Heterogeneous Multiscale Method for Optimal Control Problem Governed by Elliptic Equations with Highly Oscillatory Coefficients (Journal of Computational Mathematics, 2018)
In this paper, we investigate heterogeneous multiscale method (HMM) for the optimalcontrol problem with distributed control constraints governed by elliptic equations withhighly oscillatory coefficients. The state variable and co-state variable are approximated bythe multiscale discretization scheme that relies on coupled macro and micro finite elements,whereas the control variable is discretized by the piecewise constant. By applying the well-knownLions' Lemma to the discretized optimal control problem, we obtain the necessaryand sufficient optimality conditions. A priori error estimates in both $L^2$ and $H^1$ normsare derived for the state, co-state and the control variable with uniform bound constants.Finally, numerical examples are presented to illustrate our theoretical results.