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

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 approximation space, it is applied to discontinuous Galerkin methods upwind scheme for 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 presented verify theoretical estimates, well approximate boundary layers and/or internal layers.