Polynomial robust stability analysis for H(div)-conforming finite elements for the Stokes equations

In this article, we consider a discontinuous Galerkin method for the discretization of the Stokes problem. We use H(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 ||u - u_h||1_h + ||Π^Qhp - p_h||0 ≤ c (h/k)^s ||u||_s+1. To derive these estimates, we prove a k-robust Ladyženskaja-Babuška-Brezzi (LBB) condition. This proof is based on a polynomial H²-stable extension operator. This extension operator itself is of interest for the numerical analysis of C⁰-continuous discontinuous Galerkin methods for fourth-order problems.