A robust weak Galerkin method for singularly perturbed fourth-order convection–diffusion problems

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.