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

Abstract 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 $$\varvec{\mathcal {O}}(\varvec{\varepsilon }^{\varvec{1/2}}(\varvec{N}^{\varvec{-1}}\varvec{\ln N})^{\varvec{r-1}} \varvec{+ N}^{\varvec{-r}})$$ in the $$\varvec{H^2}$$ discrete energy norm when piecewise polynomials of order $$\varvec{r \ge 2}$$ are used on the interior of the intervals and constant polynomials at the mesh points, where $$\varvec{N}$$ is the number of subintervals. Numerical examples are presented to support the theoretical findings and confirm the sharpness of the error estimate.