Article
Kyungpook Mathematical Journal 2021; 61(3): 591-612
Published online September 30, 2021
Copyright © Kyungpook Mathematical Journal.
Higher Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Reaction-Diffusion Problems
Worku Tilahun Anilay, Gemechis File Duressa, Mesfin Mekuria Woldaregay*
Department of Mathematics, Jimma University, Jimma, Ethiopia
e-mail : workutil12@gmail.com
Department of Mathematics, Jimma University, Jimma, Ethiopia
e-mail : gammeef@gmail.com
Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia
e-mail : msfnmkr02@gmail.com
Received: April 10, 2020; Revised: January 14, 2021; Accepted: February 8, 2021
In this paper, a uniformly convergent numerical scheme is designed for solving singularly perturbed reaction-diffusion problems. The problem is converted to an equivalent weak form and then a Galerkin finite element method is used on a piecewise uniform Shishkin mesh with linear basis functions. The convergence of the developed scheme is proved and it is shown to be almost fourth order uniformly convergent in the maximum norm. To exhibit the applicability of the scheme, model examples are considered and solved for different values of a singular perturbation parameter ε and mesh elements. The proposed scheme approximates the exact solution very well.
Keywords: Finite Element, Fitted Mesh, Parameter Uniform, Singularly Perturbed.