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Title: Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

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 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.

Authors:
 [1]; ORCiD logo [1]
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  1. Univ. of California, Santa Barbara, CA (United States)
Publication Date:
Research Org.:
Univ. of California, Santa Barbara, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
OSTI Identifier:
1633907
Grant/Contract Number:  
SC0009254; DMS-0956210; DMS-1616353
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Scientific Computing
Additional Journal Information:
Journal Volume: 76; Journal Issue: 1; Journal ID: ISSN 0885-7474
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Manifolds; Laplace–Beltrami; Numerical methods; Partial differential equations

Citation Formats

Gross, B., and Atzberger, P. J. Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds. United States: N. p., 2017. Web. doi:10.1007/s10915-017-0617-2.
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Gross, B., & Atzberger, P. J. Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds. United States. https://doi.org/10.1007/s10915-017-0617-2
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Gross, B., and Atzberger, P. J. Mon . "Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds". United States. https://doi.org/10.1007/s10915-017-0617-2. https://www.osti.gov/servlets/purl/1633907.
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@article{osti_1633907,
title = {Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds},
author = {Gross, B. and Atzberger, P. J.},
abstractNote = {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.},
doi = {10.1007/s10915-017-0617-2},
journal = {Journal of Scientific Computing},
number = 1,
volume = 76,
place = {United States},
year = {Mon Dec 04 00:00:00 EST 2017},
month = {Mon Dec 04 00:00:00 EST 2017}
}
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Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1007/s10915-017-0617-2
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Cited by: 5 works
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Figures / Tables:

Figure 1 Figure 1: Radial Manifolds: A radial manifold is defined as a surface where each point can be connected by a line segment to the origin without intersecting the surface. Shown are three radial manifolds which for discussions we refer to interchangeably as the (i) Sphere / Manifold A, (ii) Dimplemore » / Manifold B, and (iii) Fountain / Manifold C. The manifolds are generated by the radial functions (i) r($\theta$, $\phi$) = 1.0, (ii) r($\theta$, $\phi$) = 1 + r0 sin(3$\phi$) cos($\theta$) with r0 = 0.4, and (iii) r($\theta$, $\phi$) = 1 + r0 sin(7$\phi$) cos($\theta$) with r0 = 0.4. For additional discussion of the differential geometry of radial manifolds see Appendix B.« less
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    A review of some geometric integrators
    journal, June 2018

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    • Advanced Modeling and Simulation in Engineering Sciences, Vol. 5, Issue 1
    • DOI: 10.1186/s40323-018-0110-y
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