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

TL;DR

This paper investigates the stability of the $L^2$-projection in various norms for finite element spaces, providing new stability results under specific mesh conditions and proposing improved bisection strategies.

Contribution

It establishes stability of the $L^2$-projection in weighted $L^p$ and $W^{1,p}$-norms across different dimensions and polynomial degrees, including new results for meshes generated by newest vertex bisection.

Findings

$L^2$-projection is stable in weighted $L^p$ and $W^{1,p}$-norms under certain mesh conditions.

W^{1,2}-stability achieved in 2D for all polynomial degrees with newest vertex bisection.

Proposed modified bisection improves $W^{1,p}$-stability.

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.