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Spectral numerical exterior calculus methods for differential equations on radial manifolds. (English) Zbl 1404.65225

Summary: 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.

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
58A14 Hodge theory in global analysis
65D32 Numerical quadrature and cubature formulas
58A15 Exterior differential systems (Cartan theory)

Software:

SymPy; SHTns
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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