Heterogeneous Multiscale Method for Optimal Control Problem Governed by Elliptic Equations with Highly Oscillatory Coefficients
Year: 2018
Author: Liang Ge, Ningning Yan, Lianhai Wang, Wenbin Liu, Danping Yang
Journal of Computational Mathematics, Vol. 36 (2018), Iss. 5 : pp. 644–660
Abstract
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' 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1703-m2015-0433
Journal of Computational Mathematics, Vol. 36 (2018), Iss. 5 : pp. 644–660
Published online: 2018-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 17
Keywords: Constrained convex optimal control Heterogeneous multiscale finite element A priori error estimate Elliptic equations with highly oscillatory coefficients.
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