On the stability of the 𝐿² projection in 𝐻¹(Ω)

Bramble, James; Pasciak, Joseph; Steinbach, Olaf

doi:10.1090/S0025-5718-01-01314-X

On the stability of the $L^2$ projection in $H^1(\Omega )$
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by James H. Bramble, Joseph E. Pasciak and Olaf Steinbach;

Math. Comp. 71 (2002), 147-156

DOI: https://doi.org/10.1090/S0025-5718-01-01314-X

Published electronically: May 7, 2001

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Abstract:

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.

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