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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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), no. 3, 329–345. MR**641313**, 10.1093/imanum/1.3.329**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,*𝜀-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems*, IMA J. Numer. Anal.**20**(2000), no. 1, 99–121. MR**1736952**, 10.1093/imanum/20.1.99**5.**H.-G. Roos, M. Stynes, and L. Tobiska,*Numerical methods for singularly perturbed differential equations*, Springer Series in Computational Mathematics, vol. 24, Springer-Verlag, Berlin, 1996. Convection-diffusion and flow problems. MR**1477665****6.**P. W. Hemker,*A numerical study of stiff two-point boundary problems*, Mathematisch Centrum, Amsterdam, 1977. Mathematical Centre Tracts, No. 80. MR**0488784****7.**Jerrold E. Marsden and Michael J. Hoffman,*Basic complex analysis*, 2nd ed., W. H. Freeman and Company, New York, 1987. MR**913736****8.**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****9.**M. V. Fedoryuk,*Metod perevala*, Izdat. “Nauka”, Moscow, 1977 (Russian). MR**0507923****10.**Richard S. Varga,*Matrix iterative analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR**0158502****11.**Claes Johnson, Uno Nävert, and Juhani Pitkäranta,*Finite element methods for linear hyperbolic problems*, Comput. Methods Appl. Mech. Engrg.**45**(1984), no. 1-3, 285–312. MR**759811**, 10.1016/0045-7825(84)90158-0

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

Retrieve articles in all journals with MSC (2000): 65F10, 65N22, 65R10, 65R20

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