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IMA Journal of Numerical Analysis Advance Access published online on August 11, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drp033
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A robust grid equidistribution method for a one-dimensional singularly perturbed semilinear reaction–diffusion problem

Naresh M. Chadha{dagger} and Natalia Kopteva{ddagger}

Mathematics and Statistics Department, University of Limerick, Limerick, Ireland

{dagger} Email: naresh.chadha{at}ul.ie

{ddagger} Corresponding author. Email: natalia.kopteva{at}ul.ie

Received on 30 May 2007. Revised on 13 February 2008.


  Abstract

The numerical solution of a singularly perturbed semilinear reaction–diffusion two-point boundary-value problem is addressed. The method considered is adaptive movement of a fixed number (N + 1) of mesh points by equidistribution of a monitor function that uses discrete second-order derivatives. We extend the analysis by Kopteva & Stynes (2001, SIAM J. Numer. Anal., 39, 1446–1467) to a new equation and a more intricate monitor function. It is proved that there exists a solution to the fully discrete equidistribution problem, i.e. a mesh exists that equidistributes the discrete monitor function computed from the discrete solution on this mesh. Furthermore, in the case when the boundary-value problem is linear, it is shown that after O(| ln {epsilon}|/ ln N) iterations of the algorithm, the piecewise linear interpolant of the computed solution achieves second-order accuracy in the maximum norm, uniformly in the diffusion coefficient {epsilon}2. Numerical experiments are presented that support our theoretical results.

Key Words: grid equidistribution; reaction–diffusion; singular perturbation; adaptive mesh; finite differences; maximum norm


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