Abstract
We prove error estimates in the maximum norm, namely in \(W^{1,\infty }(\Omega )^3\times L^\infty (\Omega )\), for the Stokes and Navier–Stokes equations in convex, three-dimensional domains \(\Omega \) with simplicial boundaries. We modify the weighted \(L^2\) estimates for regularized Green functions used earlier by us, which impose restrictions on the domain beyond convexity. The new ingredient is a Hölder regularity estimate proved recently by V. Maz’ya and J. Rossmann for the Stokes system on polyhedra. We also extend the error analysis to \(W^{1,r}(\Omega )^3\times L^r(\Omega )\) for \(1<r<\infty \).
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V. Girault was partially supported by the Aziz Lecture Fund at the University of Maryland. R. H. Nochetto was partially supported by NSF Grant DMS-1109325. L. R. Scott was partially supported by NSF Grant DMS-1226019.
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Girault, V., Nochetto, R.H. & Scott, L.R. Max-norm estimates for Stokes and Navier–Stokes approximations in convex polyhedra. Numer. Math. 131, 771–822 (2015). https://doi.org/10.1007/s00211-015-0707-8
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DOI: https://doi.org/10.1007/s00211-015-0707-8