Stabilized Finite Element Solution of Control Problem of Convection Diffusion Equation

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.

Keywords

• Optimal control,Convection-diffusion equation,Stabilized fem

References

1. Abergel F. and Temam R.(1990), On some optimal control problems in fluidmechanics, Theoret. Comput. Fluid Mech. 1 (6) 303-325. Hecht F.(2012), New development in FreeFem++, J. Numer. Math. 20, no. 3-4, 251-265. John V., Kaya S. and Layton W.(2005), A two-level variational multiscale methodfor convection-diffusion equations, Comput. Meth. Appl. Mech. Engrg., 195, 4594-4603, 2005.