Parameter-free superconvergent $H(\mathrm{div})$-conforming HDG methods for the Brinkman equations
This work provides a unified numerical method for the Brinkman equations that is parameter-free and achieves superconvergence, benefiting computational fluid dynamics applications involving both Stokes and Darcy flows.
The authors present new parameter-free superconvergent H(div)-conforming HDG methods for the Brinkman equations, achieving optimal error estimates in L^2-norms for both Stokes- and Darcy-dominated regimes, with superconvergent L^2-estimates for a projection of the velocity error. Numerical results on 2D meshes confirm the theory.
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 error estimates in $L^2$-norms for all the variables in both the Stokes-dominated regime (high viscosity/permeability ratio) and Darcy-dominated regime (low viscosity/permeability ratio). We also obtain superconvergent L^2-estimate of one order higher for a suitable projection of the velocity error, which is typical for (hybrid) mixed methods for elliptic problems. Moreover, thanks to H(div)-conformity of the velocity, our velocity error estimates are independent of the pressure regularity. Preliminary numerical results on both triangular and rectangular meshes in two-space dimensions confirm our theoretical predictions.