Parameter-Uniform Convergence Of A Finite Element Method For A Partially Singularly Perturbed Linear Reaction-Diffusion System

The reaction-diffusion kind of a partially singularly perturbed linear system of ‘𝒏’ second order ordinary differential equations is considered. The leading terms first ‘𝒎’ equation's are multiplied by a small positive parameters and the remaining ‘𝒏−𝒎’ equations are not singularly perturbed. It is assumed that these ‘𝒎’ singular perturbation parameters are distinct. First ‘𝒎’ solution's elements have overlapping boundary layers and remaining ‘𝒏-m’ solution's elements have less serve overlapping layers. 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 uniformly with respect to all perturbation parameters. In support of the theory, numerical illustrations are given.