Justification of the fast multipole method for the stokes system. Part II. Exterior domain problems

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We consider the exterior domain problems of Dirichlet and Neumann type of the two-dimensional Stokes equations. For the solution of this boundary value problem we choose a potential ansatz and show that for the reduction of the computational costs, the fast multipole method of Greengard and Rokhlin can be used. Therefore, we find a complex representation of the hydrodynamical potentials and provide statements about the corresponding multipole and Taylor expansions, as well as the appropriate translation, rotation and conversion operators. The theoretical statements are illustrated by numerical experiments. Bibliography: 15 titles.

Keywords

Stokes Equation Boundary Integral Equation Exterior Domain Stokes System Multipole Expansion 
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References

  1. 1.
    M. A. Jasnow and G. T. Symm, Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, New York (1977).Google Scholar
  2. 2.
    S. G. Mikhlin, Integral Equations, Pergamon, London (1957).MATHGoogle Scholar
  3. 3.
    B. N. Khoromskij and G. Wittum, Numerical Solution of Elliptic Differential Equations by Reduction to the Interface, Springer, Berlin (2004).MATHGoogle Scholar
  4. 4.
    W. Borchers and W. Varnhorn, “On the boundedness of the Stokes semigroup in twodimensional exterior domains,” Math. Z. 213, 275–300 (1993).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York (1969).MATHGoogle Scholar
  6. 6.
    A. N. Popov, “ Application of potential theory to the solution of a Linearized System of Navier–Stokes Equations in the two-dimensional case,” In: Proceedings of the Steklov Institute of Mathematics 116, pp. 167–186, Am. Math. Soc., Providence, RI (1973).Google Scholar
  7. 7.
    W. Varnhorn, The Stokes Equations, Akademie-Verlag, Berlin (1994).MATHGoogle Scholar
  8. 8.
    J. Carrier, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm for particle simulations,” SIAM J. Sci. Statist. Comput. 9, No. 4, 669–686 (1988).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    L. Greengard and V. Rokhlin, “The rapid evaluation of potential fields in three dimensions,” In: Vortex Methods, 121–141, Lect. Notes Math. 1360, Springer, Berlin (1988).CrossRefGoogle Scholar
  10. 10.
    V. Rokhlin, “Rapid solution of integral equations of classical potential theory,” J. Comp. Phys. 60, 187 (1985).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    A. Mayo and A. Greebaum, “Fast parallel iterative solution of Poisson’s and the biharmonic equations on irregular regions,” SIAM J. Sci. Statist. Comput. 13, No. 1, 101–118 (1992).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge Univ. Press, Cambridge (1992).MATHCrossRefGoogle Scholar
  13. 13.
    T. Samrowski, “Justification of the Fast Multipole Method for the Stokes System,” J. Math. Sci. (N.Y.) 178, No. 6, (2011).Google Scholar
  14. 14.
    L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comp. Phys. 135, 280–292 (1997).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    L. Greengard and V. Rokhlin, “A new version of the fast multipole method for the Laplace equation in three dimensions,” Acta Numerica 6, 229 (1997).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.Institute of MathematicsZurich UniversityZurichSwitzerland

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