Parameter-free superconvergent $H(\mathrm{div})$-conforming HDG methods for the Brinkman equations

TL;DR

This paper introduces new parameter-free, superconvergent H(div)-conforming HDG methods for the Brinkman equations, achieving optimal error estimates and superconvergence in both regimes, validated by numerical experiments.

Contribution

The paper develops a novel parameter-free HDG method for Brinkman equations that extends existing H(div)-conforming techniques to both simplicial and rectangular meshes with proven superconvergence.

Findings

Optimal error estimates in $L^2$-norms for all variables.

Superconvergent $L^2$-estimate of one order higher for velocity.

Numerical results confirm theoretical predictions.

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 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.