Abstract
In this paper we propose an adaptive multilevel correction scheme to solve optimal control problems discretized with finite element method. Different from the classical adaptive finite element method (AFEM for short) applied to optimal control which requires the solution of the optimization problem on new finite element space after each mesh refinement, with our approach we only need to solve two linear boundary value problems on current refined mesh and an optimization problem on a very low dimensional space. The linear boundary value problems can be solved with well-established multigrid method designed for elliptic equation and the optimization problems are of small scale corresponding to the space built with the coarsest space plus two enriched bases. Our approach can achieve the similar accuracy with standard AFEM but greatly reduces the computational cost. Numerical experiments demonstrate the efficiency of our proposed algorithm.
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Acknowledgements
The authors would like to thank two anonymous referees for their careful reviews and many valuable suggestions. The first author was supported by the National Basic Research Program of China under Grant 2012CB821204, the National Natural Science Foundation of China under Grant 11201464 and 91530204. The second author gratefully acknowledges the support of the National Natural Science Foundation of China (91330202, 11371026, 11001259, 11031006, 2011CB309703), Science Challenge Project (No. JCKY2016212A502), the National Center for Mathematics and Interdisciplinary Science, CAS and the President Foundation of AMSS-CAS. The third author was supported by the National Natural Science Foundation of China under Grant 11571356 and 91530204.
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Gong, W., Xie, H. & Yan, N. Adaptive Multilevel Correction Method for Finite Element Approximations of Elliptic Optimal Control Problems. J Sci Comput 72, 820–841 (2017). https://doi.org/10.1007/s10915-017-0386-y
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DOI: https://doi.org/10.1007/s10915-017-0386-y
Keywords
- Optimal control problems
- Elliptic equation
- Control constraints
- A posteriori error estimates
- Adaptive finite element method
- Multilevel correction method