Skip to main content
SpringerLink
Log in

Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions

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

Abstract

The Laplace-Beltrami problem ΔΓψ = f has several applications in mathematical physics, differential geometry, machine learning, and topology. In this work, we present novel second-kind integral equations for its solution which obviate the need for constructing a suitable parametrix to approximate the in-surface Green’s function. The resulting integral equations are well-conditioned and compatible with standard fast multipole methods and iterative linear algebraic solvers, as well as more modern fast direct solvers. Using layer-potential identities known as Calderón projectors, the Laplace-Beltrami operator can be pre-conditioned from the left and/or right to obtain second-kind integral equations. We demonstrate the accuracy and stability of the scheme in several numerical examples along surfaces described by curvilinear triangles.

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. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia (1999)

    Book  Google Scholar 

  2. Anselone, P.M.: Collectively Comact Operator Approximation Theory and Applications to Integral Equations. Prentice-Hall, Englewood Cliffs (1971)

    MATH  Google Scholar 

  3. Askham, T., Cerfon, A.J.: An adaptive fast multipole accelerated Poisson solver for complex geometries. J. Comput. Phys. 344, 1–22 (2017)

    Article  MathSciNet  Google Scholar 

  4. Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, New York (1997)

    Book  Google Scholar 

  5. Autodesk: Fusion 360. http://www.autodesk.com/products/fusion-360 (2017)

  6. Belkin, M., Niyogi, P., Sindhwani, V.: Manifold regularization: a geometric framework for learning from labeled and unlabeled examples. J. Mach. Learn. Res. 7, 2399–2434 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Bertalmío, M., Cheng, L.-T., Osher, S., Sapiro, G.: Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174, 759–780 (2001)

    Article  MathSciNet  Google Scholar 

  8. Bonito, A., Cascón, J.M., Morin, P., Nochetto, R.H.: AFEM for Geometric PDE: the Laplace-Beltrami Operator, pp. 257–306. Springer Milan, Milano (2013)

    MATH  Google Scholar 

  9. Boubendir, Y., Turc, C.: Well-conditioned boundary integral equation formulations for the solution of high-frequency electromagnetic scattering problems. Comput. Math. Appl. 67, 1772–1805 (2014)

    Article  MathSciNet  Google Scholar 

  10. Bremer, J., Gimbutas, Z.: A Nyström method for weakly singular integral operators on surfaces. J. Comput. Phys. 231, 4885–4903 (2012)

    Article  MathSciNet  Google Scholar 

  11. Bremer, J., Gimbutas, Z.: On the numerical evaluation of singular integrals of scattering theory. J. Comput. Phys. 251, 327–343 (2013)

    Article  MathSciNet  Google Scholar 

  12. Bremer, J., Gimbutas, Z., Rokhlin, V.: A nonlinear optimization procedure for generalized Gaussian quadratures. SIAM J. Sci. Comput. 32(4), 1761–1788 (2010)

    Article  MathSciNet  Google Scholar 

  13. Carmo, M.P.D.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976)

    MATH  Google Scholar 

  14. Chao, I., Pinkall, U., Sanan, P., Schröder, P.: A simple geometric model for elastic deformations. ACM Trans. Graph. 29, 1–6 (2010)

    Article  Google Scholar 

  15. Chen, Y., Macdonald, C.B.: The closest point method and multigrid solvers for elliptic equations on surfaces. SIAM J. Sci. Comput. 37, A134–A155 (2015)

    Article  MathSciNet  Google Scholar 

  16. Colton, D., Kress, R.: Integral Equation Methods in Scattering Theory. Wiley, New York (1983)

    MATH  Google Scholar 

  17. Contopanagos, H., Dembart, B., Epton, M., Ottusch, J.J., Rokhlin, V., Visher, J.L., Wandzura, S.M.: Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering. IEEE Trans. Antennas Propag. 50(12), 1824–1830 (2002)

