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


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.


Manifolds Laplace–Beltrami Numerical methods Partial differential equations 



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.


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Authors and Affiliations

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

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