  The Brinkman equations can be regarded as a combination of the Stokes and
Darcy equations which model transitions between the fast flow in channels
(governed by Stokes equations) and the slow flow in porous media (governed by
Darcy's law). The numerical challenge for this model is the designing of a
numerical scheme which is stable for both the Stokes-dominated (high
permeability) and the Darcy-dominated (low permeability) equations. In this
paper, we solve the Brinkman model in $n$ dimensions ($n = 2, 3$) by using the
mixed discontinuous Galerkin (MDG) method, which meets this challenge. This MDG
method is based on the pseudostress-velocity formulation and uses a
discontinuous piecewise polynomial pair
$\underline{\bm{\mathcal{P}}}_{k+1}^{\mathbb{S}}$-$\bm{\mathcal{P}}_k$ $(k \geq
0)$, where the stress field is symmetric. The main unknowns are the
pseudostress and the velocity, whereas the pressure is easily recovered through
a simple postprocessing. A key step in the analysis is to establish the
parameter-robust inf-sup stability through specific parameter-dependent norms
at both continuous and discrete levels. Therefore, the stability results
presented here are uniform with respect to the permeability. Thanks to the
parameter-robust stability analysis, we obtain optimal error estimates for the
stress in broken $\underline{\bm{H}}(\bm{{\rm div}})$-norm and velocity in
$\bm{L}^2$-norm. Furthermore, the optimal $\underline{\bm{L}}^2$ error estimate
for pseudostress is derived under certain conditions. Finally, numerical
experiments are provided to support the theoretical results and to show the
robustness, accuracy, and flexibility of the MDG method.