    Article  Google Scholar 

  18. Dai, Q.I., Chew, W.C., Jiang, L.J., Wu, Y.: Differential-forms-motivated discretizations of electromagnetic differential and integral equations. IEEE Antennas Wirel. Propag. Lett. 13, 1223–1226 (2014)

    Article  Google Scholar 

  19. Demlow, A., Dziuk, G.: An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces. SIAM J. Numer. Anal. 45, 421–442 (2007)

    Article  MathSciNet  Google Scholar 

  20. Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numerica 22, 289–396 (2013)

    Article  MathSciNet  Google Scholar 

  21. Epstein, C.L., Greengard, L.: Debye sources and the numerical solution of the time harmonic Maxwell equations. Commun. Pure Appl. Math. 63(4), 413–463 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Epstein, C.L., Greengard, L., O’Neil, M.: Debye sources and the numerical solution of the time harmonic Maxwell equations II. Commun. Pure Appl. Math. 66(5), 753–789 (2013)

    Article  MathSciNet  Google Scholar 

  23. Epstein, C.L., Greengard, L., O’Neil, M.: Debye sources, beltrami fields, and a complex structure on Maxwell fields. Commun. Pure Appl. Math. 68, 2237–2280 (2015)

    Article  MathSciNet  Google Scholar 

  24. Folland, G.B.: Introduction to Partial Differential Equations. Princeton University Press, Princeton (1995)

    MATH  Google Scholar 

  25. Frankel, T.: The Geometry of Physics. Cambridge University Press, New York (2011)

    Book  Google Scholar 

  26. Frittelli, M., Sgura, I.: Virtual element method for the Laplace-Beltrami equation on surfaces. arXiv:1612.02369 [math.NA] (2016)

  27. Geuzaine, C., Remacle, J.-F.: Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Num. Methods Eng. 79, 1309–1331 (2009)

    Article  MathSciNet  Google Scholar 

  28. Greengard, L., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numerica 6, 229–269 (1997)

    Article  MathSciNet  Google Scholar 

  29. Greer, J.B., Bertozzi, A.L., Sapiro, G.: Fourth order partial differential equations on general geometries. J. Comput. Phys. 216, 216–246 (2006)

    Article  MathSciNet  Google Scholar 

  30. Hansbo, P., Larson, M.-G., Larsson, K.: Analysis of finite element methods for vector Laplacians on surfaces. arXiv:1610.06747 [math.NA] (2016)

  31. Imbert-Gerard, L.-M., Greengard, L.: Pseudo-spectral methods for the Laplace-Beltrami equation and the Hodge decomposition on surfaces of genus one. Numer. Methods Partial Differ. Equ. 33(3), 941–955 (2017)

    Article  MathSciNet  Google Scholar 

  32. Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1999)

    Google Scholar 

  33. Koornwinder, T.: Two-variable analogues of the classical orthogonal polynomials. In: Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 435–495. Academic Press, New York (1975)

  34. Kress, R.: Linear Integral Equations. Springer, New York (2014)

    Book  Google Scholar 

  35. Kress, R., Roach, G.: Transmission problems for the Helmholtz equation. J. Math. Phys. 19, 1433–1437 (1978)

    Article  MathSciNet  Google Scholar 

  36. Kropinski, M.C.A., Nigam, N.: Fast integral equation methods for the Laplace-Beltrami equation on the sphere. Adv. Comput. Math. 40(2), 577–596 (2014)

    Article  MathSciNet  Google Scholar 

  37. Kropinski, M.C.A., Nigam, N., Quaife, B.: Integral equation methods for the Yukawa-Beltrami equation on the sphere. Adv. Comput. Math. 42(2), 469–488 (2016)

    Article  MathSciNet  Google Scholar 

  38. Macdonald, C.B., Ruuth, S.J.: Level set equations on surfaces via the closest point method. J. Sci. Comput. 35, 219–240 (2008)

