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Adaptive Virtual Element Method for Optimal Control Problem Governed by General Elliptic Equation

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Abstract

In this paper a posteriori error analysis of virtual element method (VEM) for the optimal control problem governed by general elliptic equation is presented. The virtual element discrete scheme is constructed with virtual element approximation of the state equation and variational discretization of the control variable. Based on the a posteriori error estimates of virtual element method for general elliptic equation and approximated error equivalence of the solution of the optimal control problem to solutions of the state and adjoint problems we build up upper and lower a posteriori error estimates of the optimal control problem. Under the Dörfler’s marking strategy, the traditional projected gradient algorithm and adaptive VEM algorithm drived by the state and adjoint error estimators are used to solve the optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.

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Acknowledgements

The funding was provided by National Natural Science Foundation of China (Grant Nos. 11971276, 11301311). Natural Science Foundation of Shandong Province (Grant Nos. ZR2016JL004, ZR2017MA020).

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  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250000, China

    Qiming Wang & Zhaojie Zhou

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

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Wang, Q., Zhou, Z. Adaptive Virtual Element Method for Optimal Control Problem Governed by General Elliptic Equation. J Sci Comput 88, 14 (2021). https://doi.org/10.1007/s10915-021-01528-6

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