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, <!-- MATH $\hat{W}_p^1$ --> , for <!-- MATH $2 \leqslant p \leqslant \infty$ --> . This implies that for functions in <!-- MATH $\hat{W}_p^1 \cap W_p^2$ --> the error in approximation behaves like in , for <!-- MATH $2 \leqslant p \leqslant \infty$ --> , and like in , for <!-- MATH $2 \leqslant p < \infty$ --> <img width="99" height="37" align="MIDDLE" border="0" src="images/img13.gif" alt="$ 2 \leqslant p < \infty $">. In all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur.