Mathematics of Computation

A class of singularly perturbed semilinear differential equations with interior layers

Author(s): P. A. Farrell; E. O'Riordan; G. I. Shishkin.
Journal: Math. Comp. 74 (2005), 1759-1776.
MSC (2000): Primary 65L70, 65L20, 65L10, 65L12
Posted: June 7, 2005
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Abstract: In this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined. A numerical method is constructed for these problems which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented that validate the theoretical results.

1.: K. W. Chang and F. A. Howes, Nonlinear Singular Perturbation Phenomena. Springer-Verlag, New York, (1984). MR 0764395 (86e:34090)
2.: C. M. D'Annunzio, Numerical analysis of a singular perturbation problem with multiple solutions, Ph. D. thesis, University of Maryland, College Park, (1986).
3.: J. Lorenz, Nonlinear singular perturbation problems and the Engquist-Osher difference scheme, Report 8115, University of Nijmegen, 1981.
4.: P. A. Farrell, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation. SIAM J. Numer. Anal., 33, (3), 1135-1149, 1996. MR 1393906 (97b:65086)
5.: P. A. Farrell, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, On the non-existence of $\varepsilon$ -uniform finite difference methods on uniform meshes for semilinear two-point boundary value problems. Math. Comp., 67, (222), 603-617, 1998. MR 1451321 (98g:65072)
6.: P. A. Farrell, A.F. Hegarty, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Robust Computational Techniques for Boundary Layers, Chapman and Hall/CRC Press, Boca Raton, U.S.A., (2000). MR 1750671 (2001c:65003)
7.: P. A. Farrell, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Singularly perturbed differential equations with discontinuous source terms, Proceedings of ``Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems", Lozenetz, Bulgaria, 1998, J.J.H. Miller, G. I. Shishkin and L.Vulkov eds., Nova Science Publishers, Inc., New York, USA, 23-32, 2000.
8.: J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Fitted numerical methods for singular perturbation problems, World-Scientific (Singapore), (1996).MR 1439750 (98c:65002)
9.: J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Fitted mesh methods for the singularly perturbed reaction diffusion problem, In Proc. of V-th International Colloquium on Numerical Analysis, Aug. 13-17, 1996, Plovdiv, Bulgaria, Academic Publications, ed. E. Minchev, 99- 105.
10.: D. O'Regan, Existence Theory for nonlinear ordinary differential equations. Kluwer Academic Publishers, (1997).MR 1449397 (98h:34042)
11.: H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems, Springer-Verlag, New York, (1996). MR 1477665 (99a:65134)
12.: G. I. Shishkin, Grid approximation of singularly perturbed boundary value problem for quasi-linear parabolic equations in the case of complete degeneracy in the spatial variables. Sov. J. Numer. Anal. Math. Modelling, 6, (3), 243-261, 1991. MR 1126678 (92i:65135)
13.: G. I. Shishkin, Discrete approximation of singularly perturbed elliptic and parabolic equations, Russian Academy of Sciences, Ural section, Ekaterinburg, (1992). (in Russian)
14.: G. Sun and M. Stynes, A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions. Math. Comp., 65, (215), 1085-1109, 1996. MR 1351205 (96j:65067)

Retrieve articles in Mathematics of Computation with MSC (2000): 65L70, 65L20, 65L10, 65L12

P. A. Farrell
Affiliation: Department of Computer Science, Kent State University, Kent, Ohio 44242, U.S.A.
Email: farrell@cs.kent.edu

E. O'Riordan
Affiliation: School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
Email: eugene.oriordan@dcu.ie

G. I. Shishkin
Affiliation: Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ekaterinburg, Russia
Email: shishkin@imm.uran.ru

DOI: 10.1090/S0025-5718-05-01764-3
PII: S 0025-5718(05)01764-3
Keywords: Semilinear, reaction-diffusion, interior layer, piecewise-uniform mesh
Received by editor(s): October 13, 2003
Received by editor(s) in revised form: June 11, 2004
Posted: June 7, 2005
Additional Notes: This research was supported in part by the Albert College Fellowship Scheme of Dublin City University, by the Enterprise Ireland grant SC--2000--070 and by the Russian Foundation for Basic Research under grant No. 04--01--00578.
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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