    Article  MathSciNet  Google Scholar 

  39. Macdonald, C.B., Ruuth, S.J.: The implicit closest point method for the numerical solution of partial differential equations on surfaces. SIAM J. Sci. Comput. 31, 4330–4350 (2009)

    Article  MathSciNet  Google Scholar 

  40. McKean, H.P., Singer, I.M.: Curvature and the eigenvalues of the Laplacian. J. Diff. Geom. 1, 43–69 (1967)

    Article  MathSciNet  Google Scholar 

  41. Nedelec, J.-C.: Acoustic and Electromagnetic Equations. Springer, New York (2001)

    Book  Google Scholar 

  42. Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions, 1st edn. Cambridge University Press, New York (2010)

    MATH  Google Scholar 

  43. Papas, C.H.: Theory of Electromagnetic Wave Propagation. Dover, New York (1988)

    Google Scholar 

  44. Reuter, M.: Hierarchical shape segmentation and registration via topological features of Laplace-Beltrami Eigenfunctions. Int. J. Comput. Vis. 89, 287–308 (2010)

    Article  Google Scholar 

  45. Rokhlin, V.: Solution of acoustic scattering problems by means of second kind integral equations. Wave Motion 5, 257–272 (1983)

    Article  MathSciNet  Google Scholar 

  46. Ruuth, S.J., Merriman, B.: A simple embedding method for solving partial differential equations on surfaces. J. Comput. Phys. 227, 1943–1961 (2008)

    Article  MathSciNet  Google Scholar 

  47. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)

    Article  MathSciNet  Google Scholar 

  48. Schwartz, P., Adalsteinsson, D., Colella, P., Arkin, A.P., Onsum, M.: Numerical computation of diffusion on a surface. Proc. Natl. Acad. Sci. 102, 11151–11156 (2005)

    Article  Google Scholar 

  49. Sifuentes, J., Gimbutas, Z., Greengard, L.: Randomized methods for rank-deficient linear systems. Elec. Trans. Num. Anal. 44, 177–188 (2015)

    MathSciNet  MATH  Google Scholar 

  50. Sokolowski, J., Zolesio, J.-P.: Introduction to Shape Optimization. Springer, New York (1992)

    Book  Google Scholar 

  51. Veerapaneni, S.K., Rahimian, A., Biros, G., Zorin, D.: A fast algorithm for simulating vesicle flows in three dimensions. J. Comput. Phys. 230, 5610–5634 (2011)

    Article  MathSciNet  Google Scholar 

  52. Vico, F., Greengard, L., Gimbutas, Z.: Boundary integral equation analysis on the sphere. Numer. Math. 128, 463–487 (2014)

    Article  MathSciNet  Google Scholar 

  53. Vioreanu, B., Rokhlin, V.: Spectra of multiplication operators as a numerical tool. SIAM J. Sci. Comput. 36, A267–A288 (2014)

    Article  MathSciNet  Google Scholar 

  54. Weiss, Y., Torralba, A., Fergus, R.: Spectral hashing. In: Advances in Neural Information Processing Systems, pp. 1753–1760 (2009)

Download references

Acknowledgements

The author would like to thank Jim Bremer and Zydrunas Gimbutas for sharing generalized Gaussian quadrature routines, and Charles L. Epstein, Leslie Greengard, Lise-Marie Imbert-Gérard, and Tonatiuh Sanchez-Vizuet for several useful discussions.

Author information

Authors and Affiliations

  1. Courant Institute, New York University, New York, NY, USA

    Michael O’Neil

Authors
  1. Michael O’Neil
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael O’Neil.

Additional information

Communicated by: Alexander Barnett

Research supported in part by the Office of Naval Research under Award N00014-15-1-2669.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

O’Neil, M. Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions. Adv Comput Math 44, 1385–1409 (2018). https://doi.org/10.1007/s10444-018-9587-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-018-9587-7

Keywords

Mathematics Subject Classification (2010)

Search

Navigation