Let O ⸦ R ^d be a bounded domain of class C ^2p. In L ₂(O; C _n), we study a self-adjoint strongly elliptic operator A _N,ε of order 2p given by the expression b(D) ^*g(x/ε)b(D), ε > 0, with Neumann boundary conditions. Here, g(x) is a bounded and positive definite matrix-valued function in R ^d, periodic with respect to some lattice; b(D) = Σ|α|=p ^bαDα is a differential operator of order p. The symbol b(ξ) is subject to some condition ensuring strong ellipticity of the operator A _N,ε. We find approximations for the resolvent (A _N,ε − ζ ^I) ⁻¹ in different operator norms with error estimates depending on ε and ζ.