An $hp$ Weak Galerkin FEM for singularly perturbed problems

TL;DR

This paper introduces an $hp$ weak Galerkin finite element method for one-dimensional singularly perturbed reaction-convection-diffusion problems, achieving robust exponential convergence with minimal layer-adapted meshes under analytic data assumptions.

Contribution

The paper develops a novel $hp$ weak Galerkin FEM that attains exponential convergence on spectral boundary layer meshes for singularly perturbed problems, enhancing existing methods.

Findings

Achieves robust exponential convergence in energy norm

Uses minimal layer-adapted spectral boundary layer mesh

Numerical examples confirm theoretical results

Abstract

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.