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Boundary Control Problems in Convective Heat Transfer with Lifting Function Approach and Multigrid Vanka-Type Solvers

Part of: Existence theories Partial differential equations, boundary value problems

Published online by Cambridge University Press:  14 September 2015

Eugenio Aulisa ,
Giorgio Bornia  and
Sandro Manservisi
Eugenio Aulisa
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 70409-1042, USA
Giorgio Bornia*
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 70409-1042, USA
Sandro Manservisi
Affiliation:
Dipartimento di Ingegneria Industriale, Universita’ di Bologna, Bologna, Italy
*
*Corresponding author. Email addresses: eugenio.aulisa@ttu.edu (E. Aulisa), giorgio.bornia@ttu.edu (G. Bornia), sandro.manservisi@unibo.it (S. Manservisi)
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Abstract

This paper deals with boundary optimal control problems for the heat and Navier-Stokes equations and addresses the issue of defining controls in function spaces which are naturally associated to the volume variables by trace restriction. For this reason we reformulate the boundary optimal control problem into a distributed problem through a lifting function approach. The stronger regularity requirements which are imposed by standard boundary control approaches can then be avoided. Furthermore, we propose a new numerical strategy that allows to solve the coupled optimality system in a robust way for a large number of unknowns. The optimality system resulting from a finite element discretization is solved by a local multigrid algorithm with domain decomposition Vanka-type smoothers. The purpose of these smoothers is to solve the optimality system implicitly over subdomains with a small number of degrees of freedom, in order to achieve robustness with respect to the regularization parameters in the cost functional. We present the results of some test cases where temperature is the observed quantity and the control quantity corresponds to the boundary values of the fluid temperature in a portion of the boundary. The control region for the observed quantity is a part of the domain where it is interesting to match a desired temperature value.

Keywords

Boundary optimal controlmultigrid and domain decomposition algorithmsheat and Navier-Stokes equations

MSC classification

Secondary: 49J20: Optimal control problems involving partial differential equations 65N55: Multigrid methods; domain decomposition
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 3 , September 2015 , pp. 621 - 649
Copyright
Copyright © Global-Science Press 2015 

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