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The local Green's function method in singularly perturbed convection-diffusion problems


Authors: Owe Axelsson, Evgeny Glushkov and Natalya Glushkova
Journal: Math. Comp. 78 (2009), 153-170
MSC (2000): Primary 65F10, 65N22, 65R10, 65R20
Published electronically: July 10, 2008
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Abstract: Previous theoretical and computational investigations have shown high efficiency of the local Green's function method for the numerical solution of singularly perturbed problems with sharp boundary layers. However, in several space variables those functions, used as projectors in the Petrov-Galerkin scheme, cannot be derived in a closed analytical form. This is an obstacle for the application of the method when applied to multi-dimensional problems. The present work proposes a semi-analytical approach to calculate the local Green's function, which opens a way to effective practical application of the method. Besides very accurate approximation, the matrix stencils obtained with these functions allow the use of fast and stable iterative solutions of the large sparse algebraic systems that arise from the grid-discretization. The advantages of the method are illustrated by numerical examples.


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Additional Information

Owe Axelsson
Affiliation: Faculty of Natural Sciences, Mathematics and Informatics, The University of Nijmegen, Toernooiveld 1, NL 6525 ED Nijmegen, The Netherlands
Address at time of publication: Department of Informatics Technology, Uppsala University, Box 337, 75105 Uppsala, Sweden
Email: axelsson@sci.kun.nl, Owe.Axelsson@it.uu.se

Evgeny Glushkov
Affiliation: Department of Applied Mathematics, Kuban State University, P.O. Box 4102, Krasnodar, 350080, Russia
Email: evg@math.kubsu.ru

Natalya Glushkova
Affiliation: Department of Applied Mathematics, Kuban State University, P.O. Box 4102, Krasnodar, 350080, Russia

DOI: https://doi.org/10.1090/S0025-5718-08-02161-3
Keywords: Convection-diffusion equation, Petrov-Galerkin discretization, Fourier transform, integral equations, iterative solution.
Received by editor(s): March 25, 2003
Received by editor(s) in revised form: January 9, 2007
Published electronically: July 10, 2008
Article copyright: © Copyright 2008 American Mathematical Society