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Preconditioners for Some Matrices of Two-by-Two Block Form, with Applications, I

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Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 45))

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Abstract

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.

Mathematics Subject Classification (2010): 65F10, 65F35, 76T10, 49J20

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Acknowledgements

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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Authors and Affiliations

  1. IT4 Innovations Department, Institute of Geonics AS CR, Ostrava, Czech Republic

    Owe Axelsson

  2. King Abdulaziz University, Jeddah, Saudi Arabia

    Owe Axelsson

Authors
  1. Owe Axelsson
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Correspondence to Owe Axelsson .

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Editors and Affiliations

  1. , Dept. of Flow and Material Simulation, Fraunhofer Institute for Industrial Math, Fraunhofer-Platz 1, Kaiserslautern, 67663, Germany

    Oleg P. Iliev

  2. Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. 25A, Sofia, 1113, Bulgaria

    Svetozar D. Margenov

  3. , Dept. of Mathematical and Statistical Sc, University of Alberta, 677 Central Academic Building, Edmonton, T6G 2G1, Alberta, Canada

    Peter D Minev

  4. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, East Avenue 7000, Livermore, 94550, California, USA

    Panayot S. Vassilevski

  5. , Department of Mathematics, Pennsylvania State University, 310 McAllister Building, University Park, 16802, Pennsylvania, USA

    Ludmil T Zikatanov

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Axelsson, O. (2013). Preconditioners for Some Matrices of Two-by-Two Block Form, with Applications, I. In: Iliev, O., Margenov, S., Minev, P., Vassilevski, P., Zikatanov, L. (eds) Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications. Springer Proceedings in Mathematics & Statistics, vol 45. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7172-1_3

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