A multiscale finite element method for optimal control problems governed by the elliptic homogenization equations

In this article, we develop and analyze a priori estimates for optimal control problems with multiscale governed by the elliptic homogenization equations. The multiscale finite element is applied to capture the effect of microscale through modification of finite element basis functions without resolving all the small scale features. The optimal estimate is derived for elliptic homogenization problems without resonance effect O(ϵ/h)O(ϵ/h) by using an over-sampling technique and the boundary layer assumption. Furthermore, the a priori estimate is obtained for the optimal control problems governed by the elliptic homogenization equations.