Efficient preconditioning techniques for velocity tracking of Stokes control problem

Abstract

We are concerned with robust iterative solution methods for solving the Stokes optimal control problems. Two efficient preconditioners are proposed for the discretized saddle point linear systems arising from the velocity tracking of the Stokes control problem. The proposed preconditioners are similar in structure to and can be viewed as modifications of the preconditioner in Axelsson et al. (2017) [2], which are economic to implement in an inner-outer framework within Krylov acceleration. They can lead to similar tight and problem independent eigenvalue distribution results for the preconditioned matrices, which yield rates of convergence independent of both the regularization parameter and refinement level. Moreover, we also give inexact variants of the proposed preconditioners, which avoid the inner-outer implementations utilizing preconditioned GMRES methods as inner loops. Numerical experiments indicate that the proposed preconditioners demonstrate robust performance and comparable to some existing preconditioners when used to accelerate the Krylov subspace methods.