Let O ⊂ ℝ ^d be a bounded domain of class C ^2p. The object under study is a selfadjoint strongly elliptic operator AD, e of order 2p, p ≤ 2, in L ₂(O, ⁿ), given by the expression b(D)* g(x/ε)b(D), ε > 0, with the Dirichlet boundary conditions. Here g(x) is a bounded and positive definite (m×m)-matrix-valued function in ℝ ^d, periodic with respect to some lattice; b(D) = ∑ |α| = p bαD ^α is a differential operator of order p with constant coefficients; and the ba are constant (m × n)-matrices. It is assumed that m ≤ n and the symbol b(Ξ) has maximal rank. Approximations are found for the resolvent (A _D,ε - ζI) ^-1 in the L ₂(O; ⁿ)-operator norm and in the norm of operators acting from L ₂(O; ⁿ) to H ^p(O; ⁿ), with error estimates depending on ε and ζ.