On a multigrid solver for stationary Navier–Stokes velocity–pressure tracking-type control problems

In this article, a multigrid solver for distributed optimal control problems governed by time-independent Navier–Stokes equations is presented. A mixed (velocity–pressure) tracking-type control problem is considered and first-order optimality conditions are discussed. We investigate a full multigrid method with coarsening by a factor-of-three strategy to stationary Navier–Stokes control problems. The potential advantage of multigrid with coarsening by a factor-of-three strategy is that it results in nested hierarchy of staggered grids and thus simplifies the inter-grid transfer operators, reduces the number of levels, and hence the CPU time. The construction of the multigrid algorithm for Stokes control problems of our earlier work gives us a natural extension but still significant challenges are rooted in the nonlinear part of the Navier–Stokes equations (constraints) and mixed (velocity–pressure) tracking-type control formulation. Numerical experiments are reported to show the behavior and efficiency of the proposed multigrid algorithm for small Reynolds numbers and moderate values of regularization parameter.