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Adaptive finite element method for elliptic optimal control problems: convergence and optimality

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Abstract

In this paper we consider the convergence analysis of adaptive finite element method for elliptic optimal control problems with pointwise control constraints. We use variational discretization concept to discretize the control variable and piecewise linear and continuous finite elements to approximate the state variable. Based on the well-established convergence theory of AFEM for elliptic boundary value problems, we rigorously prove the convergence and quasi-optimality of AFEM for optimal control problems with respect to the state and adjoint state variables, by using the so-called perturbation argument. Numerical experiments confirm our theoretical analysis.

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Acknowledgments

The authors would like to thank two anonymous referees for their careful reviews and many helpful suggestions. The first author was supported by the National Basic Research Program of China under Grant 2012CB821204 and the National Natural Science Foundation of China under Grants 11201464 and 91530204. The second author acknowledged the support of the National Natural Science Foundation of China under Grants 11171337 and 91530204.

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Authors and Affiliations

  1. NCMIS, LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Wei Gong

  2. NCMIS, LSEC, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Ningning Yan

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  1. Wei Gong
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Correspondence to Wei Gong.

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Gong, W., Yan, N. Adaptive finite element method for elliptic optimal control problems: convergence and optimality. Numer. Math. 135, 1121–1170 (2017). https://doi.org/10.1007/s00211-016-0827-9

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  • DOI: https://doi.org/10.1007/s00211-016-0827-9

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