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arXiv · 2008.01801

On the Sobolev and $L^p$-Stability of the $L^2$-projection

Lars Diening·Johannes Storn·Tabea Tscherpel
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Abstract

We show stability of the $L^2$-projection onto Lagrange finite element spaces with respect to (weighted) $L^p$ and $W^{1,p}$-norms for any polynomial degree and for any space dimension under suitable conditions on the mesh grading. This includes $W^{1,2}$-stability in two space dimensions for any polynomial degree and meshes generated by newest vertex bisection. Under realistic but conjectured assumptions on the mesh grading in three dimensions we show $W^{1,2}$-stability for all polynomial degrees. We also propose a modified bisection strategy that leads to better $W^{1,p}$-stability. Moreover, we investigate the stability of the $L^2$-projection onto Crouzeix-Raviart elements.

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Lars Diening, Johannes Storn, Tabea Tscherpel. 2020-08-04. On the Sobolev and $L^p$-Stability of the $L^2$-projection. https://doi.org/10.1137/20m1358013

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