Block triangular preconditioning for time-dependent Stokes control

We consider the numerical solution of time-dependent Stokes control problems, an important class of flow control problems within the field of PDE-constrained optimization. The problems we examine lead to large and sparse matrix systems which, with suitable rearrangement, can be written in block tridiagonal form, with the diagonal blocks given by saddle point systems. Using previous results for preconditioning PDE-constrained optimization and fluid dynamics problems, along with well-studied saddle point theory, we construct a block triangular preconditioner for the matrix systems. Numerical experiments verify the effectiveness of our solver.