Advertisement

Journal of Scientific Computing

, Volume 76, Issue 1, pp 145–165 | Cite as

Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

  • B. Gross
  • P. J. Atzberger
Article
  • 182 Downloads

Abstract

We develop exterior calculus approaches for partial differential equations on radial manifolds. We introduce numerical methods that approximate with spectral accuracy the exterior derivative \(\mathbf {d}\), Hodge star \(\star \), and their compositions. To achieve discretizations with high precision and symmetry, we develop hyperinterpolation methods based on spherical harmonics and Lebedev quadrature. We perform convergence studies of our numerical exterior derivative operator \(\overline{\mathbf {d}}\) and Hodge star operator \(\overline{\star }\) showing each converge spectrally to \(\mathbf {d}\) and \(\star \). We show how the numerical operators can be naturally composed to formulate general numerical approximations for solving differential equations on manifolds. We present results for the Laplace–Beltrami equations demonstrating our approach.

Keywords

Manifolds Laplace–Beltrami Numerical methods Partial differential equations 

Notes

Acknowledgements

We acknowledge support to P. J. Atzberger and B. Gross from research Grants DOE ASCR CM4 DE-SC0009254, NSF CAREER Grant DMS-0956210, and NSF Grant DMS-1616353.

References

  1. 1.
    Gillette, A., Holst, M., Zhu, Y.: Finite element exterior calculus for evolution problems. J. Comput. Math. 35(2), 187–212 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47(2), 281–354 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    de Goes, F., Crane, K., Desbrun, M., Schroder, P.: Digital geometry processing with discrete exterior calculus. In: SIGGRAPH (2013)Google Scholar
  4. 4.
    Hirani, A.N.: Discrete exterior calculus. Ph.D. thesis, Caltech (2003)Google Scholar
  5. 5.
    Arroyo, M., DeSimone, A.: Relaxation dynamics of fluid membranes. Phys. Rev. E 79(3), 031915 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Sigurdsson, J.K., Atzberger, P.J.: Hydrodynamic coupling of particle inclusions embedded in curved lipid bilayer membranes. Soft Matter 12(32), 6685–6707 (2016)CrossRefGoogle Scholar
  7. 7.
    Stern, A., Tong, Y., Desbrun, M., Marsden, J.E.: Geometric computational electrodynamics with variational integrators and discrete differential forms. In: Chang, D.E., Holm, D.D., Patrick, G., Ratiu, T. (eds.) Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden, pp. 437–475. Springer, New York (2015)Google Scholar
  8. 8.
    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    de Goes, F., Desbrun, M., Meyer, M., DeRose, T.: Subdivision exterior calculus for geometry processing. ACM Trans. Graph. 35(4), 1–11 (2016)CrossRefGoogle Scholar
  10. 10.
    Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete exterior calculus. Technical report (2003)Google Scholar
  11. 11.
    Wang, K., Tong, Y., Desbrun, M., Schroder, P.: Edge subdivision schemes and the construction of smooth vector fields. In: ACM SIGGRAPH 2006 Papers, pp. 1041–1048. ACM, Boston (2006)Google Scholar
  12. 12.
    Kanso, E., Arroyo, M., Tong, Y., Yavari, A., Marsden, J.G., Desbrun, M.: On the geometric character of stress in continuum mechanics. Zeitschrift fr angewandte Mathematik und Physik 58(5), 843–856 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, Mineola (1994)zbMATHGoogle Scholar
  14. 14.
    Eells, J.: Geometric aspects of currents and distributions. Proc. Natl. Acad. Sci. 41(7), 493–496 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957)CrossRefzbMATHGoogle Scholar
  16. 16.
    Rufat, D., Mason, G., Mullen, P., Desbrun, M.: The chain collocation method: a spectrally accurate calculus of forms. J. Comput. Phys. 257(Part B), 1352–1372 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Atkinson, K., Han, W.: Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Springer, Berlin (2010)zbMATHGoogle Scholar
