Numerical comparison of different solution methods for optimal boundary control problems in thermal fluid dynamics

In this paper we propose and compare different methods for the solution of the control-adjoint-state optimality system which minimizes an objective functional in temperature. The minimization is constrained by the energy convection-diffusion equation with velocity field defined by the incompressible Navier- Stokes system. Three methods, based on different solution spaces, for solving the adjoint-state optimality system are compared. In the first one, as in the standard approach, the controlled temperature field is assumed to belong to a regular class of solutions with smooth derivatives and the resulting control-adjoint-state optimality system is solved in a segregated way. In the second one we introduce a fully coupled solution approach, where, in order to obtain a more robust numerical algorithm, the boundary control is extended to the interior and Dirichlet conditions are implicitly enforced through a volumetric force term. In the last approach we introduce Discontinuous Galerkin formulation for the energy equation in order to seek discontinuous solutions. Numerical two and three-dimensional test cases arenreported in order to show the validity of the proposed approaches. The results are compared in term of solution smoothness and achievement of low values of the objective functional