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Adaptive finite element approximation of optimal control problems with the integral fractional Laplacian

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Abstract

In this paper, we study an adaptive finite element approximation of optimal control problems with integral fractional Laplacian and pointwise control constraints. The state variable is approximated by piecewise linear polynomials, and the control variable is implicitly discretized. Upper and lower bounds of a posteriori error estimates for finite element approximation of the optimal control problem are derived. An h-adaptive algorithm driven by the a posterior error estimator is presented with Dörfler’s marking criterion. We prove that the adaptive algorithm yields a sequence of approximations that converge at the optimal algebraic rate. Numerical examples are given to illustrate the theoretical findings.

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Funding

The work was supported by the National Natural Science Foundation of China under Grant No. 11971276 and 12171287.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, 250014, Jinan, China

    Zhaojie Zhou

  2. School of Mathematical Sciences, Beijing Normal University, 519087, Zhuhai, China

    Qiming Wang

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  1. Zhaojie Zhou
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  2. Qiming Wang
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Correspondence to Zhaojie Zhou.

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Communicated by: Peter Benner

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The original online version of this article was revised: the given and family names of Zhaojie Zhou and Qiming Wang were incorrectly structured.

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Zhou, Z., Wang, Q. Adaptive finite element approximation of optimal control problems with the integral fractional Laplacian. Adv Comput Math 49, 59 (2023). https://doi.org/10.1007/s10444-023-10064-w

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