Perturbed Reaction-Diffusion Problem
Authors
José Luis Gracia
Niall Madden
Thái Anh Nhan
Copyright Year
2017
Book
Modeling, Simulation and Optimization of Complex Processes HPSC 2015

Print ISBN: 978-3-319-67167-3

Electronic ISBN: 978-3-319-67168-0

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-67168-0_4

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