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 RT1 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 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.