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

TL;DR

This paper introduces two novel mixed finite element methods for weakly imposing Neumann boundary conditions in Darcy flow, demonstrating their stability and optimal convergence through rigorous analysis and numerical validation.

Contribution

The paper presents two new finite element formulations for Neumann boundary conditions in Darcy flow, with proven stability and optimal convergence, including error estimates and numerical validation.

Findings

Both methods are stable and optimally convergent.

Optimal a priori L2-error estimates are achieved for velocity.

Numerical examples confirm theoretical results.

Abstract

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.