In this paper we introduce and analyze new mixed finite volume methods for second order elliptic problemswhich are based on H(div)-conforming approximations for the vector variable anddiscontinuous approximations for the scalar variable.The discretization is fulfilled by combining the ideas of the traditional finite volume box method andthe local discontinuous Galerkin method.We propose two different types of methods, called Methods I and II, and show that they have distinct advantagesover the mixed methods used previously.In particular, a clever elimination of the vector variable leads to a primal formulation for the scalar variablewhich closely resembles discontinuous finite element methods.We establish error estimates for these methods that are optimal for the scalar variable in both methodsand for the vector variable in Method II.
