# On the Discontinuous Galerkin Finite Element Method for Reaction-Diffusion Problems: Error Estimates in Energy and Balanced Norms

## TL;DR

This paper develops and analyzes a nonsymmetric discontinuous Galerkin finite element method with interior penalties for reaction-diffusion problems, demonstrating robust convergence in energy and balanced norms through theoretical proofs and numerical experiments.

## Contribution

It introduces a new DG FEM approach with higher order splines on layer-adapted meshes, providing rigorous error estimates for singularly perturbed reaction-diffusion problems.

## Key findings

- Robust convergence in energy norm
- Robust convergence in balanced norm
- Numerical results confirm theoretical error estimates

## Abstract

A nonsymmetric discontinuous Galerkin FEM with interior penalties has been applied to one-dimensional singularly perturbed reaction-diffusion problems. Using higher order splines on Shishkin-type layer-adapted meshes and certain graded meshes, robust convergence has been proved in the corresponding energy norm and in a balanced norm. Numerical experiments support theoretical findings.