Numerical solution of the unsteady Brinkman equations in the framework of $H$(div)-conforming finite element methods

We present projection-based mixed finite element methods for the solution of the unsteady Brinkman equations for incompressible single-phase flow with fixed in space porous solid inclusions. At each time step the method requires the solution of a predictor and a projection problem. The predictor problem, which uses a stress-velocity mixed formulation, accounts for the momentum balance, while the projection problem, which is based on a velocity-pressure mixed formulation, accounts for the incompressibility. The spatial discretization is $H$(div)-conforming and the velocity computed at the end of each time step is pointwise divergence-free. Unconditional stability of the fully-discrete scheme and first order in time accuracy are established. Due to the $H$(div)-conformity of the formulation, the methods are robust in both the Stokes and the Darcy regimes. In the specific code implementation, we discretize the computational domain using the Raviart--Thomas space $RT_1$ in two and three dimensions, applying a second-order accurate multipoint flux mixed finite element scheme with a quadrature rule that samples the flux degrees of freedom. In the predictor problem this allows for a local elimination of the viscous stress and results in element-based symmetric and positive definite systems for each velocity component with $\left(d+1\right)$ degrees of freedom per simplex (where $d$ is the dimension of the problem). In a similar way, we locally eliminate the corrected velocity in the projection problem and solve an element-based system for the pressure. Numerical experiments are presented to verify the convergence of the proposed scheme and illustrate its performance for several challenging applications, including one-domain modeling of coupled free fluid and porous media flows and heterogeneous porous media with strong discontinuity of the porosity and permeability values.