Numerical analysis of a singularly perturbed nonlinear reaction-diffusion problem with multiple solutions

Applied Numerical Mathematics

Volumn 51, Issue 2-3, 2004, Pages 273-288

  Kopteva, Natalia ^a   Stynes, Martin ^b  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

^b UNIVERSITY COLLEGE CORK   (Ireland)

Author keywords

Dynamical systems; Error estimates; Layer adapted mesh; Nonlinear reaction diffusion; Singular perturbation

Indexed keywords

ERROR ANALYSIS; MATHEMATICAL MODELS; PARAMETER ESTIMATION; PERTURBATION TECHNIQUES; PROBLEM SOLVING;

DYNAMICAL SYSTEMS; ERROR ESTIMATES; SINGULAR PERTURBATION;

NONLINEAR EQUATIONS;

EID: 5144222419     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2004.07.001     Document Type: Article

References (20)

1. 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Perturbation Problem with Multiple Solutions, Ph.D. Dissertation, University of Maryland at College Park, 1986 (unpublished).
2. 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).
3. P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O'Riordan, G.I. Shishkin, Robust Computational Techniques for Boundary Layers, CRC, Boca Raton, FL, 2000.
4. P.C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal. 52 (1973) 205-232.
5. D. Herceg, Uniform fourth order difference scheme for a singular perturbation problem, Numer. Math. 56 (1990) 675-693.
6. F.A. Howes, Boundary-interior layer interactions in nonlinear singular perturbation theory, Mem. Amer. Math. Soc. 15 (203) (1978).
7. W.L. Kath, Slowly varying phase planes and boundary-layer theory, Stud. Appl. Math. 72 (1985) 221-239.
8. N. Kopteva, Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes, Computing 66 (2001) 179-197.
9. T. Linß, Sufficient conditions for uniform convergence on layer-adapted grids, Appl. Numer. Math. 37 (2001) 241-255.
10. J. Lorenz, Nonlinear singular perturbation problems and the Enquist-Osher scheme, Report 8115, Mathematical Institute, Catholic University of Nijmegen, 1981 (unpublished).
11. J.J.H. Miller, E. O'Riordan, G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, Singapore, 1996.
12. J.D. Murray, Mathematical Biology, second corrected ed., Springer, Berlin, 1993.
13. N.N. Nefedov, The method of differential inequalitites for some classes of nonlinear singularly perturbed problems with internal layers, Differential Equations 31 (1995) 1077-1085.
14. R.E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Springer, New York, 1991.
15. C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, FL, 1995.
16. H.-G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer Ser. Comput. Math., vol. 24, Springer, Berlin, 1996.
17. G. Sun, M. Stynes, A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions, Math. Comp. 65 (1996) 1085-1109.
18. L. Tobiska, Die asymptotische Lösung singulär gestörter elliptischer Randwertaufgaben, 2nd Doctoral Thesis, Technische Hochschule Otto von Guericke, Magdeburg, 1983.
19. A.B. Vasil'eva, V.F. Butuzov, Asimptoticheskie razlozheniya resheniǐ singulyarno-vozmushchennykh uravneniǐ (Asymptotic expansions of the solutions of singularly perturbed equations), Nauka, Moscow, 1973 (in Russian).
20. A.B. Vasil'eva, V.F. Butuzov, L.V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, With a foreword by Robert E. O'Malley Jr, SIAM Stud. Appl. Math., vol. 14, SIAM, Philadelphia, PA, 1995.