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.
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.
A
but conjectured
D
greater than one