Asymptotic homogenization of a problem for an elliptic equation with locally microperiodic continuously differentiable coefficient

Abstract

In this work, the asymptotic homogenization method (AHM) is applied to a Dirichlet boundary problem for a non-homogeneous, one-dimensional, elliptic, second-order differential equation with continuously differentiable coefficient and continuous non-homogeneity. Both the coefficient and non-homogeneity exhibit locally periodic and rapidly oscillating behaviors. As alternatives to the exact solution, three formal asymptotic solutions (FASs) are presented in the form of two-scale series in terms of powers of the small geometric parameter characterizing the separation of structural scales of the locally microperiodic medium modeled by the described problem. Such FASs are constructed from the solutions of the recurrent sequence of problems for the coefficients of the powers of the small parameter, which is form by the so-called homogenized problem and local problems over the periodicity cell. Finally, an example is presented in order to illustrate the fact that the three FASs obtained by applying the AHM are good approximations of the exact solution.