On the stability of the $L^2$ projection in $H^1(\Omega)$
James H. Bramble, Joseph E. Pasciak, Olaf Steinbach · Mathematics of Computation · 2001
We prove the stability in $H^1(\Omega )$ of the $L^2$ projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions. We give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions. In particular, stability of the $L^2$ projection in $H^1(\Omega )$ holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically.