Abstract
Solving the algebraic linear systems proceeding from the discretization on some condensed meshes of 2D singularly perturbed problems, is a difficult task. In this work we present numerical experiments obtained with the multigrid method for this class of linear systems. On Shishkin meshes, the classical multigrid algorithm is not convergent. We see that modifying only the restrict on operator in an appropriate form, the algorithm is convergent, the CPU time ncreases linearly with the discretization parameter and the number of cycles is independent of the mesh sizes.
This research was supported by the projects DGES-PB97-1013 and P226-68
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
A. Brandt, Mult-level adaptive solutions to boundary-value problems, Math. Comput. 31 333–390 (1977).
C. Clavero, J.L. Gracia, F. Lisbona, G.I. Shishkin, A method of high order for convection-diffusion problems in domains with characteristic boundaries, submitted (2000).
P.A. Farrell, A.F. Hegarty, On the determinat on of the order of uniform convergence, in R. Wichnevetsky and J.J.H. Miller (Eds): Proceeding of the 13th IMACS World Congress for Computat on and Applied Mathematics, IMACS, 501–502,(1991).
F.J. Gaspar, C. Clavero, F. Lisbona, Some numerical experiments with multigrid methods on Shishkin meshes, submitted (2000).
W. Hackbusch, Multi-grid methods and applications, Berlin: Springer-Verlag 1985.
P.W. Hemker, Multigrid methods for problems with a small parameter in the highest derivative, Lecture Notes in Mathematics 1066 106–121 (1983).
T. Linß, M. Stynes, A hybrid difference scheme on a Shishkin mesh for linear convection-diffusion problems, Appl. Numer. Math. 31 255–270 (1999).
J.J. Miller, E. O’ Riordan, G. I. Shishkin, Fitted numerical methods for singular perturbation problems. Error estimates in the maximun error for linear problems in one and two dimensions, Singapore: World Scientific, 1996.
H.G. Roos, A note on the conditioning of upwind schemes on Shishkin meshes. IMA J. Numer. Anal. 16 529–538 (1996).
G. I. Shishkin, Grid approximat on of singularly perturbed boundary value problems with convective terms, Sov. J. Numer. Anal. Math. Modelling 5 173–187 (1990).
G.I. Shishkin, Grid approximation of singularly perturbed elliptic equations in domains with characteristics faces, Sov. J. Numer. Anal. Math. Modelling 5 327–343 (1990).
H.A. Van Der Vorst, BI-CGSTAB:A fast and smoolhly converging variant of BI-CG for the solut on of nonsymetric linear systems, SIAM J. Sci. Stat. Comput. 3 v.2, 631–644 (1992).317
P. Wesseling, An introduction to multigrid methods, Chichester: Wiley 1992. 317
P. M. De Zeeuw, Matrix-dependent prolongations and restrict ons n a blackbox multigrid solver, Journal of Computational and Applied Mathematics 3 1–27 (1990).317
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gaspar, F., Lisbona, F., Clavero, C. (2001). Multigrid Methods and Finite Difference Schemes for 2D Singularly Perturbed Problems. In: Vulkov, L., Yalamov, P., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2000. Lecture Notes in Computer Science, vol 1988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45262-1_37
Download citation
DOI: https://doi.org/10.1007/3-540-45262-1_37
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41814-6
Online ISBN: 978-3-540-45262-1
eBook Packages: Springer Book Archive