Polynomial robust stability analysis for $H(\textrm{div})$-conforming finite elements for the Stokes equations

TL;DR

This paper develops a polynomial-robust stability analysis for an $H(\textrm{div})$-conforming finite element method applied to the Stokes equations, ensuring uniform stability and optimal error estimates for high order discretizations.

Contribution

It introduces a $k$-robust LBB condition using a polynomial $H^2$-stable extension operator, advancing the analysis of high order $H(\textrm{div})$-conforming methods for Stokes.

Findings

Method is uniformly stable with respect to polynomial order $k$.

Provides optimal error estimates for high order approximations.

Proves a $k$-robust LBB condition based on a polynomial $H^2$-stable extension operator.

Abstract

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} + \| \Pi^{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.