Petrov-Galerkin method with local Green's functions in singularly perturbed convection-diffusion problems
Owe Axelsson 1, Evgeny Glushkov 2, Natalia Glushkova 21 Faculty of Natural Sciences, Mathematics and Informatics, University of Nijmegen, Toernooiveld 1, NL 6525 ED Nijmegen, The Netherlands
2 Department of Applied Mathematics, Kuban State University, P.O.Box 4102, Krasnodar, 350080, Russia
Received by the editors February 4, 2004 and, in revised form, May 12, 2004.
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 solution of the large sparse algebraic systems that arise from the grid-discretization. The advantages of the method are illustrated by numerical examples.
AMS subject classifications: 65F10, 65N22, 65R10, 65R20
Key words: convection-diffusion equation; Petrov-Galerkin discretization; Fourier transform; integral equations; iterative solution
Email: firstname.lastname@example.org (O. Axelsson), email@example.com (E. Glushkov), firstname.lastname@example.org (N. Glushkova)