Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlocal vector calculus on the sphere

Hadrien Montanelli, Richard Mikael Slevinsky, Qiang Du

TL;DR

This work develops a nonlocal vector calculus on the unit sphere ${\mathbb S}^2$ using weakly singular integral operators. By adopting spherical-harmonic bases, the nonlocal surface divergence, gradient, and curl become diagonally scaled operators, enabling a nonlocal Stokes theorem on a curved surface and establishing strong convergence to the classical local operators as the horizon $\delta\to0$. The framework relies on antisymmetric kernels $\boldsymbol{\beta}$ and geodesically symmetric kernels $\boldsymbol{\alpha}$, together with an averaging operator $\mathcal{A}^\delta$ to connect nonlocal and local quantities. These results provide a spectrally clean, stable, and asymptotically compatible approach to solving nonlocal advection-diffusion-type problems on the sphere, with potential extensions to other closed surfaces and applications in atmospheric modeling and related kernel-based methods.

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.

Nonlocal vector calculus on the sphere

TL;DR

This work develops a nonlocal vector calculus on the unit sphere using weakly singular integral operators. By adopting spherical-harmonic bases, the nonlocal surface divergence, gradient, and curl become diagonally scaled operators, enabling a nonlocal Stokes theorem on a curved surface and establishing strong convergence to the classical local operators as the horizon . The framework relies on antisymmetric kernels and geodesically symmetric kernels , together with an averaging operator to connect nonlocal and local quantities. These results provide a spectrally clean, stable, and asymptotically compatible approach to solving nonlocal advection-diffusion-type problems on the sphere, with potential extensions to other closed surfaces and applications in atmospheric modeling and related kernel-based methods.

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.
Paper Structure (5 sections, 17 theorems, 192 equations)

This paper contains 5 sections, 17 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Nonlocal operators and their adjoints
  3. Diagonalization and nonlocal Stokes theorem
  4. Analysis of the scaling and convergence to the operators of local vector calculus
  5. Discussion

Key Result

lemma 1

For any (twice) differentiable $u:\mathbb{S}^2\to\mathbb{R}$ and differentiable $\boldsymbol{V}:\mathbb{S}^2\to\mathbb{R}^3$, where $\mathcal{L}_S^0=\Delta_S$ is the Laplace--Betlrami operator defined by Moreover, i.e., and, in particular, Finally, i.e.,

Theorems & Definitions (35)

  • definition 1: Local operators
  • lemma 1: Local vector calculus
  • theorem 1: Local adjoints
  • proof
  • definition 2: Geodesic symmetry
  • definition 3: Nonlocal operators
  • theorem 2: Nonlocal adjoints
  • proof
  • theorem 3: Nonlocal compositions
  • proof
  • ...and 25 more