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.
- K. W. Chang and F. A. Howes, Nonlinear singular perturbation phenomena: theory and applications, Applied Mathematical Sciences, vol. 56, Springer-Verlag, New York, 1984. MR 764395, DOI 10.1007/978-1-4612-1114-3
- C. M. D’Annunzio, Numerical analysis of a singular perturbation problem with multiple solutions, Ph. D. thesis, University of Maryland, College Park, (1986).
- J. Lorenz, Nonlinear singular perturbation problems and the Engquist-Osher difference scheme, Report 8115, University of Nijmegen, 1981.
- Paul A. Farrell, John J. H. Miller, Eugene O’Riordan, and Grigori I. Shishkin, A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1135–1149. MR 1393906, DOI 10.1137/0733056
- Paul A. Farrell, John J. H. Miller, Eugene O’Riordan, and Grigorii I. Shishkin, On the non-existence of $\epsilon$-uniform finite difference methods on uniform meshes for semilinear two-point boundary value problems, Math. Comp. 67 (1998), no. 222, 603–617. MR 1451321, DOI 10.1090/S0025-5718-98-00922-3
- P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Robust computational techniques for boundary layers, Applied Mathematics (Boca Raton), vol. 16, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR 1750671
- 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.
- J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. Error estimates in the maximum norm for linear problems in one and two dimensions. MR 1439750, DOI 10.1142/2933
- 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.
- Donal O’Regan, Existence theory for nonlinear ordinary differential equations, Mathematics and its Applications, vol. 398, Kluwer Academic Publishers Group, Dordrecht, 1997. MR 1449397, DOI 10.1007/978-94-017-1517-1
- 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, DOI 10.1007/978-3-662-03206-0
- G. I. Shishkin, Grid approximation of singularly perturbed boundary value problem for quasi-linear parabolic equations in the case of complete degeneracy in spatial variables, Soviet J. Numer. Anal. Math. Modelling 6 (1991), no. 3, 243–261. MR 1126678
- G. I. Shishkin, Discrete approximation of singularly perturbed elliptic and parabolic equations, Russian Academy of Sciences, Ural section, Ekaterinburg, (1992). (in Russian)
- Guang Fu Sun and Martin Stynes, A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions, Math. Comp. 65 (1996), no. 215, 1085–1109. MR 1351205, DOI 10.1090/S0025-5718-96-00753-3
- P. A. Farrell
- Affiliation: Department of Computer Science, Kent State University, Kent, Ohio 44242, U.S.A.
- Email: firstname.lastname@example.org
- E. O’Riordan
- Affiliation: School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
- Email: email@example.com
- G. I. Shishkin
- Affiliation: Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ekaterinburg, Russia
- Email: firstname.lastname@example.org
- Received by editor(s): October 13, 2003
- Received by editor(s) in revised form: June 11, 2004
- Published electronically: 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 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Math. Comp. 74 (2005), 1759-1776
- MSC (2000): Primary 65L70, 65L20, 65L10, 65L12
- DOI: https://doi.org/10.1090/S0025-5718-05-01764-3
- MathSciNet review: 2164095