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

Lars Diening, Johannes Storn, Tabea Tscherpel

Published 2020-08-04Version 1

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 assumptions on the mesh grading in three dimensions we show $W^{1,2}$-stability for all polynomial degrees greater than one. 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.