Mixed finite element projection methods for the unsteady Brinkman equations

Abstract: We present $H(\text{div})$-conforming mixed finite element methods for the unsteady Brinkman equations for incompressible single-phase flow with fixed in space porous solid inclusions. We employ a projection scheme with incremental pressure correction, which requires at each time step the solution of a predictor and a projection problem. The predictor problem, which uses a stress-velocity mixed formulation, accounts for the viscous effects, while the projection problem, which is based on a velocity-pressure mixed formulation, accounts for the incompressibility. The spatial discretization is based on the Raviart-Thomas or Brezzi-Douglas-Marini mixed finite element spaces on simplicial grids. 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 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 generally unstructured triangular (in 2D) and tetrahedral (in 3D) grids, and we use the Raviart--Thomas space $RT_1$, applying a second order 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. A series of challenging numerical experiments is presented to verify the convergence and performance of the proposed scheme