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P. Lederer, J. Schöberl:
"Polynomial robust stability analysis for H(div)-conforming finite elements for the Stokes equations";
IMA J. Numer. Anal., 38 (2018), 4; S. 1832 - 1860.

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−uh∥1h+∥ΠQhp−ph∥0≤c(h/k)s∥u∥s+1⁠. To derive these estimates, we prove a k-robust Ladyzenskaja-Babuska-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.

http://dx.doi.org/10.1093/imanum/drx051