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In Theorem 6.2 of their 2003 paper on "Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal., 40(6), 2171–2194", D. Schötzau, Ch. Schwab, and A. Toselli prove a bound of the $\inf$-$\sup$ constant of the Stokes problem which decreases by $1/k$ if $k$ is the polynomial degree of an hp-discontinuous Galerkin method.

Numerical examples suggest that this estimate may not be sharp.

Is anybody aware of an improved estimate, possibly independent of $k$?

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  • $\begingroup$ Is this the paper you are referring to in your question? $\endgroup$ – Paul Jul 21 '13 at 13:06
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    $\begingroup$ Paul, if I remember correctly it is: epubs.siam.org/doi/abs/10.1137/… Schötzau, D., Schwab, C. & Toselli, A. (2003) Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal., 40(6), 2171–2194. $\endgroup$ – Christian Waluga Jul 21 '13 at 16:59
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For interior penalty type HDG methods we have recently shown that $1/\sqrt{k}$ is possible (Proposition 6.10) and this bound is valid for several element types. In Remark 6.11 there is also a discussion about known results from the literature for other methods.

To my knowledge there exist no uniform results for hp-DG methods yet. I would however also be interested in new ideas and recent developments.

References

Egger, Waluga. hp-Analysis of a Hybrid DG Method for Stokes Flow, 2013 (link).

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The problem was solved recently: Lederer/Schöberl: Polynomial robust stability analysis for H(div)-conforming finite elements for the Stokes equations

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The inf-sup constant for a continuous-velocity space is a lower bound for the corresponding DG space, therefore one can take the discontinuous analogue of any uniformly stable continuous space. Two natural choices are $Q_k - Q_{\min(k-2, \lambda k)}$ where $0 < \lambda < 1$ and $Q_k - P_{k-1}$. This was pointed out in Section 9 of Toselli, $hp$ Discontinuous Galerkin Approximations for the Stokes Problem, 2002. Note that the latter space is not uniformly stable with respect to aspect ratio. Recall also the anisotropically-enriched spaces of Ainsworth and Coggins (2000) that provide a logarithmic stability bound, though I think these are of limited practical utility.

The spaces above are not very satisfying because normally one would hope to raise the pressure approximation order when going to a discontinuous velocity. One practical reason for this is to simultaneously obtain optimal approximation order and uniform inf-sup stability with respect to aspect ratio. For example, DG using $Q_k - Q_{k-1}$ is uniformly stable with respect to aspect ratio (see the sequel, Schötzau, Schwab, Toselli, Mixed $hp$-DGFEM for incompressible flows II: Geometric edge meshes, 2004), for which there is no known analogue in the continuous Galerkin world, causing one to use the suboptimal space $Q_k - Q_{k-2}$ to obtain uniform stability with respect to aspect ratio.

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