On Some Mixed Finite Element Methods with Numerical Integration

Abstract

In this paper a new family of mixed finite volume methods is analyzed for the approximation of a reaction-diffusion problem. All the methods are obtained starting from the dual mixed formulation of the problem and then employing the lowest-order Raviart--Thomas finite element spaces plus a suitable quadrature 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 ${\cal O}(h)$ convergence rate with respect to the standard mixed finite element norm in the case of both smooth and piecewise smooth coefficients. Numerical results on test problems with both smooth and nonsmooth coefficients support the theoretical estimates.