Some optimal error estimates for piecewise linear finite element approximations

It is shown that the Ritz projection onto spaces of piecewise linear finite elements is bounded in the Sobolev space, $\hat{W}_p^1$, for $2 \leqslant p \leqslant \infty$. This implies that for functions in $\hat{W}_p^1 \cap W_p^2$ the error in approximation behaves like $O(h)$ in $W_p^1$, for $2 \leqslant p \leqslant \infty$, and like $O({h^2})$ in ${L_p}$, for $2 \leqslant p < \infty$. In all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur.