A fast multipole method for the three-dimensional Stokes equations

  • Authors:

  • Anna-Karin Tornberg;Leslie Greengard

  • Affiliations:

  • Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA and Royal Institute of Technology (KTH), Linne Flow Centre/Numerical Analysis, SE-100 44 ...;Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2008

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Abstract

Many problems in Stokes flow (and linear elasticity) require the evaluation of vector fields defined in terms of sums involving large numbers of fundamental solutions. In the fluid mechanics setting, these are typically the Stokeslet (the kernel of the single layer potential) or the Stresslet (the kernel of the double layer potential). In this paper, we present a simple and efficient method for the rapid evaluation of such fields, using a decomposition into a small number of Coulombic N-body problems, following an approach similar to that of Fu and Rodin [Y. Fu, G.J. Rodin, Fast solution methods for three-dimensional Stokesian many-particle problems, Commun. Numer. Meth. En. 16 (2000) 145-149]. While any fast summation algorithm for Coulombic interactions can be employed, we present numerical results from a scheme based on the most modern version of the fast multipole method [H. Cheng, L. Greengard, V. Rokhlin, A fast adaptive multipole algorithm in three dimensions, J. Comput. Phys. 155 (1999) 468-498]. This approach should be of value in both the solution of boundary integral equations and multiparticle dynamics.