(A) theoretical study on Bramble Hilbert Lemma

Abstract

This paper is concerned with the proof of so called Bramble Hilbert Lemma. We propose that Poincare` s inequality in [3] implies one of results of Morrey which is used importantly in the proof. In this point of view, we recognize that to remove the average term in Poincare`s inequality fulfills a crucial rule in the proof of Bramble-Hilbert Lemma. Another result of Morrey accomplishes it by adding some polynomial of degree k-1 less one than the degree of the Sobolev space in the outset. So, the condition annihilating the set of polynomials $P_{k-1}$ of degree k-1 is needed in Bramble-Hilbert Lemma.