Divergence-conforming HDG methods for Stokes flows

Abstract:

In this paper, we show that by sending the normal stabilization function to infinity in the hybridizable discontinuous Galerkin methods previously proposed in [Comput. Methods Appl. Mech. Engrg. 199 (2010), 582–597], for Stokes flows, a new class of divergence-conforming methods is obtained which maintains the convergence properties of the original methods. Thus, all the components of the approximate solution, which use polynomial spaces of degree $k$, converge with the optimal order of $k+1$ in $L^2$ for any $k \ge 0$. Moreover, the postprocessed velocity approximation is also divergence-conforming, exactly divergence-free and converges with order $k+2$ for $k\ge 1$ and with order $1$ for $k=0$. The novelty of the analysis is that it proceeds by taking the limit when the normal stabilization goes to infinity in the error estimates recently obtained in [Math. Comp., 80 (2011) 723–760].