Nonlocal vector calculus on the sphere

2025/05/18 by Montanelli, Hadrien, Slevinsky, Richard Mikael, Du, Qiang
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.12372

Abstract

We introduce a nonlocal vector calculus on the unit two-sphere using weakly singular integral operators. Within this framework, the operators are diagonalizable in terms of scalar and vector spherical harmonics, a property that facilitates the proof of a nonlocal Stokes theorem. This constitutes the first instance of such a theorem on a curved surface. Furthermore, our analysis demonstrates the strong convergence of these nonlocal operators to the classical differential operators of vector calculus as the interaction range tends to zero.

Citations

• A rapid and well-conditioned algorithm for the Helmholtz--Hodge decomposition of vector fields on the sphere
• A spectral method for nonlocal diffusion operators on the sphere
• Conquering the pre-computation in two-dimensional harmonic polynomial transforms
• Fast and backward stable transforms between spherical harmonic\n expansions and bivariate Fourier series
• Fourth-order time-stepping for stiff PDEs on the sphere
• Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential\n integrators
• Vector spherical harmonics and their application to magnetostatics
• Reformulation of elasticity theory for discontinuities and long-range forces

Related

• Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition
• Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications
• Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications
• Nonlocal Ordered Mean Curvature with Integrable Kernel
• A Generalized Nonlocal Calculus with Application to the Peridynamics Model for Solid Mechanics