A unified approach to handle convection terms in Finite Volumes and Mimetic Discretization Methods for elliptic problems

We study the numerical approximation to the solution of the steady convection–diffusion equation. The diffusion term is discretized by using the hybrid mimetic method (HMM), which is the unified formulation for the hybrid finite-volume (FV) method, the mixed FV method and the mimetic finite-difference method recently proposed in Droniou et al. (2010, Math. Models Methods Appl. Sci., 20, 265–295). In such a setting we discuss several techniques to discretize the convection term that are mainly adapted from the literature on FV or FV schemes. For this family of schemes we provide a full proof of convergence under very general regularity conditions of the solution field and derive an error estimate when the scalar solution is in H2(Ω). Finally, we compare the performance of these schemes on a set of test cases selected from the literature in order to document the accuracy of the numerical approximation in both diffusion- and convection-dominated regimes. Moreover, we numerically investigate the behaviour of these methods in the approximation of solutions with boundary layers or internal regions with strong gradients.