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

Lourenco Beirao da Veiga (1) , Jerome Droniou (2) , Gianmarco Manzini (3)

1 UNIMI - Università degli Studi di Milano = University of Milan
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier
3 Istituto di Matematica Applicata e Tecnologie Informatiche--CNR

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.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Lourenco Beirao da Veiga, Jerome Droniou, Gianmarco Manzini. A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems. IMA Journal of Numerical Analysis, 2011, 31 (4), pp.1357-1401. ⟨10.1093/imanum/drq018⟩. ⟨hal-00808695⟩