A parameter robust numerical method for a two dimensional reaction-diffusion problem
HTML articles powered by AMS MathViewer
- by C. Clavero, J. L. Gracia and E. O’Riordan PDF
- Math. Comp. 74 (2005), 1743-1758 Request permission
Abstract:
In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximations are almost second order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method.References
- Thomas Apel and Gert Lube, Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem, Appl. Numer. Math. 26 (1998), no. 4, 415–433. MR 1612364, DOI 10.1016/S0168-9274(97)00106-2
- V. F. Butuzov, Asymptotic behavior of the solution of the equation $\mu ^{2}\,\Delta u-k^{2}(x,\,y)u=f(x,\,y)$ in a rectangular domain, Differencial′nye Uravnenija 9 (1973), 1654–1660, 1740 (Russian). MR 0340781
- 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
- H. Han and R. B. Kellogg, Differentiability properties of solutions of the equation $-\epsilon ^2\Delta u+ru=f(x,y)$ in a square, SIAM J. Math. Anal. 21 (1990), no. 2, 394–408. MR 1038899, DOI 10.1137/0521022
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
- J. Li and I. M. Navon, Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems. I. Reaction-diffusion type, Comput. Math. Appl. 35 (1998), no. 3, 57–70. MR 1605555, DOI 10.1016/S0898-1221(97)00279-4
- Torsten Linß, Layer-adapted meshes for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 192 (2003), no. 9-10, 1061–1105. MR 1960975, DOI 10.1016/S0045-7825(02)00630-8
- 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, G. I. Shishkin, and L. P. Shishkina, Fitted mesh methods for problems with parabolic boundary layers, Math. Proc. R. Ir. Acad. 98A (1998), no. 2, 173–190. MR 1759430
- Eugene O’Riordan and Martin Stynes, A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem, Math. Comp. 47 (1986), no. 176, 555–570. MR 856702, DOI 10.1090/S0025-5718-1986-0856702-7
- 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
- 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), no. 161, 47–89. MR 679434, DOI 10.1090/S0025-5718-1983-0679434-4
- G.I. Shishkin, Discrete approximation of singularly perturbed elliptic and parabolic equations, Russian Academy of Sciences, Ural section, Ekaterinburg, 1992. (In Russian)
- Martin Stynes and Eugene O’Riordan, A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem, J. Math. Anal. Appl. 214 (1997), no. 1, 36–54. MR 1645503, DOI 10.1006/jmaa.1997.5581
- Guang Fu Sun and Martin Stynes, Finite-element methods for singularly perturbed high-order elliptic two-point boundary value problems. I. Reaction-diffusion-type problems, IMA J. Numer. Anal. 15 (1995), no. 1, 117–139. MR 1311341, DOI 10.1093/imanum/15.1.117
- E. A. Volkov, On differential properties of solutions of boundary value problems for the Laplace and Poisson equations on a rectangle, Trudy Mat. Inst. Steklov 77 (1965), 89–112 (Russian). MR 0192077
- Christos Xenophontos and Scott R. Fulton, Uniform approximation of singularly perturbed reaction-diffusion problems by the finite element method on a Shishkin mesh, Numer. Methods Partial Differential Equations 19 (2003), no. 1, 89–111. MR 1946805, DOI 10.1002/num.10034
Additional Information
- C. Clavero
- Affiliation: Departamento de Matemática Aplicada, Universidad de Zaragoza, Zaragoza, Spain
- Email: clavero@unizar.es
- J. L. Gracia
- Affiliation: Departamento de Matemática Aplicada, Universidad de Zaragoza, Teruel, Spain
- Email: jlgracia@unizar.es
- E. O’Riordan
- Affiliation: School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
- Email: eugene.oriordan@dcu.ie
- Received by editor(s): May 19, 2004
- Published electronically: June 7, 2005
- Additional Notes: This research was partially supported by the Diputación General de Aragón and the project MCYT/FEDER BFM2001–2521
- © Copyright 2005 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 1743-1758
- MSC (2000): Primary 65N06, 65N12, 65N15; Secondary 35J25
- DOI: https://doi.org/10.1090/S0025-5718-05-01762-X
- MathSciNet review: 2164094