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IMA Journal of Numerical Analysis Advance Access published online on July 4, 2008
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn024
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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.

Fitted mesh numerical methods for singularly perturbed elliptic problems with mixed derivatives

R. K. Dunne{dagger}

School of Mathematical Sciences, Dublin Institute of Technology, Dublin 2, Ireland

E. O'Riordan

School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland

G. I. Shishkin

Institute for Mathematics and Mechanics, Russian Academy of Sciences, Ekaterinburg, Russia

{dagger} Email: raymond.dunne{at}dit.ie

Received on 19 April 2006.


  Abstract

A class of singularly perturbed convection–diffusion problems is considered which contain a mixed derivative term. We consider the case when exponential boundary layers are present in the solutions of problems from this class. Under appropriate assumptions on the data of the problem, we construct a decomposition of the solution into regular and layer components. We then introduce a numerical method on a piecewise-uniform fitted mesh. Excluding a neighbourhood of one of the corners, it is shown that in the perturbed case (i.e. when the perturbation parameter is small relative to the inverse of the number of mesh intervals in both coordinate directions), the approximations generated by the method converge uniformly with respect to the singular perturbation parameter. Finally, numerical examples are presented that illustrate the theoretical result.

Key Words: singularly perturbed elliptic; mixed derivatives; fitted mesh


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