The weak Galerkin method for solving the incompressible Brinkman flow

The Brinkman equations are used to describe the dynamics of fluid flows in complex porous media, with the high variability in the permeability coefficients, which may take extremely large or small values. This paper is devoted to the numerical analysis of a family of weak Galerkin (WG) finite element methods for solving the time-dependent Brinkman problems. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k≥1k≥1 for the velocity and polynomials of degree k−1k−1 for the pressure. The velocity element is enhanced by polynomials of degree kk on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. We further establish a priori error estimates in L2L2 norm and H1H1 norm, and we provide a few numerical experiments to illustrate the behavior of the proposed scheme and confirm our theoretical findings regarding optimal convergence of the approximate solutions.