Parameter uniform convergence of a finite element method for a singularly perturbed linear reaction diffusion system with discontinuous source terms

Vinoth Maruthamuthu, Joseph Paramasivam Mathiyazhagan

Abstract


A linear system of ’n’ second order ordinary differential equations of reaction-diffusion type with discontinuous source terms is considered. On a piecewise uniform Shishkin mesh, a numerical system is built that employs the finite element method. The numerical approximations obtained by this approach are proven to be effectively almost
second order convergent.


Keywords


singular perturbation problems, system of differential equations, reaction - diffusion equations, overlapping boundary and interior layers, finite element method, Shishkin mesh, parameter - uniform convergence, discontinuous source terms

Full Text:

PDF

References


E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders. Uniform numerical methods for problems with initial and boundary layers. Boole Press, Dublin, Ireland, 1980.

T. Lin and N. Madden. Layer-adapted meshes for a linear system of coupled singularly perturbed reaction-diffusion problems. IMA J. Num. Anal., 29:109–125, 2009.

J. J. H. Miller, E. O. Riordan, and 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 publishing CO.Pvt.Ltd., Singapore, 1996.

M. J. Paramasivam, S. Valarmathi, and J. J. H. Miller. Second order parameter uniform convergence for a finite difference method for a singularly perturbed linear reaction diffusion system. Mathematical Communications, 15:587–612, 2010.

M. J. Paramasivam, S. Valarmathi, and J. J. H. Miller. Second order parameteruniform numerical method for a partially singularly perturbed linear system of reaction-diusion type. Mathematical Communications, 18:271–295, 2013.

M. J. Paramasivam, S. Valarmathi, and J. J. H. Miller. Parameter-uniform convergence for a finite difference method for a singularly perturbed linear reaction diffusion system with discontinuous source terms. Int. J. Numer. Anal. Model, 11:385–399, 2014.

H. G. Roos, M. Stynes, and L. Tobiska. Numerical methods for singularly perturbed differential equations. Springer Verlag, 1996.




DOI: http://dx.doi.org/10.23755/rm.v50i0.1547

Refbacks

  • There are currently no refbacks.


Copyright (c) 2023 Vinoth Maruthamuthu, Joseph Paramasivam Mathiyazhagan

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.