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We propose two different discrete formulations for the weak imposition of theNeumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finiteelement on both triangular and quadrilateral meshes is considered for bothmethods. One is a consistent discretization depending on a weighting parameterscaling as $\mathcal O(h^{-1})$, while the other is a penalty-type formulationobtained as the discretization of a perturbation of the original problem andrelies on a parameter scaling as $\mathcal O(h^{-k-1})$, $k$ being the order ofthe Raviart-Thomas space. We rigorously prove that both methods are stable andresult in optimal convergent numerical schemes with respect to appropriatemesh-dependent norms, although the chosen norms do not scale as the usual$L^2$-norm. However, we are still able to recover the optimal a priori$L^2$-error estimates for the velocity field, respectively, for high-order andthe lowest-order Raviart-Thomas discretizations, for the first and secondnumerical schemes. Finally, some numerical examples validating the theory areexhibited.

Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flow