Connections between nonlocal operators: From vector calculus identities to a fractional Helmholtz decomposition
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)
- Univ. of Tennessee, Knoxville, TN (United States)
- Columbia Univ., New York, NY (United States)
Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields such as hydrology, mechanics, and image processing. In this work, we study the analytical underpinnings of these operators. We rigorously treat compositions of nonlocal operators, prove nonlocal vector calculus identities, and connect weighted and unweighted variational frameworks. We combine these results to obtain a weighted fractional Helmholtz decomposition which is valid for sufficiently smooth vector fields. Our approach identifies the function spaces in which the stated identities and decompositions hold, providing a rigorous foundation to the nonlocal vector calculus identities that can serve as tools for nonlocal modeling in higher dimensions.
- Research Organization:
- Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
- DOE Contract Number:
- NA0003525
- OSTI ID:
- 1855046
- Report Number(s):
- SAND2021-15379R; 702662
- Country of Publication:
- United States
- Language:
- English
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