Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

Gross, B.; Atzberger, P. J.

doi:10.1007/s10915-017-0617-2

Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

Journal Article · Mon Dec 04 00:00:00 EST 2017 · Journal of Scientific Computing

DOI:https://doi.org/10.1007/s10915-017-0617-2· OSTI ID:1633907

Gross, B. ^[1]; ^[2]

Univ. of California, Santa Barbara, CA (United States); US Department of Energy
Univ. of California, Santa Barbara, CA (United States)

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 demonstrate how the numerical operators can be naturally composed to formulate general numerical approximations for solving differential equations on manifolds. We introduce findings for the Laplace–Beltrami equations demonstrating our approach.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of California, Santa Barbara, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

Grant/Contract Number:: SC0009254

OSTI ID:: 1633907

Journal Information:: Journal of Scientific Computing, Journal Name: Journal of Scientific Computing Journal Issue: 1 Vol. 76; ISSN 0885-7474

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

References (23)

Hyperinterpolation on the Sphere at the Minimal Projection Order Reimer, Manfred Journal of Approximation Theory, Vol. 104, Issue 2 https://doi.org/10.1006/jath.2000.3454	journal	June 2000
Geometric Aspects of Currents and Distributions Eells, J. Proceedings of the National Academy of Sciences, Vol. 41, Issue 7 https://doi.org/10.1073/pnas.41.7.493	journal	July 1955
Isogeometric Analysis Cottrell, J. Austin; Hughes, Thomas J. R.; Bazilevs, Yuri John Wiley & Sons, Ltd https://doi.org/10.1002/9780470749081	book	January 2009
Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations: EFFICIENT SPHERICAL HARMONIC TRANSFORM Schaeffer, Nathanaël Geochemistry, Geophysics, Geosystems, Vol. 14, Issue 3 https://doi.org/10.1002/ggge.20071	journal	March 2013
Computing Fourier Transforms and Convolutions on the 2-Sphere Driscoll, J. R.; Healy, D. M. Advances in Applied Mathematics, Vol. 15, Issue 2 https://doi.org/10.1006/aama.1994.1008	journal	June 1994
Polynomial Interpolation and Hyperinterpolation over General Regions Sloan, I. H. Journal of Approximation Theory, Vol. 83, Issue 2 https://doi.org/10.1006/jath.1995.1119	journal	November 1995
Constructive Polynomial Approximation on the Sphere Sloan, Ian H.; Womersley, Robert S. Journal of Approximation Theory, Vol. 103, Issue 1 https://doi.org/10.1006/jath.1999.3426	journal	March 2000
Efficient Spherical Designs with Good Geometric Properties Womersley, Robert S. Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan https://doi.org/10.1007/978-3-319-72456-0_57	book	January 2018
On the geometric character of stress in continuum mechanics Kanso, Eva; Arroyo, Marino; Tong, Yiying Zeitschrift für angewandte Mathematik und Physik, Vol. 58, Issue 5 https://doi.org/10.1007/s00033-007-6141-8	journal	June 2007
FFTs for the 2-Sphere-Improvements and Variations Healy, D. M.; Rockmore, D. N.; Kostelec, P. J. Journal of Fourier Analysis and Applications, Vol. 9, Issue 4 https://doi.org/10.1007/s00041-003-0018-9	journal	July 2003
Quadratures on a sphere Lebedev, V. I. USSR Computational Mathematics and Mathematical Physics, Vol. 16, Issue 2 https://doi.org/10.1016/0041-5553(76)90100-2	journal	January 1976
Delaunay Hodge star Hirani, Anil N.; Kalyanaraman, Kaushik; VanderZee, Evan B. Computer-Aided Design, Vol. 45, Issue 2 https://doi.org/10.1016/j.cad.2012.10.038	journal	February 2013
Comparison of discrete Hodge star operators for surfaces Mohamed, Mamdouh S.; Hirani, Anil N.; Samtaney, Ravi Computer-Aided Design, Vol. 78 https://doi.org/10.1016/j.cad.2016.05.002	journal	September 2016
The chain collocation method: A spectrally accurate calculus of forms Rufat, Dzhelil; Mason, Gemma; Mullen, Patrick Journal of Computational Physics, Vol. 257 https://doi.org/10.1016/j.jcp.2013.08.011	journal	January 2014
Finite element exterior calculus, homological techniques, and applications Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar Acta Numerica, Vol. 15 https://doi.org/10.1017/S0962492906210018	journal	May 2006
Hydrodynamic coupling of particle inclusions embedded in curved lipid bilayer membranes Sigurdsson, Jon Karl; Atzberger, Paul J. Soft Matter, Vol. 12, Issue 32 https://doi.org/10.1039/C6SM00194G	journal	January 2016
Finite element exterior calculus: from Hodge theory to numerical stability Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar Bulletin of the American Mathematical Society, Vol. 47, Issue 2 https://doi.org/10.1090/S0273-0979-10-01278-4	journal	January 2010
Relaxation dynamics of fluid membranes Arroyo, Marino; DeSimone, Antonio Physical Review E, Vol. 79, Issue 3 https://doi.org/10.1103/PhysRevE.79.031915	journal	March 2009
Numerical Linear Algebra Trefethen, Lloyd N.; Bau, David https://doi.org/10.1137/1.9780898719574	book	January 1997
Regularized Least Squares Approximations on the Sphere Using Spherical Designs An, Congpei; Chen, Xiaojun; Sloan, Ian H. SIAM Journal on Numerical Analysis, Vol. 50, Issue 3 https://doi.org/10.1137/110838601	journal	January 2012
Subdivision exterior calculus for geometry processing de Goes, Fernando; Desbrun, Mathieu; Meyer, Mark ACM Transactions on Graphics, Vol. 35, Issue 4 https://doi.org/10.1145/2897824.2925880	journal	July 2016
Finite Element Exterior Calculus for Evolution Problems Gillette, Andrew Journal of Computational Mathematics, Vol. 35, Issue 2 https://doi.org/10.4208/jcm.1610-m2015-0319	journal	June 2017
SymPy: symbolic computing in Python Meurer, Aaron; Smith, Christopher P.; Paprocki, Mateusz PeerJ Computer Science, Vol. 3 https://doi.org/10.7717/peerj-cs.103	journal	January 2017

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