Abstract:

In this paper, we present new parameter-free superconvergent H(div)-conforming HDG methods for the Brinkman equations on both simplicial and rectangular meshes. The methods are based on a velocity gradient-velocity-pressure formulation, which can be considered as a natural extension of the H(div)-conforming HDG method (defined on simplicial meshes) for the Stokes flow [Math. Comp. 83(2014), pp. 1571-1598].

We obtain optimal  L2-error estimate for the velocity in both the Stokes-dominated regime (high viscosity/permeability ratio) and Darcy-dominated regime (low viscosity/permeability ratio). We also obtain superconvergent L2-estimate of one order higher for a suitable projection of the velocity error in the Stokes-dominated regime. Moreover, thanks to H(div)-conformity of the velocity, our velocity error estimates are independent of the pressure regularity. Furthermore, we  provide a discrete H1-stability result of the velocity field, which is essential in the error analysis of the natural generalization of these new HDG methods to the incompressible Navier-Stokes equations.

Preliminary numerical results on both triangular and rectangular meshes in two dimensions confirm our theoretical predictions.