A discontinuous Galerkin method by patch reconstruction for convection-diffusion-reaction problems over polytopic meshes

TL;DR

This paper introduces a novel discontinuous Galerkin method using patch reconstruction with weighted discrete least-squares, achieving optimal error estimates for convection-diffusion-reaction problems on polytopic meshes.

Contribution

It develops a new finite element space with one degree of freedom per element and applies it to DG methods, providing optimal error estimates in various regimes.

Findings

Optimal error estimates in diffusion- and convection-dominated regimes

Effective approximation of boundary and internal layers

Numerical experiments confirm theoretical results

Abstract

In this article, using the weighted discrete least-squares, we propose a patch reconstruction finite element space with only one degree of freedom per element. As the approximation space, it is applied to the discontinuous Galerkin methods with the upwind scheme for the steady-state convection-diffusion-reaction problems over polytopic meshes. The optimal error estimates are provided in both diffusion-dominated and convection-dominated regimes. Furthermore, several numerical experiments are presented to verify the theoretical error estimates, and to well approximate boundary layers and/or internal layers.