Nonlocal Helmholtz Decompositions and Connections to Classical Counterparts

Authors

Andrew Haar, University of Nebraska - LincolnFollow
Petronela Radu, University of Nebraska - LincolnFollow

Date of this Version

Spring 5-10-2020

Document Type

Poster

Citation

Poster presentation, UCARE Research Fair, Spring 2020, University of Nebraska-Lincoln.

Comments

Copyright 2020 by the authors.

Abstract

In recent years nonlocal models have been successfully introduced in a variety of applications, such as dynamic fracture, nonlocal diffusion, flocking, and image processing. Thus, the development of a nonlocal calculus theory, together with the study of nonlocal operators has become the focus of many theoretical investigations. Our work focuses on a Helmholtz decomposition in the nonlocal (integral) framework. In the classical (differential) setting the Helmholtz decomposition states that we can decompose a three dimensional vector field as a sum of an irrotational function and a solenoidal function. We will define new nonlocal gradient and curl operators that allow us to create a similar nonlocal decomposition.