Analysis of the hybrid finite element/finite volume methods for linear hyperbolic and convection-dominated convection–diffusion problems

In this paper we introduce a generalized hybrid finite element/finite volume methods. We then establish the mathematical foundations of the hybrid finite element/finite volume methods for linear hyperbolic, convection-dominated convection–diffusion, and convection–diffusion problems. More precisely, we study the stability and convergence properties of this hybrid scheme for such problems. This analysis is performed for general mesh of a bounded polygonal domain of Rn (n=2 or 3) satisfying the minimum angle condition. Our stability results are completely new and solve important open problems related to whether or not there exist approximations of hyperbolic and convection dominated problems having such stability properties.