2020/05/07 by da Silva, Clauson Carvalho, Lessig, Christian, Dodov, Boyko +2
#58J90 #65M70 #65T60 #FOS: Mathematics #FOS: Physical sciences #Geophysics (physics.geo-ph) #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2005.03598
We introduce Ψec, a local spectral exterior calculus for the two-sphere S2. Ψec provides a discretization of Cartan's exterior calculus on S2 formed by spherical differential r-form wavelets. These are well localized in space and frequency and provide (Stevenson) frames for the homogeneous Sobolev spaces H-r+1( Ωνr , S2 ) of differential r-forms. At the same time, they satisfy important properties of the exterior calculus, such as the de Rahm complex and the Hodge-Helmholtz decomposition. Through this, Ψec is tailored towards structure preserving discretizations that can adapt to solutions with varying regularity. The construction of Ψec is based on a novel spherical wavelet frame for L2(S2) that we obtain by introducing scalable reproducing kernel frames. These extend scalable frames to weighted sampling expansions and provide an alternative to quadrature rules for the discretization of needlet-like scale-discrete wavelets. We verify the practicality of Ψec for numerical computations using the rotating shallow water equations. Our numerical results demonstrate that a Ψec-based discretization of the equations attains accuracy comparable to those of spectral methods while using a representation that is well localized in space and frequency.