Multigrid methods for optimal control problems governed by PDEs

Optimal control problems governed by partial differential equations (PDEs) has received much intention in the last two decades [16]. In this project, we shall consider one such important optimal control problem which has a lot of application in applied mathematics and to fluid dynamics. In particular, we shall invesitigate the control problem with the field of flow problems known as Navier-Stokes control problem. First, we shall describe the distributed optimal control problems constrained by stationary and/or time dependent Navier-Stokes equations in a two-dimensional computational domain [15-16]. A velocity and mixed (velocity-pressure) tracking-type control problem shall be considered and first-order optimality conditions shall be discussed. We shall investigate a full multigrid method with coarsening by a factor-of-three strategy on staggered grids which results in nested hierarchy of staggered grids and simplified the inter-grid transfer operators c.f. [7-8]. To smooth the residual of the state and adjoint variables, i.e., for smoothing, a distributive-Gauss-Seidel scheme (DGS) [5] shall be employed with a line search strategy to gradient update step for the control variable. In the last part of this project, we shall perform numerical experiments using MATLAB and/or FORTRAN. At the end we shall compile these results together with analysis in the form of manuscripts for possible publication in ISI International Journals.