Multiscale asymptotic expansion and finite element methods for the mixed boundary value problems of second order elliptic equation in perforated domains

In this paper, we will discuss the mixed boundary value problems for the second order elliptic equation with rapidly oscillating coefficients in perforated domains, and will present the higher-order multiscale asymptotic expansion of the solution for the problem, which will play an important role in the numerical computation . The convergence theorems and their rigorous proofs will be given. Finally a multiscale finite element method and some numerical results will be presented. This work is Supported by National Natural Science Foundation of China (grant # 10372108, # 90405016), and Special Funds for Major State Basic Research Projects( grant # TG2000067102)