Pure vorticity formulation and Galerkin discretization for the Brinkman equations

We introduce a new finite element method for the approximation of the three-dimensional Brinkman problem formulated in terms of the velocity, vorticity, and pressure fields. The proposed strategy exhibits the advantage that, at the continuous level, a complete decoupling of vorticity and pressure can be established under the assumption of sufficient regularity. The velocity is then obtained as a simple postprocess from vorticity and pressure, using the momentum equation. Well-posedness follows straightforwardly by the Lax-Milgram theorem. The Galerkin scheme is based on Nédélec and piecewise continuous finite elements of degree k > 1 for vorticity and pressure, respectively. The discrete setting uses the very same ideas as in the continuous case, and the error analysis for the vorticity scheme is carried out first. As a byproduct of these error bounds and the problem decoupling, the convergence rates for the pressure and velocity are readily obtained in the natural norms with constants independent of the viscosity. We also present details about how the analysis of the method is modified for axisymmetric, meridian Brinkman flows; and modify the decoupling strategy to incorporate the case of Dirichlet boundary conditions for the velocity. A set of numerical examples in two and three spatial dimensions illustrate the robustness and accuracy of the finite element method, as well as its competitive computational cost compared to recent fully-mixed and augmented formulations of incompressible flows.