Mixed and hybrid finite element methods for convection-diffusion equations and their relationships with finite volume

We study the relationship between finite volume and mixed finite element methods for the the hyperbolic conservation laws, and the closely related convection-diffusion equations.A general framework is proposed for the derivation and a functional framework is developed which could allow the analysis of relating finite volume (FV) schemes. We show via two nonstandard formulations, that numerous FV schemes, including centred, upwind, Lax-Friedrichs, Roe, Engquist-Osher, the central Nessyahu-Tadmor schemes, etc., can be recovered in the unique dual mixed and hybrid (DMH) finite element framework. That makes possible a better understanding of these FV schemes. Moreover, the large number of DMH finite element results available can then give the analysis of these FV methods in a unified fashion. Furthermore, stabilized methods are proposed. In particular, interpretation in terms of the Lagrange multiplier of flux-limiter is given. We end by presenting numerical results to validate the newly proposed stabilized schemes.