Error estimates of the weak Galerkin method for singularly perturbed fourth-order reaction-diffusion problems

This paper investigates the weak Galerkin finite element method (WG-FEM) for a singularly perturbed fourth-order reaction-diffusion problem. Based on a solution decomposition and a Shishkin mesh, we establish a parameter-robust convergence of the WG-FEM. Specifically, we prove that the method achieves an error estimate of order O(ε ^1/2(N^-1ln N)^r-1 + N^-r) in the H² discrete energy norm when piecewise polynomials of order r ≥ 2 are used on the interior of the intervals and constant polynomials at the mesh points, where N is the number of subintervals. Numerical examples are presented to support the theoretical findings and confirm the sharpness of the error estimate.