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IMA Journal of Numerical Analysis Advance Access originally published online on February 16, 2008
IMA Journal of Numerical Analysis 2009 29(1):109-125; doi:10.1093/imanum/drm053
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Layer-adapted meshes for a linear system of coupled singularly perturbed reaction–diffusion problems

Torsten Linß{dagger}

Institut für Numerische Mathematik, Technische Universität Dresden, 01062 Dresden, Germany

Niall Madden{ddagger}

Department of Mathematics, National University of Ireland, Galway, Ireland

{dagger} Email: torsten.linss{at}tu-dresden.de

{ddagger} Email: niall.madden{at}nuigalway.ie

Received on 4 April 2007. Revised on 12 October 2007.


  Abstract

We consider a system of {ell} ≥ 2 one-dimensional singularly perturbed reaction–diffusion equations coupled at the zero-order term. The second derivative of each equation is multiplied by a distinct small parameter. We show how to decompose the solution to the problem into regular and layer parts. Properties of the discretized operator are established using discrete Green's functions. We prove that a central difference scheme on certain layer-adapted meshes converges independently of the perturbation parameters. Supporting numerical examples confirm our theoretical results.

Key Words: reaction–diffusion; singular perturbation; solution decomposition; layer-adapted mesh


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