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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.