ABSTRACT: We prove the stability in $H^1(\Omega)$ of the $L^2$ projection onto the finite element space of piecewise linear basis functions assuming appropriate local mesh conditions. We give explicit formulae to check these conditions locally 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 quasi--uniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically.
ACKNOWLEGEMENTS: This work was supported by the National Science Foundation under grants numbered DMS-9626567 and DMS-9973328 and by the State of Texas under ARP/ATP grant \#010366-168. This work was done while the third author was a Postdoctoral Research Associate at the Institute for Scientific Computation (ISC), Texas A & M University. The financial support by the ISC is gratefully acknowledged.