A Parameter-Uniform Essentially First Order Convergent Fitted Mesh Method for a Singularly Perturbed Robin Problem

janet Rajaiah; Valarmathi Sigamani

doi:https://doi.org/10.14445/22315373/IJMTT-V59P503

International Journal of Mathematics Trends and Technology

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Volume 59 | Number 1 | Year 2018 | Article Id. IJMTT-V59P503 | DOI : https://doi.org/10.14445/22315373/IJMTT-V59P503

A Parameter-Uniform Essentially First Order Convergent Fitted Mesh Method for a Singularly Perturbed Robin Problem

janet Rajaiah, Valarmathi Sigamani

Abstract

In this paper, a class of linear systems of singularly perturbed second order ordinary differential equations of reaction-diffusion type with Robin boundary conditions is considered. The components of the solution of this system exhibit boundary layers with sublayers. A piecewise-uniform Shishkin mesh is introduced and is used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that the numerical approximations obtained with this method are essentially first order convergent uniformly with respect to all of the parameters. Numerical illustration is provided to support the theory.

Keywords

Singular perturbations, boundary layers, system of differential equations, Robin boundary conditions, finite difference scheme, Shishkin meshes, parameter-uniform convergence

References

[1] J.J.H.Miller, E.O’Riordan, G.I.Shishkin, Fitted Numerical Methods for Singular Perturbation Problems. World Scientific Publishing Co., Singapore, New Jersey, London, Hong Kong, (1996).
[2] H.-G.Roos, M.Stynes, L.Tobiska, Numerical methods for singularly perturbed differential equations, Springer- Verlag, Berlin, (1996).
[3] E.P.Doolan, J.J.H.Miller, W.H.Schilders, Uniform Numerical methods for problems with initial and boundary layers, Boole Press, Dublin, Ireland, (1980).
[4] P.A.Farell, A.F.Hegarty, J.J.H.Miller, E.O’Riordan, G.I.Shishkin, Robust Computational Techniques for Boundary Layers. Chapman and Hall / CRC Press LLC, New York,(2000).
[5] Zhao Weili, Singular perturbation for non linear problems, Journal of Computational and Applied Mathematics,81,pp 59-74 (1997).
[6] Sonlin Chen, Weigen Hou, Xiaohui Jiang, Finite element analysis for singularly perturbed advection- diffusion Robin boundary values problem, Advances in Pure Mathematics, No.3,pp 643-646 (2013).
[7] Ni Shou Ping, Diagonalization method for a singularly perturbed vector Robin problem, Applied Mathematics and Mechanics, Volume 10, 329-336, (1989).
[8] Paramasivam Mathiyazhagan, Valarmathi Sigamani and John J.H. Miller, Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear reaction-diffusion system, Mathematical Communications, Vol.15, No.2, pp 587-612 (2010).
[9] Pratibhamoy Das and Srinivasan Natesan, Higher-Order parameter uniform convergent schemes for Robin type reaction-diffusion problems using adaptively generated grid, International Journal of Computational Methods, Vol.9,No.4, (2012).

Citation :

janet Rajaiah, Valarmathi Sigamani, "A Parameter-Uniform Essentially First Order Convergent Fitted Mesh Method for a Singularly Perturbed Robin Problem," International Journal of Mathematics Trends and Technology (IJMTT), vol. 59, no. 1, pp. 8-21, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V59P503