Boundary Control Problems in Convective Heat Transfer with Lifting Function Approach and Multigrid Vanka-Type Solvers (Communications in Computational Physics, 2018)
This paper deals with boundary optimal control problems for the heat andNavier-Stokes equations and addresses the issue of defining controls in function spaceswhich are naturally associated with the volume variables by trace restriction. For thisreason we reformulate the boundary optimal control problem into a distributed problemthrough a lifting function approach. The stronger regularity requirements whichare imposed by standard boundary control approaches can then be avoided. Furthermore,we propose a new numerical strategy that allows solving the coupled optimalitysystem in a robust way for a large number of unknowns. The optimality systemresulting from a finite element discretization is solved by a local multigrid algorithmwith domain decomposition Vanka-type smoothers. The purpose of these smoothersis to solve the optimality system implicitly over subdomains with a small number ofdegrees of freedom, in order to achieve robustness with respect to the regularizationparameters in the cost functional. We present the results of some test cases where temperatureis the observed quantity and the control quantity corresponds to the boundaryvalues of the fluid temperature in a portion of the boundary. The control region forthe observed quantity is a part of the domain where it is interesting to match a desiredtemperature value.