A robust multigrid method for the time-dependent Stokes problem

In the present paper we propose an all-at-once multigrid method for generalized Stokes flow problems. Such problems occur as subproblems in implicit time-stepping approaches for time-dependent Stokes problems. The discretized optimality system is a large scale linear system whose condition number depends on the grid size of the spacial discretization and of the length of the time step. Recently, for this problem an all-at-once multigrid method has been proposed, where in each smoothing step the Poisson problem has to be solved (approximatively) for the pressure field. In the present paper, we propose an all-at-once multigrid method where the solution of such subproblems is not needed. We prove that the proposed method shows robust convergence behavior in the grid size of the spacial discretization and of the length of the time-step

A robust all-at-once multigrid method for the Stokes control problem

In this paper we present an all-at-once multigrid method for a distributed Stokes control problem (velocity tracking problem). For solving such a problem, we use the fact that the solution is characterized by the optimality system (Karush-Kuhn-Tucker-system). The discretized optimality system is a large-scale linear system whose condition number depends on the grid size and on the choice of the regularization parameter forming a part of the problem. Recently, block-diagonal preconditioners have been proposed, which allow to solve the problem using a Krylov space method with convergence rates that are robust in both, the grid size and the regularization parameter or cost parameter. In the present paper, we develop an all-at-once multigrid method for a Stokes control problem and show robust convergence, more precisely, we show that the method converges with rates which are bounded away from one by a constant which is independent of the grid size and the choice of the regularization or cost parameter

On the role of commutator arguments in the development of parameter-robust preconditioners for Stokes control problems

The development of preconditioners for PDE-constrained optimization problems is a field of numerical analysis which has recently generated much interest. One class of problems which has been investigated in particular is that of Stokes control problems, that is the problem of minimizing a functional with the Stokes (or Navier-Stokes) equations as constraints. In this manuscript, we present an approach for preconditioning Stokes control problems using preconditioners for the Poisson control problem and, crucially, the application of a commutator argument. This methodology leads to two block diagonal preconditioners for the problem, one of which was previously derived by W. Zulehner in 2011 (SIAM. J. Matrix Anal. & Appl., v.32) using a nonstandard norm argument for this saddle point problem, and the other of which we believe to be new. We also derive two related block triangular preconditioners using the same methodology, and present numerical results to demonstrate the performance of the four preconditioners in practice

Preconditioning and fast solvers for incompressible flow

We give a brief description with references of work on fast solution methods for incompressible Navier-Stokes problems which has been going on for about a decade. Specifically we describe preconditioned iterative strategies which involve the use of simple multigrid cycles for subproblems

ParMooN - a modernized program package based on mapped finite elements

{\sc ParMooN} is a program package for the numerical solution of elliptic and parabolic partial differential equations. It inherits the distinct features of its predecessor {\sc MooNMD} \cite{JM04}: strict decoupling of geometry and finite element spaces, implementation of mapped finite elements as their definition can be found in textbooks, and a geometric multigrid preconditioner with the option to use different finite element spaces on different levels of the multigrid hierarchy. After having presented some thoughts about in-house research codes, this paper focuses on aspects of the parallelization for a distributed memory environment, which is the main novelty of {\sc ParMooN}. Numerical studies, performed on compute servers, assess the efficiency of the parallelized geometric multigrid preconditioner in comparison with some parallel solvers that are available in the library {\sc PETSc}. The results of these studies give a first indication whether the cumbersome implementation of the parallelized geometric multigrid method was worthwhile or not.Comment: partly supported by European Union (EU), Horizon 2020, Marie Sk{\l}odowska-Curie Innovative Training Networks (ITN-EID), MIMESIS, grant number 67571

Preconditioned iterative methods for Navier-Stokes control problems

PDE-constrained optimization problems are a class of problems which have attracted much recent attention in scientific computing and applied science. In this paper, we discuss preconditioned iterative methods for a class of Navier-Stokes control problems, one of the main problems of this type in the field of fluid dynamics. Having detailed the Oseen-type iteration we use to solve the problems and derived the structure of the matrix system to be solved at each step, we utilize the theory of saddle point systems to develop efficient preconditioned iterative solution techniques for these problems. We also require theory of solving convection-diffusion control problems, as well as a commutator argument to justify one of the components of the preconditioner