$Ψ$ec: A Local Spectral Exterior Calculus
This work provides a new computational framework for exterior calculus, potentially benefiting numerical simulations in physics and engineering that require accurate discretization of differential forms.
The paper introduces Ψec, a wavelet-based discretization of Cartan's exterior calculus for differential forms in ℝ² and ℝ³, achieving tight frames that satisfy the de Rham complex, Hodge decomposition, and Stokes' theorem. The construction leverages Fourier-domain geometric simplicity, enabling directionally localized forms like curvelets and ridgelets.
We introduce $Ψ\mathrm{ec}$, 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 $Ω^r(\mathbb{R}^n)$ of forms in $\mathbb{R}^2$ and $\mathbb{R}^3$. 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 $Ψ\mathrm{ec}$ 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.