On the stability of the \(L^2\) projection in \(H^1(\omega)\) (Q2759089)

The authors prove the \(H^1\)-stability of the \(L^2\)-projection onto a family of finite element spaces \(V_h\), where the spaces \(V_h\) are spanned by continuous piecewise linear functions on triangulations of a polyhedral domain \(\Omega\) in \(R^n\), \(n = 1,2, \ldots\), satisfying certain local mesh conditions. An explicit formula for checking these conditions for a given finite element mesh is presented. Especially, the \(L^2\)-projection is \(H^1\)-stable for locally quasiuniform geometrically refined triangulations, where the measure of neighbouring elements does not change too drastically. NEWLINENEWLINENEWLINEThe presented approach can also be used in the case of higher order piecewise polynomial finite element spaces.