H-1 Least Squares Method for the Navier-Stokes Equations

Separated incompressible viscous flows around wings and airfoils are simulated by the numerical solution of the unsteady Navier-Stokes equations at low Reynolds number. The nonlinear equations are minimized in the Sobolev-Space H** minus **1 ( OMEGA ). It is shown that this method amounts to a sequence of optimal control problems with the Stokes equations as dynamics (state system). A fast solver is used for the Stokes equations based on a new mixed formulation. The formulation is discretized in finite elements. The mixed formulation can be decomposed into elementary standard Dirichlet problems. The optimal control problem is solved by a conjugate gradient method. Thus, although the method is fully implicit in time, it amounts to solving a few Dirichlet equations at each iteration.