Let Ω be a C^1,s bounded domain (s > 1/2) in ℝ^d, and let A^ε= − div A(x, x/ ε) ∇ be a matrix elliptic operator on Ω with Dirichlet boundary condition. We suppose that ε is small and the function A is Lipschitz in the first variable and periodic in the second one, so the coefficients of A^ε are locally periodic. For μ in the resolvent set, we are interested in finding the rates of approximations, as ε → 0, for (Aε−μρε)−1 and ∇(Aε−μρε)−1 in the operator topology on L₂. Here ρ^ε(x)= ρ(x,x/ε) is a positive definite locally periodic function with ρ satisfying the same assumptions as A. Keeping track of the rate dependence on both ε and μ, we then proceed to similar questions for the solution to the initial boundary-value problem ρ^ε∂ _tv_ε= − A^εv_ε.