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We consider large scale sparse linear systems in saddle point form. A natural property of such indefinite 2-by-2 block systems is the positivity of the (1,1) block on the kernel of the (2,1) block. Many solution methods, however, require that the positivity of the (1,1) block is satisfied everywhere. To enforce the positivity everywhere, an augmented… (More)

In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate… (More)

For a general class of saddle point problems sharp estimates for Babuška's inf-sup stability constants are derived in terms of the constants in Brezzi's theory. In the finite-dimensional Hermitian case more detailed spectral properties of preconditioned saddle point matrices are presented, which are helpful for the convergence analysis of common Krylov… (More)

We consider the discretized optimality system of a special class of elliptic optimal control problems and propose an all-at-once multigrid method for solving this discretized system. Under standard assumptions the convergence of the multigrid method and the robustness of the convergence rates with respect to the involved parameter are shown. Numerical… (More)

For the optimal control problem with time-periodic Stokes equations a practical robust preconditioner is presented. The discretization of the corresponding optimality system leads to a linear system with a large, sparse and complex 4-by-4 block matrix in saddle point form. We present a decoupling strategy, which reduces the system to two linear systems with… (More)

The Boundary Element Tearing and Interconnecting (BETI) methods have recently been introduced as boundary element counterparts of the well–established Finite Element Tearing and Interconnecting (FETI) methods. In this paper we present inexact data–sparse versions of the BETI methods which avoid the elimination of the primal unknowns and dense matrices.… (More)