@Article{JCM-36-5,
author = {Liang, Ge and Yan, Ningning and Wang, Lianhai and Wenbin, Liu and Yang, Danping},
title = {Heterogeneous Multiscale Method for Optimal Control Problem Governed by Elliptic Equations with Highly Oscillatory Coefficients},
journal = {Journal of Computational Mathematics},
year = {2018},
volume = {36},
number = {5},
pages = {644--660},
abstract = {<p style="text-align: justify;">In this paper, we investigate heterogeneous multiscale method (HMM) for the optimal
control problem with distributed control constraints governed by elliptic equations with
highly oscillatory coefficients. The state variable and co-state variable are approximated by
the 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-known
Lions&#39; Lemma to the discretized optimal control problem, we obtain the necessary
and sufficient optimality conditions. A priori error estimates in both $L^2$ and $H^1$ norms
are derived for the state, co-state and the control variable with uniform bound constants.
Finally, numerical examples are presented to illustrate our theoretical results.</p>},
issn = {1991-7139},
doi = {https://doi.org/10.4208/jcm.1703-m2015-0433},
url = {https://global-sci.com/article/84488/heterogeneous-multiscale-method-for-optimal-control-problem-governed-by-elliptic-equations-with-highly-oscillatory-coefficients}
}