Mathematics of Computation

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

Author(s): Owe Axelsson; Evgeny Glushkov; Natalya Glushkova.
Journal: Math. Comp. 78 (2009), 153-170.
MSC (2000): Primary 65F10, 65N22, 65R10, 65R20
Posted: July 10, 2008
Retrieve article in: PDF

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.

1.: O. Axelsson, Stability and error estimates of Galerkin finite element approximations for convection-diffusion equations, IMA J. Numer. Anal., 1(1981), pp. 329-345. MR 641313 (83a:65105)
2.: U. Nävert, A finite element method for convection-diffusion problems, Ph.D. thesis, Chalmers University of Technology, Göteborg, Sweden, 1982.
3.: O. Axelsson, A survey of numerical methods for convection-diffusion equations, In Proceedings of XIV National Summer School on Numerical Solution Methods,Varna, 29.8-3.9, 1988.
4.: P. W. Hemker, G. I. Shishkin, and L. P. Shishkina, $\varepsilon$ -uniform schemes with high-order time-accuracy for parabolic singular perturbation problems, IMA J. Numer. Anal., 20(2000), no. 1, pp. 99-121. MR 1736952 (2000k:65139)
5.: H. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer, Heidelberg, 1996. MR 1477665 (99a:65134)
6.: P. W. Hemker, A numerical study of stiff two-point boundary problems, Ph.D. thesis, Mathematical Center, Amsterdam, 1977. MR 0488784 (58:8294)
7.: J. E. Marsden and M. J. Hoffman, Basic Complex Analysis, Second ed., W.H. Freeman, New York, 1987. MR 913736 (88m:30001)
8.: M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions. Applied Mathematics Series 55, National Bureau of Standards, Washington DC, 1964. MR 0167642 (29:4914)
9.: M. V. Fedorjuk, The Method of Steepest Descent, Nauka, Moscow, 1977. MR 0507923 (58:22580)
10.: R.S.Varga, Matrix Iterative Analysis, Englewood Cliffs, NJ, Prentice Hall, 1962. MR 0158502 (28:1725)
11.: C. Johnson, U. Nävert, and J. Pitkäranta, Finite element methods for linear hyperbolic problems, Comp. Meth. Appl. Mech. Eng., 45 (1984), pp. 285-312. MR 759811 (86a:65103)

Retrieve articles in Mathematics of Computation with MSC (2000): 65F10, 65N22, 65R10, 65R20

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: 10.1090/S0025-5718-08-02161-3
PII: S 0025-5718(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
Posted: July 10, 2008
Copyright of article: Copyright 2008, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS