Let be the fundamental solution of a divergence operator of the following form: Two types of asymptotics of are considered in the paper: the asymptotic behavior at infinity, i.e. as , and the asymptotic behavior of at . In the first case, for operators with smooth, quasiperiodic coefficients the principal term of the asymptotic expression is found, and a power estimate of the remainder term is established. In the second case the principal term in the asymptotic expression for as is found for an operator with arbitrary bounded and measurable coefficients . These results are obtained by means of the concept of the -convergence of elliptic differential operators. Further, applications of the results are given to the asymptotics of the spectrum of the operator in a bounded domain .Bibliography: 13 titles.