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Adaptive Finite Element Method for Dirichlet Boundary Control of Elliptic Partial Differential Equations

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Abstract

In this paper, we consider the Dirichlet boundary control problem of elliptic partial differential equations, and get a coupling system of the state and adjoint state by cancelling the control variable in terms of the control rule, and prove that this coupling system is equivalent to the known Karush–Kuhn–Tucker (KKT) system. For corresponding finite element approximation, we find a measure of the numerical errors by employing harmonic extension, based on this measure, we develop residual-based a posteriori error analytical technique for the Dirichlet boundary control problem. The derived estimators for the coupling system and the KKT system are proved to be reliable and efficient over adaptive mesh. Numerical examples are presented to validate our theory.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, China

    Shaohong Du

  2. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907-2067, USA

    Zhiqiang Cai

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  1. Shaohong Du
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  2. Zhiqiang Cai
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Correspondence to Shaohong Du.

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This work is supported in part by the Natural Science Foundation of Chongqing (cstc2018jcyjAX490), the Education Science Foundation of Chongqing (KJZD-K201900701), and the Team Building Projection for Graduate Tutors in Chongqing (JDDSTD201802).

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Du, S., Cai, Z. Adaptive Finite Element Method for Dirichlet Boundary Control of Elliptic Partial Differential Equations. J Sci Comput 89, 36 (2021). https://doi.org/10.1007/s10915-021-01644-3

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  • DOI: https://doi.org/10.1007/s10915-021-01644-3

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