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On the other hand, in many situations,<jats:italic>nonlocal models<\/jats:italic>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.<\/jats:p>","DOI":"10.1017\/s096249292000001x","type":"journal-article","created":{"date-parts":[[2020,11,30]],"date-time":"2020-11-30T14:55:12Z","timestamp":1606748112000},"page":"1-124","source":"Crossref","is-referenced-by-count":170,"title":["Numerical methods for nonlocal and fractional models"],"prefix":"10.1017","volume":"29","author":[{"given":"Marta","family":"D\u2019Elia","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qiang","family":"Du","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christian","family":"Glusa","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Max","family":"Gunzburger","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaochuan","family":"Tian","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhi","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2020,11,30]]},"reference":[{"key":"S096249292000001X_r167","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(96)01082-1"},{"key":"S096249292000001X_r78","first-page":"667","article-title":"Optimal distributed control of nonlocal steady diffusion problems","volume":"55","author":"D\u2019Elia","year":"2014","journal-title":"SIAM J. 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