Some optimal error estimates for piecewise linear finite element approximations

Abstract

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