This paper introduces a weak Galerkin finite element method for a system of $\ell \ge 2$ coupled singularly perturbed reaction-diffusion problems. The proposed method is independent of parameter and uses piecewise discontinuous polynomials on interior of each element and constant on the boundary of each element. By the Schur complement technique, the interior unknowns can be locally efficiently eliminated from the resulting linear system, and the degrees of freedom of the proposed method are comparable with the classical FEM. It has been reported that the energy norm is not adequate for singularly perturbed reaction-diffusion problems since it can not efficiently reflect the behaviour of the boundary layer parts when the diffusion coefficient is very small. For the first time, the error estimates in the balanced norm has been presented for a system of coupled singularly perturbed problems when each equation has different parameter. Optimal and uniform error estimates have been established in the energy and balanced norm on an uniform Shishkin mesh. Finally, we carry out various numerical experiments to verify the theoretical findings.

We propose a uniform weak Galerkin finite element method (WG-FEM) to solve singularly perturbed fourth-order convection–diffusion problems exhibiting boundary layers. The method is designed to handle small perturbation parameters $\epsilon$, ensuring accurate resolution of sharp layers without spurious oscillations. We construct appropriate layer-adapted meshes and use uniform estimates to analyze the stability and convergence of the weak Galerkin scheme. Numerical experiments confirm the theoretical results and demonstrate that the method achieves uniform convergence with respect to the perturbation parameter, effectively capturing the boundary layer behavior on layer-adapted meshes. The proposed approach provides a reliable and efficient tool for high-order singularly perturbed problems with mixed derivative boundary conditions.