Error Estimates for Finite Element Discretizations of Elliptic Problems with Oscillatory Coefficients

In this paper we will consider elliptic boundary value problems with oscillatory diffusion coefficient, say A. We will derive regularity estimates in Sobolev norms which are weighted by certain derivatives of A. The constants in the regularity estimates then turn out to be independent of the variations in A. These regularity results will be employed for the derivation of error estimates for hp-finite element discretizations which are explicit with respect to the local variations of the diffusion coefficient.