A parameter-uniform weak Galerkin finite element method for a coupled system of singularly perturbed reaction-diffusion equations

Abstract

The aim of this paper  to investigate a    weak Galerkin finite element method (WG-FEM) for solving a system of  coupled  singularly perturbed reaction-diffusion equations.  The singular perturbed equations have  different perturbation parameters and thus their solutions present two overlapping boundary layers near each boundary of the domain. The method is applied to the system on a piecewise uniform Shishkin mesh to solve the problem numerically.  Efficiently elimination of the interior unknowns  from the discrete solution system reduces the degrees of freedom. The stability of the proposed method is presented and it is shown that the method   convergences  of order $\mathcal{O}((N^{-1} \ln N)^{k})$ in the energy  norm, uniformly with respect to  the perturbation parameters. Moreover, the optimal convergence rate of $\mathcal {O}(N^{-(k+1)})$ in the $L^2$-norm and the convergence rate of $\mathcal {O}((N^{-1}\ln N)^{2k})$ in the discrete $L^\infty$-norm is observed numerically.   Finally,  some numerical experiments are given  to verify numerically theory.