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Volume 30, Issue 4
Optimal Control of the Laplace-Beltrami Operator on Compact Surfaces-Concept and Numerical Treatment

Michael Hinze & Morten Vierling

DOI: 10.4208/jcm.1111-m3678

J. Comp. Math., 30 (2012), pp. 392-403.

Published online: 2012-08

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  • Abstract

We consider optimal control problems of elliptic PDEs on hypersurfaces $Γ$ in $\mathbb{R}^n$ for $n$=2, 3. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhedral approximation of $Γ$. The discrete optimal control problem is formulated on the approximating surface and is solved numerically with a semi-smooth Newton algorithm. We derive optimal a priori error estimates for problems including control constraints and provide numerical examples confirming our analytical findings.

  • Keywords

Elliptic optimal control problem, Laplace-Beltrami operator, Surfaces, Control constraints, Error estimates, Semi-smooth Newton method.

  • AMS Subject Headings

58J32, 49J20, 49M15.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-30-392, author = {Michael Hinze and Morten Vierling}, title = {Optimal Control of the Laplace-Beltrami Operator on Compact Surfaces-Concept and Numerical Treatment}, journal = {Journal of Computational Mathematics}, year = {2012}, volume = {30}, number = {4}, pages = {392--403}, abstract = {

We consider optimal control problems of elliptic PDEs on hypersurfaces $Γ$ in $\mathbb{R}^n$ for $n$=2, 3. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhedral approximation of $Γ$. The discrete optimal control problem is formulated on the approximating surface and is solved numerically with a semi-smooth Newton algorithm. We derive optimal a priori error estimates for problems including control constraints and provide numerical examples confirming our analytical findings.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1111-m3678}, url = {http://global-sci.org/intro/article_detail/jcm/8438.html} }
TY - JOUR T1 - Optimal Control of the Laplace-Beltrami Operator on Compact Surfaces-Concept and Numerical Treatment AU - Michael Hinze & Morten Vierling JO - Journal of Computational Mathematics VL - 4 SP - 392 EP - 403 PY - 2012 DA - 2012/08 SN - 30 DO - http://doi.org/10.4208/jcm.1111-m3678 UR - https://global-sci.org/intro/article_detail/jcm/8438.html KW - Elliptic optimal control problem, Laplace-Beltrami operator, Surfaces, Control constraints, Error estimates, Semi-smooth Newton method. AB -

We consider optimal control problems of elliptic PDEs on hypersurfaces $Γ$ in $\mathbb{R}^n$ for $n$=2, 3. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhedral approximation of $Γ$. The discrete optimal control problem is formulated on the approximating surface and is solved numerically with a semi-smooth Newton algorithm. We derive optimal a priori error estimates for problems including control constraints and provide numerical examples confirming our analytical findings.

Michael Hinze and Morten Vierling. (2012). Optimal Control of the Laplace-Beltrami Operator on Compact Surfaces-Concept and Numerical Treatment. Journal of Computational Mathematics. 30 (4). 392-403. doi:10.4208/jcm.1111-m3678
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