Two-by-two block matrices with square blocks arise in the numerical treatment of numerous applications of practical significance, such as optimal control problems, constrained by a state equation in the form of partial differential equations, multiphase models, solving complex linear systems in real arithmetics, to name a few.a
Such problems lead to algebraic systems of equations with matrices of a certain two-by-two block form. For such matrices, a number of preconditioners has been proposed, some of them with tight eigenvalue bounds. In this paper it is shown that in particular one of them, referred to as PRESB, is very efficient, not only giving robust, favourable properties of the spectrum but also enabling an efficient implementation with low computational complexity. Various applications and generalizations of this preconditioning technique, such as in time-harmonic parabolic and Stokes equations, eddy current electromagnetic problems and problems with additional box-constraints, i.e. upper and/or lower bounds of the solution, are also discussed.
The method is based on the use of coupled inner-outer iterations, where the inner iteration can be performed to various relative accuracies. This leads to variable preconditioners, thus, a flexible version of a Krylov subspace iteration method must be used. Alternatively, some version of a defect-correction iterative method can be applied.
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