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This paper is devoted to the construction and analysis of the finite elementapproximations for the $H(D)$ convection-diffusion problems, where $D$ can bechosen as ${\rm grad}$, ${\rm curl}$ or ${\rm div}$ in 3D case. An essentialfeature of these constructions is to properly average the PDE coefficients onthe sub-simplexes. The schemes are of the class of exponential fitting methodsthat result in special upwind schemes when the diffusion coefficient approachesto zero. Their well-posedness are established for sufficiently small mesh sizeassuming 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 andeffectiveness for general convection-diffusion problems.

Simplex-averaged finite element methods for $H({\rm grad})$, $H({\rm curl})$ and $H({\rm div})$ convection-diffusion problems