A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems

Abstract

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 formula- tion 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 conver- gence under very general regularity conditions of the solution field and derive an error estimate when the scalar solution is in H 2 (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.