Helmholtz decomposition in the nonlocal framework: Well-posedness analysis and applications.

D’Elia, Marta; Flores, Cynthia M. Navarro; Li, Xingjie; Radu, Petronela; Yu, Yue

doi:10.2172/1497417

Cited by 1 publication

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“…A nonlocal vector calculus has been developed (Gunzburger and Lehoucq 2010, Du et al 2012 a , Du, Gunzburger, Lehoucq and Zhou 2013 a , Alali, Gunzburger and Liu 2015, Mengesha and Du 2016, Du 2019, D’Elia et al 2019 a ) to deal with nonlocal models such as (1.5) in much the same way as the classical vector calculus is used to deal with PDE models such as (1.1). Here, mostly following Du et al (2013 a ), we provide a brief introduction to the nonlocal vector calculus, including notions that are used in the rest of the article.…”

Section: General Models For Nonlocal Diffusionmentioning

confidence: 99%

“…In addition, the operators introduced in this remark are used in D’Elia et al (2019 a ) to define the nonlocal Hodge–Helmholtz decomposition of vector-valued functions that depend on two points and . Note that the operators , , and , as operators between functions of two points and and functions of one point or , do not recover their classical counterparts in the local limit .…”

Section: General Models For Nonlocal Diffusionmentioning

confidence: 99%

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Numerical methods for nonlocal and fractional models

et al. 2020

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Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

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Section: General Models For Nonlocal Diffusionmentioning

confidence: 99%