An $hp$ Weak Galerkin FEM for singularly perturbed problems

We present the analysis for an $hp$ weak Galerkin-FEM for singularly perturbed reaction-convection-diffusion problems in one-dimension. Under the analyticity of the data assumption, we establish robust exponential convergence, when the error is measured in the energy norm, as the degree $p$ of the approximating polynomials is increased. The Spectral Boundary Layer mesh is used, which is the minimal (layer adapted) mesh for such problems. Numerical examples illustrating the theory are also presented.