Abstract
Two dimensional elliptic reaction - diffusion problem with highly anisotropic coefficients is considered. The second order derivative with respect to one of the independent variables is multiplied by a small parameter ∈. In this work, we construct and study finite difference schemes, defined on a priori Shishkin meshes, uniformly convergent with respect to the small parameter ∈, which have order three except for a logarithmic factor. Numerical experiments confirming the theoretical results are given.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bastian, P., Wittum, G.: On robust and adaptive multi-grid methods. In: Hemker, P.W. et. al (eds.): Multigrid Methods. Proceed. of the 4-th European Multigrid Conference, Basel (1994) 1–19.
Clavero, C., Gracia, J.L., Lisbona, F.: High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type. Numerical Algorithms 22, 2 (1999), 73–97.
Gartland, E.C. Jr.: Compact high-order finite differences for interface problems in one dimension. IMA J. of Num. Anal. 9 (1989) 243–260.
Gracia J.L., Lisbona F., Clavero C.: High-order ∈-uniform methods for singularly perturbed reaction-diffusion problems. In: L. Vulkov, Waśniewski, J., Yalamov, P. (eds.): Numerical Analysis and its Applications. Lecture notes in Comp. Sci., Vol. 1988. Springer-Verlag (2001) 350–359.
Han, H., Kellogg, R.B.: Differentiability properties of solutions of the equations-∈2Δu + ru = f(x, y) in a square. SIAM J. Math. Anal. 21 (1990) 394–408.
Kopteva, N.V.: Uniform difference methods for some singularly perturbed problems on condensed meshes. Phd Thesis, M.: MGU, 1996.
Li, J.: Quasioptimal uniformly convergent finite element methods for the elliptic boundary layer problem. Computers Math. Applic., Vol. 33, 10 (1997) 11–22.
Miler, J.J.H., O'Riordan, E., Shishkin, G.I.: Fitted numerical methods for singularly perturbed problems. World Scientific, Singapore (1996)
Roos, H.-G.: A note on the conditioning of upwind schemes on Shishkin meshes. IMA J. of Num. Anal., Vol. 16 (1996) 529–538.
Ross, H.G., Stynes, M., Tobiska, L.: Numerical methods for singularly perturbed differential equations. Springer Verlag (1996)
Spotz, W.F.: High-order compact finite difference schemes for computational mechanics. Ph. D. Thesis, University of Texas at Austin (1995)
Vasileva, A., Butusov, V.: Asymptotic methods in singular perturbation theory. Vyshaya Shkola, Moscow (1996) (in Russian).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Brayanov, I., Dimitrova, I. (2003). Uniformly Convergent High-Order Schemes for a 2D Elliptic Reaction-Diffusion Problem with Anisotropic Coefficients. In: Dimov, I., Lirkov, I., Margenov, S., Zlatev, Z. (eds) Numerical Methods and Applications. NMA 2002. Lecture Notes in Computer Science, vol 2542. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36487-0_44
Download citation
DOI: https://doi.org/10.1007/3-540-36487-0_44
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00608-4
Online ISBN: 978-3-540-36487-0
eBook Packages: Springer Book Archive