East Asian Journal on Applied Mathematics, 1 (2011), pp. 235-247. Numerical Methods for Constrained Elliptic Optimal Control Problems with Rapidly Oscillating Coefficients Yanping Chen 1*, Yuelong Tang 21 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China. 2 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China. Received 7 October 2010; Accepted (in revised version) 25 April 2011 Available online 27 July 2011 doi:10.4208/eajam.071010.250411a Abstract In this paper we use two numerical methods to solve constrained optimal control problems governed by elliptic equations with rapidly oscillating coefficients: one is finite element method and the other is multiscale finite element method. We derive the convergence analysis for those two methods. Analytical results show that finite element method can not work when the parameter $\varepsilon$ is small enough, while multiscale finite element method is useful for any parameter $\varepsilon$. Key words: Optimal control problems, finite element method, multiscale finite element method, homogenization, convergence analysis. *Corresponding author. Email: yanpingchen@scnu.edu.cn (Y. Chen)