Stabilized Finite Element Solution of Control Problem of Convection Diffusion Equation

Abstract

Inthis work, we consider the stabilized numerical solutions of optimal controlproblems of convection diffusion equation of this type of equations have beencommonly studied in the literature. We use finite element method (FEM). Becauseof the viscosity term in the problem, the FEM solution blows up if Reynoldnumber is large. In this case, the solution is unstabilized so that astabilization technique is needed. As for stabilization technique, we applyboth variational multiscale (VMS) and grad-div stabilization technique. Thevariational multiscale method is reviewed as a framework for developingcomputational methods for large-eddy simulation of turbulent flow. Some of themost used numerical stabilization techniques for flow problems are streamlineupwind Galerkin (SUPG) and pressure stabilization methods, large eddysimulation (LES) methods, and VMS methods. First of all, we obtain theoptimality system. We then use FEM to obtain the discrete system. We obtain thetheoretical stability results. We use the package freefem ++ to get thenumerical results. We compare the stabilized solutions.