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Strang-Type Preconditioners for Differential-Algebraic Equations

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Book cover Numerical Analysis and Its Applications (NAA 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1988))

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Abstract

We consider linear constant coefficient differential-algebraic equations (DAEs) A x’(t) + Bx(t) = f(t) where A, B are square matrices and A is singular. If det(λA + B) with λ ∈ ℂ is not identically zero, the system of DAEs is solvable and can be separated into two uncoupled subsystems. One of them can be solved analytically and the other one is a system of ordinary differential equations (ODEs). We discretize the ODEs by boundary value methods (BVMs) and solve the linear system by using the generalized minimal residual (GMRES) method with Strang-type block-circulant preconditioners. It was shown that the preconditioners are nonsingular when the BVM is Aν,μ- ν-stable, and the eigenvalues of preconditioned matrices are clustered. Therefore, the number of iterations for solving the preconditioned systems by the GMRES method is bounded by a constant that is independent of the discretization mesh. Numerical results are also given.

Authors are supported by the research grant No. RG010/99-00S/JXQ/FST from the University of Macau.

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References

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Author information

Authors and Affiliations

  1. Faculty of Science and Technology, University of Macau, Macau, P.R. China

    Siu-Long Lei & Xiao-Qing Jin

Authors
  1. Siu-Long Lei
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  2. Xiao-Qing Jin
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Rousse, 7000, Rousse, Bulgaria

    Lubin Vulkov  & Plamen Yalamov  & 

  2. UNI-C,Danish Computing Center for Research and Education, DTU,Building 304, 2800, Lyngby, Denmark

    Jerzy Waśniewski

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© 2001 Springer-Verlag Berlin Heidelberg

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Lei, SL., Jin, XQ. (2001). Strang-Type Preconditioners for Differential-Algebraic Equations. In: Vulkov, L., Yalamov, P., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2000. Lecture Notes in Computer Science, vol 1988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45262-1_59

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  • DOI: https://doi.org/10.1007/3-540-45262-1_59

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41814-6

  • Online ISBN: 978-3-540-45262-1

  • eBook Packages: Springer Book Archive

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