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Journal of Scientific Computing

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04-12-2017

Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

Authors: B. Gross, P. J. Atzberger

Published in: Journal of Scientific Computing | Issue 1/2018

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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.

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Gillette, A., Holst, M., Zhu, Y.: Finite element exterior calculus for evolution problems. J. Comput. Math. 35(2), 187–212 (2017)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

de Goes, F., Crane, K., Desbrun, M., Schroder, P.: Digital geometry processing with discrete exterior calculus. In: SIGGRAPH (2013)

Hirani, A.N.: Discrete exterior calculus. Ph.D. thesis, Caltech (2003)

Arroyo, M., DeSimone, A.: Relaxation dynamics of fluid membranes. Phys. Rev. E 79(3), 031915 (2009)MathSciNetCrossRef

Sigurdsson, J.K., Atzberger, P.J.: Hydrodynamic coupling of particle inclusions embedded in curved lipid bilayer membranes. Soft Matter 12(32), 6685–6707 (2016)CrossRef

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)

Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)MathSciNetCrossRefMATH

de Goes, F., Desbrun, M., Meyer, M., DeRose, T.: Subdivision exterior calculus for geometry processing. ACM Trans. Graph. 35(4), 1–11 (2016)CrossRef

10.

Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete exterior calculus. Technical report (2003)

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)

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)MathSciNetCrossRefMATH

13.

Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, Mineola (1994)MATH

14.

Eells, J.: Geometric aspects of currents and distributions. Proc. Natl. Acad. Sci. 41(7), 493–496 (1955)MathSciNetCrossRefMATH

15.

Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957)CrossRefMATH

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)MathSciNetCrossRefMATH

17.

Atkinson, K., Han, W.: Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Springer, Berlin (2010)MATH

18.

Cottrell, J.A., Hughes, T.J., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Hoboken (2009)CrossRefMATH

19.

Sloan, I.H.: Polynomial interpolation and hyperinterpolation over general regions. J. Approx. Theory 83(2), 238–254 (1995)MathSciNetCrossRefMATH

20.

Reimer, M.: Hyperinterpolation on the sphere at the minimal projection order. J. Approx. Theory 104(2), 272–286 (2000)MathSciNetCrossRefMATH

21.

Sloan, I.H., Womersley, R.S.: Constructive polynomial approximation on the sphere. J. Approx. Theory 103(1), 91–118 (2000)MathSciNetCrossRefMATH

22.

Womersley, R.S., Sloan, I.H.: How good can polynomial interpolation on the sphere be? Adv. Comput. Math. 14(3), 195–226 (2001)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

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)

25.

Lebedev, V.I.: Quadratures on a sphere. USSR Comput. Math. Math. Phys. 16(2), 10–24 (1976)MathSciNetCrossRefMATH

26.

Driscoll, J.R., Healy, D.M.: Computing Fourier transforms and convolutions on the 2-sphere. Adv. Appl. Math. 15(2), 202–250 (1994)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

28.

Schaeffer, N.: Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations. Geochem. Geophys. Geosyst. 14(3), 751–758 (2013)CrossRef

29.

Abraham, R., Marsden, J.E., Raiu, T.S.: Manifolds, Tensor Analysis, and Applications, vol. 75. Springer, New York (1988)CrossRef

30.

Pressley, A.: Elementary Differential Geometry. Springer, Berlin (2001)CrossRefMATH

31.

Spivak, M.: Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. Westview Press, Boulder (1971)MATH

32.

Beentjes, C.H.L.: Quadrature on spherical surface. Technical report (2015)

33.

Womersley, R.S.: Efficient spherical designs with good geometric properties. arXiv:1709.01624 (2017)

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) MathSciNetCrossRef

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)CrossRef

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)CrossRef

37.

Trefethen, L.N., Bau III, D.: Numerical Linear Algebra. SIAM, Philadelphia (1997)CrossRefMATH

38.

Strang, G.: Linear Algebra and Its Applications. Academic Press Inc., Cambridge (1980)MATH

Title: Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds
Authors: B. Gross
P. J. Atzberger
Publication date: 04-12-2017
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0617-2

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