Parameter-free superconvergent H (div)-conforming HDG methods for the Brinkman equations

In this paper, we present new parameter-free superconvergent H (div)-conforming hybridizable discontinuous Galerkin (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 a natural extension of the H (div)-conforming HDG method (defined on simplicial meshes) for the Stokes flow (Cockburn, B. & Sayas, F.-J. (2014) Divergence-conforming HDG methods for Stokes flow. Math. Comp., 83, 1571–1598). We obtain an 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 a 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 for 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.