Several procedures of mixed finite element type for solving elliptic partial differential equations are presented. The efficient implementation of these approaches using the lowest-order Raviart-Thomas approximating spaces defined on triangular elements is discussed. A quadrature rule is given which reduces a mixed method to a finite difference method on triangles. This approach substantially reduces the complexity of the mixed finite element matrix, but may also lead to a loss of accuracy in the solution. An enhancement of this method is derived which combines numerical quadrature with Lagrange multipliers on certain element edges. The enhanced method regains the accuracy of the solution, with little additional cost if the geometry is sufficiently smooth. Numerical examples in two dimensions are given comparing the accuracy of the various methods.