Simplex-averaged finite element methods for $H(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d})$, $H(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l})$ and $H(\mathrm{d}\mathrm{i}\mathrm{v})$ convection-diffusion problems

This paper is devoted to the construction and analysis of the finite element approximations for the $H(D)$ convection-diffusion problems, where $D$ can be chosen as $\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}$, $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}$ or $\mathrm{d}\mathrm{i}\mathrm{v}$ in 3D case. An essential feature of these constructions is to properly average the PDE coefficients on the sub-simplexes. The schemes are of the class of exponential fitting methods that result in special upwind schemes when the diffusion coefficient approaches to zero. Their well-posedness are established for sufficiently small mesh size assuming that the convection-diffusion problems are uniquely solvable. Convergence of first order is derived under minimal smoothness of the solution. Some numerical examples are given to demonstrate the robustness and effectiveness for general convection-diffusion problems.