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Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem

Author: Natalia Kopteva
Journal: Math. Comp. 76 (2007), 631-646
MSC (2000): Primary 65N06, 65N15, 65N30; Secondary 35B25
Published electronically: December 27, 2006
MathSciNet review: 2291831
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Abstract: A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter $ \varepsilon^2$ is arbitrarily small, which induces boundary layers. Constructing discrete sub- and super-solutions, we prove existence and investigate the accuracy of multiple discrete solutions on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the discrete maximum norm, uniformly in $ \varepsilon$ for $ \varepsilon\le Ch$. Here $ h>0$ is the maximum side length of mesh elements, while the number of mesh nodes does not exceed $ Ch^{-2}$. Numerical experiments are performed to support the theoretical results.

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  • 1. N. S. Bakhvalov, On the optimization of methods for solving boundary value problems with boundary layers, Zh. Vychisl. Mat. Mat. Fis., 9 (1969) 841-859 (in Russian). MR 0255066 (40:8273)
  • 2. I.A. Blatov, Galerkin finite-element method for elliptic quasilinear singularly perturbed boundary problems. I, (Russian) Differ. Uravn., 28 (1992), 1168-1177; translation in Differ. Eqs., 28 (1992), 931-940. MR 1201213 (94a:65056)
  • 3. I.A. Blatov, Galerkin finite element method for elliptic quasilinear singularly perturbed boundary problems. II, (Russian) Differ. Uravn., 28 (1992), 1799-1810; translation in Differ. Eqs., 28 (1992), 1469-1477. MR 1208410 (94c:65130)
  • 4. C. Clavero, J. L. Gracia, E. O'Riordan, A parameter robust numerical method for a two dimensional reaction-diffusion problem, Math. Comp., 74 (2005), 1743-1758. MR 2164094 (2006e:65192)
  • 5. C.M. D'Annunzio, Numerical analysis of a singular perturbation problem with multiple solutions, Ph.D. Dissertation, University of Maryland at College Park, 1986 (unpublished).
  • 6. P.C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal., 52 (1973), 205-232. MR 0374665 (51:10863)
  • 7. P. Grindrod, Patterns and Waves: The theory and applications of reaction-diffusion equations, Clarendon Press, Oxford, 1991. MR 1136256 (92k:35145)
  • 8. N. Kopteva and M. Stynes, Numerical analysis of a singularly perturbed nonlinear reaction-diffusion problem with multiple solutions, Appl. Numer. Math., 51 (2004), 273-288. MR 2091404 (2005e:65097)
  • 9. J. Lorenz, Nonlinear singular perturbation problems and the Enquist-Osher scheme, Report 8115, Mathematical Institute, Catholic Univerity of Nijmegen, 1981 (unpublished).
  • 10. J.M. Melenk, $ hp$-finite element methods for singular perturbations, Springer, 2002. MR 1939620 (2003i:65108)
  • 11. J.D. Murray, Mathematical Biology. Second corrected edition, Springer-Verlag, Berlin, 1993. MR 1239892 (94j:92002)
  • 12. N.N. Nefedov, The method of differential inequalities for some classes of nonlinear singularly perturbed problems with internal layers, (Russian) Differ. Uravn., 31 (1995), 1142-1149; translation in Differ. Eqs., 31 (1995), 1077-1085. MR 1429769 (97m:35018)
  • 13. J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 0273810 (42:8686)
  • 14. A.A. Samarski, Theory of Difference Schemes, Nauka, Moscow, 1989 (in Russian). MR 1196231 (93g:65004)
  • 15. A.H. Schatz and L.B. Wahlbin, On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Math. Comp., 40 (1983), 47-89. MR 0679434 (84c:65137)
  • 16. G.I. SHISHKIN, Grid approximation of singularly perturbed elliptic and parabolic equations, Ur. O. Ran, Ekaterinburg, 1992 (in Russian).
  • 17. G. Sun and M. Stynes, A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions, Math. Comp., 65 (1996), 1085-1109. MR 1351205 (96j:65067)
  • 18. A.B. Vasil'eva, V.F. Butuzov and L.V. Kalachev, The boundary function method for singular perturbation problems. SIAM Studies in Applied Mathematics, 14, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. MR 1316892 (96a:34119)
  • 19. M.I. Višik and L.A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, (Russian), Uspehi Mat. Nauk, 12 (1957), 3-122. MR 0096041 (20:2539)

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Additional Information

Natalia Kopteva
Affiliation: Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland

Keywords: Semilinear reaction-diffusion, singular perturbation, maximum norm error estimate, $Z$-field, Bakhvalov mesh, Shishkin mesh, second order
Received by editor(s): October 8, 2005
Received by editor(s) in revised form: February 23, 2006
Published electronically: December 27, 2006
Additional Notes: This publication has emanated from research conducted with the financial support of Science Foundation Ireland under the Basic Research Grant Programme 2004; Grant 04/BR/M0055.
Dedicated: Dedicated to Professor V. B. Andreev on the occasion of his 65th birthday
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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