Author : Ali Mohammed, Hussein
journal of kerbala university,
2006, Volume 2, Issue 4, Pages 137-142
The Bramble-Hilbert lemma is a fundamental result on multivariate polynomial approximation .It is frequently applied in the analysis of finite element methods used for numerical solutions. Our main result is to improve the following Bramble-Hilbert lemma to the case :let be abounded convex domain and let , ,0 ,where ( ) is the Sobolev spaces ,then there exists a polynomial P of degree m-1 for which
c(n,m)(diam ) g ,
where . = is the Sobolev semi norm of order . As a consequence we get that for f L , < .
is the rate of polynomial approximation of degree , and is the averaged modulus of smoothness, and >0.