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Ψec: A local spectral exterior calculus.

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Bibliographic Details
Title: Ψec: A local spectral exterior calculus.
Authors: Lessig, Christian1 (AUTHOR) christian.lessig@ovgu.de
Source: Applied & Computational Harmonic Analysis. Mar2021, Vol. 51, p56-103. 48p.
Subjects: Differential forms, Differential calculus, Calculus, Fourier transforms, Stokes equations, Wavelets (Mathematics), Integrals
Abstract: We introduce Ψ ec (R n) , a discretization of Cartan's exterior calculus of differential forms using wavelets. Our construction consists of differential r -form wavelets with flexible directional localization that provide tight frames for the spaces Ω r (R n) of forms in R 2 and R 3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the k -dimensional integral of an r -form is an (r − k) -form. They also verify Stokes' theorem for differential forms, with the most efficient finite dimensional approximation attained using directionally localized, curvelet- or ridgelet-like forms. The construction of Ψ ec (R n) builds on the geometric simplicity of the exterior calculus in the Fourier domain. We establish this structure by extending existing results on the Fourier transform of differential forms to a frequency description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
Abstract:We introduce Ψ ec (R n) , a discretization of Cartan's exterior calculus of differential forms using wavelets. Our construction consists of differential r -form wavelets with flexible directional localization that provide tight frames for the spaces Ω r (R n) of forms in R 2 and R 3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the k -dimensional integral of an r -form is an (r − k) -form. They also verify Stokes' theorem for differential forms, with the most efficient finite dimensional approximation attained using directionally localized, curvelet- or ridgelet-like forms. The construction of Ψ ec (R n) builds on the geometric simplicity of the exterior calculus in the Fourier domain. We establish this structure by extending existing results on the Fourier transform of differential forms to a frequency description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators. [ABSTRACT FROM AUTHOR]
ISSN:10635203
DOI:10.1016/j.acha.2020.10.003