Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains

Pablo Seleson; Max Gunzburger; Michael L. Parks

doi:10.1016/j.cma.2013.05.018

Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains

Pablo Seleson, Max Gunzburger, Michael L. Parks

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87 Scopus citations

Abstract

We investigate interface problems in nonlocal diffusion and demonstrate how to reformulate and generalize the classical treatment of interface problems in the presence of nonlocal interactions. Through formal derivations, we show that nonlocal diffusion interface problems converge to their classical local counterparts, in the limit of vanishing nonlocality. A central focus of this paper is local/nonlocal interface problems, or interface problems with sharp transitions between local and nonlocal domains. Such problems can be cast as instances of a nonlocal interface problem, with a finite horizon in certain regions and a vanishing horizon in other regions. We derive a local/nonlocal interface problem and utilize conservation principles to obtain local/nonlocal interface conditions. Comparisons between nonlocal, local, and local/nonlocal interface problems are presented, analytically and numerically, with a focus on multiscale aspects of nonlocal models induced by their inherent length scales.

Original language	English
Pages (from-to)	185-204
Number of pages	20
Journal	Computer Methods in Applied Mechanics and Engineering
Volume	266
DOIs	https://doi.org/10.1016/j.cma.2013.05.018
State	Published - Nov 1 2013
Externally published	Yes

Funding

This research was supported by DOE Grant DE-SC0004970 at Florida State University, by DOE Grant DE-FG02-05ER25701 at the University of Texas, and by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC0494-AL85000. We acknowledge helpful discussions with David Littlewood, Stewart Silling, Jakob Ostien, and Qiang Du. Pablo Seleson acknowledges support from the ICES Postdoctoral Fellowship Program and useful discussions with Serge Prudhomme and Leszek Demkowicz.

Funders	Funder number
University of Texas
U.S. Department of Energy	DE-SC0004970
Sandia National Laboratories
Florida State University	DE-FG02-05ER25701
Laboratory Directed Research and Development
Institute for Clinical Evaluative Sciences

Keywords

Interface conditions
Interface problems
Multiscale modeling
Nonlocal diffusion

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10.1016/j.cma.2013.05.018

Cite this

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abstract = "We investigate interface problems in nonlocal diffusion and demonstrate how to reformulate and generalize the classical treatment of interface problems in the presence of nonlocal interactions. Through formal derivations, we show that nonlocal diffusion interface problems converge to their classical local counterparts, in the limit of vanishing nonlocality. A central focus of this paper is local/nonlocal interface problems, or interface problems with sharp transitions between local and nonlocal domains. Such problems can be cast as instances of a nonlocal interface problem, with a finite horizon in certain regions and a vanishing horizon in other regions. We derive a local/nonlocal interface problem and utilize conservation principles to obtain local/nonlocal interface conditions. Comparisons between nonlocal, local, and local/nonlocal interface problems are presented, analytically and numerically, with a focus on multiscale aspects of nonlocal models induced by their inherent length scales.",

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AB - We investigate interface problems in nonlocal diffusion and demonstrate how to reformulate and generalize the classical treatment of interface problems in the presence of nonlocal interactions. Through formal derivations, we show that nonlocal diffusion interface problems converge to their classical local counterparts, in the limit of vanishing nonlocality. A central focus of this paper is local/nonlocal interface problems, or interface problems with sharp transitions between local and nonlocal domains. Such problems can be cast as instances of a nonlocal interface problem, with a finite horizon in certain regions and a vanishing horizon in other regions. We derive a local/nonlocal interface problem and utilize conservation principles to obtain local/nonlocal interface conditions. Comparisons between nonlocal, local, and local/nonlocal interface problems are presented, analytically and numerically, with a focus on multiscale aspects of nonlocal models induced by their inherent length scales.

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Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains

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