A Nitsche method for the Brinkman problem with mixed boundary conditions

We present the analysis of a Nitsche-based finite element method for the Brinkman equations written in terms of velocity and pressure. We provide a new discrete variational formulation that enables the weak imposition of mixed and non-standard boundary conditions, including the normal and tangential components of the velocity, through a consistent and stable Nitsche method. The method is analyzed within the framework of Babuska–Brezzi theory, ensuring the well-posedness of the discrete problem. We derive a priori error estimates for the discrete scheme with constants independent of the permeability tensor. Finally, we present some numerical experiments to validate the theoretical results and assess the robustness of the proposed scheme.