We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as $\mathcal O(h^{-1})$, while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as $\mathcal O(h^{-k-1})$, $k$ being the order of the Raviart-Thomas space. We rigorously prove that both methods are stable and result in optimal convergent numerical schemes with respect to appropriate mesh-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 and the lowest-order Raviart-Thomas discretizations, for the first and second numerical schemes. Finally, some numerical examples validating the theory are exhibited.

翻译：两种方法都考虑使用Raviart-Thomas 混合限值元素,其中一种是始终的离散化,取决于加权参数的缩放,以$mathcal O(h ⁇ )-1美元为单位;另一种是惩罚型配方,作为原问题的扰动的分解获得,并依赖以$mathcal O(h ⁇ -k-1})美元为单位的参数缩放,美元是Raviart-Thoomas空间的顺序。我们严格地证明,两种方法都稳定,在适当的网状依赖规范方面形成了最佳的趋同数字计划,尽管所选择的规范没有像通常的 $L ⁇ 2美元-诺尔姆那样的缩放。然而,我们仍能够分别从高等级和最低等级的Raviart-Thoomas离化模型中回收一些最理想的前置值($L ⁇ 2美元-eror)的参数缩放值,用于第一个和第二个数字模型。