Polynomial robust stability analysis for $H(\textrm{div})$-conforming finite elements for the Stokes equations
Provides a rigorous stability and error analysis for high-order H(div)-conforming finite elements, addressing a known bottleneck in polynomial robustness for Stokes discretizations.
This work proves that a discontinuous Galerkin method using H(div)-conforming finite elements for the Stokes problem is uniformly stable with respect to polynomial order k and achieves optimal error estimates, with convergence rate (h/k)^s.
In this work we consider a discontinuous Galerkin method for the discretization of the Stokes problem. We use $H(\textrm{div})$-conforming finite elements as they provide major benefits such as exact mass conservation and pressure-independent error estimates. The main aspect of this work lies in the analysis of high order approximations. We show that the considered method is uniformly stable with respect to the polynomial order $k$ and provides optimal error estimates $ \| \boldsymbol{u} - \boldsymbol{u}_h \|_{1_h} + \| Π^{Q_h}p-p_h \| \le c \left( h/k \right)^s \| \boldsymbol{u} \|_{s+1} $. To derive those estimates, we prove a $k$-robust LBB condition. This proof is based on a polynomial $H^2$-stable extension operator. This extension operator itself is of interest for the numerical analysis of $C^0$-continuous discontinuous Galerkin methods for $4^{th}$ order problems.