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Numerical methods for nonlocal and fractional models

Published online by Cambridge University Press:  30 November 2020

Marta D’Elia ,
Qiang Du ,
Christian Glusa ,
Max Gunzburger ,
Xiaochuan Tian  and
Zhi Zhou
Marta D’Elia
Affiliation:
Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA E-mail: mdelia@sandia.gov, caglusa@sandia.gov http://sites.google.com/site/martadeliawebsite
Qiang Du
Affiliation:
Department of Applied Physics and Applied Mathematics and Data Science Institute, Columbia University, New York, NY 10027, USA E-mail: qd2125@columbia.edu http://www.columbia.edu/~qd2125
Christian Glusa
Affiliation:
Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA E-mail: mdelia@sandia.gov, caglusa@sandia.gov http://sites.google.com/site/martadeliawebsite
Max Gunzburger
Affiliation:
Department of Scientific Computing, Florida State University, Tallahassee, Florida 32306, USA E-mail: mgunzburger@fsu.edu http://www.sc.fsu.edu/gunzburg
Xiaochuan Tian
Affiliation:
Department of Mathematics, University of Texas, Austin, Texas 78751, USA E-mail: xtian@math.utexas.edu http://web.ma.utexas.edu/users/xtian
Zhi Zhou
Affiliation:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China E-mail: zhizhou@polyu.edu.hk http://sites.google.com/site/zhizhou0125
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Abstract

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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Type
Research Article
Information
Acta Numerica , Volume 29 , May 2020 , pp. 1 - 124
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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