On some mixed finite element methods with numerical integration

In this paper a new family of mixed finite volume methods is analyzed for t he approximation of a reaction-diffusion problem. All the methods are obtai ned starting from the dual mixed formulation of the problem and then employ ing the lowest-order Raviart Thomas finite element spaces plus a suitable q uadrature formula for the mass matrix. This allows for the use of different averages of the inverse diffusion coefficient to enforce the constitutive law for the fluxes at the interelement boundary in a finite volume fashion. The soundness of the methods is supported by an error analysis which shows optimal O (h) convergence rate with respect to the standard mixed finite e lement norm in the case of both smooth and piecewise smooth coefficients. N umerical results on test problems with both smooth and nonsmooth coefficien ts support the theoretical estimates.