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We study the asymptotic behaviour, as the small parameter $\varepsilon$ tends to zero, of the resolvents of uniformly elliptic second-order differential operators with locally periodic coefficients depending on the slow variable $x$ and the fast variable $x/\varepsilon$, with periodicity only in the fast variable. We provide a construction for the leading terms in the operator asymptotics of these resolvents in the sense of $L^2$-operator-norm convergence with order $\varepsilon^2$ remainder estimates. We apply the modified method of the first approximation with the usage of the shift.

On Resolvent Approximations of Elliptic Differential Operators with Locally Periodic Coefficients