  18. 18.
    Cottrell, J.A., Hughes, T.J., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Hoboken (2009)CrossRefzbMATHGoogle Scholar
  19. 19.
    Sloan, I.H.: Polynomial interpolation and hyperinterpolation over general regions. J. Approx. Theory 83(2), 238–254 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Reimer, M.: Hyperinterpolation on the sphere at the minimal projection order. J. Approx. Theory 104(2), 272–286 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sloan, I.H., Womersley, R.S.: Constructive polynomial approximation on the sphere. J. Approx. Theory 103(1), 91–118 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Womersley, R.S., Sloan, I.H.: How good can polynomial interpolation on the sphere be? Adv. Comput. Math. 14(3), 195–226 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    An, C., Chen, X., Sloan, I.H., Womersley, R.S.: Regularized least squares approximations on the sphere using spherical designs. SIAM J. Numer. Anal. 50(3), 1513–1534 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lebedev, V.I., Laikov, D.N.: A quadrature formula for the sphere of the 131st algebraic order of accuracy. Dokl. Math. 59, 477–481 (1999)Google Scholar
  25. 25.
    Lebedev, V.I.: Quadratures on a sphere. USSR Comput. Math. Math. Phys. 16(2), 10–24 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Driscoll, J.R., Healy, D.M.: Computing Fourier transforms and convolutions on the 2-sphere. Adv. Appl. Math. 15(2), 202–250 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Healy, D.M., Rockmore, D.N., Kostelec, P.J., Moore, S.: FFTs for the 2-sphere-improvements and variations. J. Fourier Anal. Appl. 9(4), 341–385 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Schaeffer, N.: Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations. Geochem. Geophys. Geosyst. 14(3), 751–758 (2013)CrossRefGoogle Scholar
  29. 29.
    Abraham, R., Marsden, J.E., Raiu, T.S.: Manifolds, Tensor Analysis, and Applications, vol. 75. Springer, New York (1988)CrossRefGoogle Scholar
  30. 30.
    Pressley, A.: Elementary Differential Geometry. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  31. 31.
    Spivak, M.: Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. Westview Press, Boulder (1971)zbMATHGoogle Scholar
  32. 32.
    Beentjes, C.H.L.: Quadrature on spherical surface. Technical report (2015)Google Scholar
  33. 33.
    Womersley, R.S.: Efficient spherical designs with good geometric properties. arXiv:1709.01624 (2017)
  34. 34.
    Hirani, A.N., Kalyanaraman, K., VanderZee, E.B.: Delaunay Hodge star. Comput. Aided Des. 45(2), 540–544 (2013). (Solid and Physical Modeling 2012) MathSciNetCrossRefGoogle Scholar
  35. 35.
    Mohamed, M.S., Hirani, A.N., Samtaney, R.: Comparison of discrete Hodge star operators for surfaces. Comput. Aided Des. 78(C), 118–125 (2016)CrossRefGoogle Scholar
  36. 36.
    Meurer, A., Smith, C.P., Paprocki, M., Certik, O., Kirpichev, S.B., Rocklin, M., Kumar, A., Ivanov, S., Moore, J.K., Singh, S., Rathnayake, T., Vig, S., Granger, E., Muller, R.P., Bonazzi, F., Gupta, H., Vats, S., Johansson, F., Pedregosa, F., Curry, M.J., Terrel, A.R., Roučka, S., Saboo, A., Fernando, I., Kulal, S., Cimrman, R., Scopatz, A.: Sympy: symbolic computing in python. PeerJ Comput. Sci. 3, e103 (2017)CrossRefGoogle Scholar
  37. 37.
    Trefethen, L.N., Bau III, D.: Numerical Linear Algebra. SIAM, Philadelphia (1997)CrossRefzbMATHGoogle Scholar
  38. 38.
    Strang, G.: Linear Algebra and Its Applications. Academic Press Inc., Cambridge (1980)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California Santa BarbaraSanta BarbaraUSA
  2. 2.Department of Mechanical EngineeringUniversity of California Santa BarbaraSanta BarbaraUSA

Personalised recommendations