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Finite Element Method and A Priori Error Estimates for Dirichlet Boundary Control Problems Governed by Parabolic PDEs

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Abstract

Finite element approximations of Dirichlet boundary control problems governed by parabolic PDEs on convex polygonal domains are studied in this paper. The existence of a unique solution to optimal control problems is guaranteed based on very weak solution of the state equation and \(L^2(0,T;L^2(\varGamma ))\) as control space. For the numerical discretization of the state equation we use standard piecewise linear and continuous finite elements for the space discretization of the state, while a dG(0) scheme is used for time discretization. The Dirichlet boundary control is realized through a space–time \(L^2\)-projection. We consider both piecewise linear, continuous finite element approximation and variational discretization for the controls and derive a priori \(L^2\)-error bounds for controls and states. We finally present numerical examples to support our theoretical findings.

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Acknowledgments

The first author would like to thank the Alexander von Humboldt Foundation for the support during the stay in University of Hamburg, Germany where this work was initialized. This work was supported by the National Basic Research Program of China under Grant 2012CB821204, the National Natural Science Foundation of China under Grant 11201464 and 91330115, and the scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. The second author gratefully acknowledges the support of the DFG Priority Program 1253 entitled “Optimization with Partial Differential Equations”. The third author was supported by National Natural Science Foundation of China under Grant 11301311. The authors also would like to thank two anonymous referees for their valuable suggestions which lead to an improved paper.

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Authors and Affiliations

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

    Wei Gong

  2. Schwerpunkt Optimierung und Approximation, Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Michael Hinze

  3. School of Mathematics Sciences, Shandong Normal University, Jinan, 250014, China

    Zhaojie Zhou

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  1. Wei Gong
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  2. Michael Hinze
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  3. Zhaojie Zhou
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Correspondence to Wei Gong.

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Gong, W., Hinze, M. & Zhou, Z. Finite Element Method and A Priori Error Estimates for Dirichlet Boundary Control Problems Governed by Parabolic PDEs. J Sci Comput 66, 941–967 (2016). https://doi.org/10.1007/s10915-015-0051-2

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  • DOI: https://doi.org/10.1007/s10915-015-0051-2

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