English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We introduce $\Psi\mathrm{ec}$, a local spectral exterior calculus for thetwo-sphere $S^2$. $\Psi\mathrm{ec}$ provides a discretization of Cartan'sexterior calculus on $S^2$ formed by spherical differential $r$-form wavelets.These are well localized in space and frequency and provide (Stevenson) framesfor the homogeneous Sobolev spaces $\dot{H}^{-r+1}( \Omega_{\nu}^{r} , S^2 )$of differential $r$-forms. At the same time, they satisfy important propertiesof the exterior calculus, such as the de Rahm complex and the Hodge-Helmholtzdecomposition. Through this, $\Psi\mathrm{ec}$ is tailored towards structurepreserving discretizations that can adapt to solutions with varying regularity.The construction of $\Psi\mathrm{ec}$ is based on a novel spherical waveletframe for $L_2(S^2)$ that we obtain by introducing scalable reproducing kernelframes. These extend scalable frames to weighted sampling expansions andprovide an alternative to quadrature rules for the discretization ofneedlet-like scale-discrete wavelets. We verify the practicality of$\Psi\mathrm{ec}$ for numerical computations using the rotating shallow waterequations. Our numerical results demonstrate that a $\Psi\mathrm{ec}$-baseddiscretization of the equations attains accuracy comparable to those ofspectral methods while using a representation that is well localized in spaceand frequency.

A Local Spectral Exterior Calculus for the Sphere and Application to the Shallow Water Equations