  Convection-diffusion equations arise in a variety of applications such as
particle transport, electromagnetics, and magnetohydrodynamics. Simulation of
the convection-dominated regime for these problems, even with high-fidelity
techniques, is particularly challenging due to the presence of sharp boundary
layers and shocks causing jumps and discontinuities in the solution, and
numerical issues such as loss of the maximum principle in the discretization.
These complications cause instabilities, admitting large oscillations in the
numerical solution when using traditional methods. Drawing connections to the
simplex-averaged finite-element method (S. Wu and J. Xu, 2020), this paper
develops a mimetic finite-difference (MFD) discretization using
exponentially-averaged coefficients to overcome instability of the numerical
solution as the diffusion coefficient approaches zero. The finite-element
framework allows for transparent analysis of the MFD, such as proving
well-posedness and deriving error estimates. Numerical tests are presented
confirming the stability of the method and verifying the error estimates.
