Skip to main content
SpringerLink
Log in

Fast integral equation methods for the Laplace-Beltrami equation on the sphere

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

Integral equation methods for solving the Laplace-Beltrami equation on the unit sphere in the presence of multiple “islands” are presented. The surface of the sphere is first mapped to a multiply-connected region in the complex plane via a stereographic projection. After discretizing the integral equation, the resulting dense linear system is solved iteratively using the fast multipole method for the 2D Coulomb potential in order to calculate the matrix-vector products. This numerical scheme requires only O(N) operations, where N is the number of nodes in the discretization of the boundary. The performance of the method is demonstrated on several examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bertalmio, M., Cheng, L.T., Osher, S., Sapiro, G.: Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174(2), 759–780 (2001). doi:10.1006/jcph.2001.6937. http://www.sciencedirect.com/science/article/pii/S0021999101969372

    Article  MATH  MathSciNet  Google Scholar 

  2. Bonner, B.D., Graham, I.G., Smyshlyaev, V.P.: The computation of conical diffraction coefficients in high-frequency acoustic wave scattering. SIAM J. Numer. Anal. 43(3) (2005). doi:10.1137/040603358

  3. Carrier, J., Greengard, L., Rokhlin, V.: A fast adaptive multipole algorithm for particle simulations. SIAM. J. Sci. Stat. Comput. 9, 669–686 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chaplain, M., Ganesh, M., Graham, I.: Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. J. Math. Biology 42, 387–423 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Crowdy, D., Cloke, M.: Analytical solutions for distributed multipolar vortex equilibria on a sphere. Phys. Fluids 15(1), 22–34 (2003). doi:10.1063/1.1521727

    Article  MathSciNet  Google Scholar 

  6. Duduchava, L.R., Mitrea, D., Mitrea, M.: Differential operators and boundary value problems on hypersurfaces. Math. Nachr. 279(9-10), 996–1023 (2006). doi:10.1002/mana.200410407

    Article  MATH  MathSciNet  Google Scholar 

  7. Firey, W.J.: The determination of convex bodies from their mean radius of curvature functions. Mathematika 14, 1–13 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  8. Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Dodgson, N.A., Floater, M.S., Sabin, M.A. (eds.) Advances in Multiresolution for Geometric Modelling, Mathematics and Visualization, pp. 157–186. Springer Berlin Heidelberg (2005). doi:10.1007/3-540-26808-1_9

  9. Gemmrich, S., Nigam, N.: A boundary integral strategy for the Laplace-Beltrami-Dirichlet problem on the sphere \(S^{2}\). In: Frontiers of Applied and Computational Mathematics, pp. 222–230. World Science Publisher, Hackensack, NJ (2008). doi:10.1142/9789812835291_0024

  10. Gemmrich, S., Nigam, N., Steinbach, O.: Boundary integral equations for the Laplace-Beltrami operator. In: Mathematics and Computation, A Contemporary View, Abel Symposium, vol. 3, pp. 21–37. Springer, Berlin (2008). doi:10.1007/978-3-540-68850-1_2

  11. Greenbaum, A., Greengard, L., McFadden, G.B.: Laplace’s equation and the Dirichlet-Neumann map in multiply connected domains. J. Comput. Phys. 105(2), 267–278 (1993). doi:10.1006/jcph.1993.1073

    Article  MATH  MathSciNet  Google Scholar 

  12. Greengard, L.: The Rapid Evaluation of Potential Fields in Particle Systems. MIT Press, Cambridge (1988)

  13. Greengard, L., Kropinski, M.C.A., Mayo, A.: Integral equation methods for Stokes flow and isotropic elasticity in the plane. J. Comp. Phys. 125, 403–414 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comp. Phys 73, 325–348 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hsiao, G.C., Wendland, W.L.: Boundary integral equations, Applied Mathematical Sciences, vol. 164. Springer-Verlag, Berlin (2008). doi:10.1007/978-3-540-68545-6

  16. Kidambi, R., Newton, P.K.: Motion of three point vortices on a sphere. Phys. D 116(1–2), 143–175 (1998). doi:10.1016/S0167-2789(97)00236-4

    Article  MATH  MathSciNet  Google Scholar 

  17. Lindblom, L., Szilagyi, B.: Solving partial differential equations numerically on manifolds with arbitrary spatial topologies (2012). arXiv:comp-ph.1210/5016

  18. Mikhlin, S.: Integral Equations and their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, International Series of Monographs on Pure and Applied Mathematics, vol. 4. Pergamon Press (1957)

  19. Mitrea, M., Nistor, V.: Boundary value problems and layer potentials on manifolds with cylindrical ends. Czechoslovak Math. J. 57(132)(4), 1151–1197 (2007). doi:10.1007/s10587-007-0118-9

    Article  MathSciNet  Google Scholar 

  20. Mitrea, M., Taylor, M.: Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem. J. Funct. Anal. 176(1), 1–79 (2000). doi:10.1006/jfan.2000.3619

    Article  MATH  MathSciNet  Google Scholar 

  21. Boutet de Monvel, L.: Boundary problems for pseudo-differential operators. Acta Math 126(1-2), 11–51 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  22. Myers, T., Charpin, J.: A mathematical model for atmospheric ice accretion and water flow on a cold surface. Int. J. Heat Mass Transfer 25(5483–5500) (2004). doi:10.1016/j.ijheatmasstransfer.2004.06.037. http://www.sciencedirect.com/science/article/pii/S0017931004002807

  23. Nishimura, N.: Fast multipole accelerated boundary integral equation methods. Appl. Mech. Rev. 55(4), 299–324 (2002)

    Article  Google Scholar 

  24. Ruuth, S.J., Merriman, B.: A simple embedding method for solving partial differential equations on surfaces. J. Comput. Nonlinear Dyn. 227(3), 1943–1961 (2008). doi:10.1016/j.jcp.2007.10.009. http://www.sciencedirect.com/science/article/pii/S002199910700441X

    MATH  MathSciNet  Google Scholar 

  25. Surana, A., Crowdy, D.: Vortex dynamics in complex domains on a spherical surface. J. Comput. Phys. 227(12), 6058–6070 (2008). doi:10.1016/j.jcp.2008.02.027

    Article  MATH  MathSciNet  Google Scholar 

  26. Witkin, A., Kass, M.: Reaction-diffusion textures. SIGGRAPH Comput. Graph. 25(4), 299–308 (1991). doi:10.1145/127719.122750

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, Burnaby, Canada

    Mary Catherine A. Kropinski & Nilima Nigam

Authors
  1. Mary Catherine A. Kropinski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nilima Nigam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mary Catherine A. Kropinski.

Additional information

Communicated by: Leslie Greengard

Supported in part by grants from the Natural Sciences and Engineering Research Council of Canada. NN gratefully acknowledges support from the Canada Research Chairs Council, Canada

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kropinski, M.C.A., Nigam, N. Fast integral equation methods for the Laplace-Beltrami equation on the sphere. Adv Comput Math 40, 577–596 (2014). https://doi.org/10.1007/s10444-013-9319-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-013-9319-y

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation