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 steady-state convection-diffusion-reaction problems over polytopic meshes. The optimal error estimates are proved in both diffusion-dominated and convection-dominated regimes. Finally, several numerical experiments are presented to verify the error estimates, and further to well approximate boundary layers and/or internal layers.
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