Preconditioners for Some Matrices of Two-by-Two Block Form, with Applications, I
Matrices of two-by-two block form with matrix blocks of equal order arise in various important applications, such as when solving complex-valued systems in real arithmetics, in linearized forms of the Cahn–Hilliard diffusive phase-field differential equation model and in constrained partial differential equations with distributed control. It is shown how an efficient preconditioner can be constructed which, under certain conditions, has a resulting spectral condition number of about 2. The preconditioner avoids the use of Schur complement matrices and needs only solutions with matrices that are linear combinations of the matrices appearing in each block row of the given matrix and for which often efficient preconditioners are already available.
KeywordsTwo-by-two block-structured matrices Preconditioning Complex-valued system Cahn–Hilliard phase-field model Optimal control Distributed control
This work was supported by the European Regional Development Fund in the IT4 Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070).
Discussions with Maya Neytcheva on implementation aspects of the method are gratefully acknowledged.
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