The Eurasia Proceedings of Science Technology Engineering and Mathematics

Abstract

In
this work, we consider the stabilized numerical solutions of optimal control
problems of convection diffusion equation of this type of equations have been
commonly studied in the literature. We use finite element method (FEM). Because
of the viscosity term in the problem, the FEM solution blows up if Reynold
number is large. In this case, the solution is unstabilized so that a
stabilization technique is needed. As for stabilization technique, we apply
both variational multiscale (VMS) and grad-div stabilization technique. The
variational multiscale method is reviewed as a framework for developing
computational methods for large-eddy simulation of turbulent flow. Some of the
most used numerical stabilization techniques for flow problems are streamline
upwind Galerkin (SUPG) and pressure stabilization methods, large eddy
simulation (LES) methods, and VMS methods. First of all, we obtain the
optimality system. We then use FEM to obtain the discrete system. We obtain the
theoretical stability results. We use the package freefem ++ to get the
numerical results. We compare the stabilized solutions.