{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:19:13Z","timestamp":1760239153590,"version":"build-2065373602"},"reference-count":57,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T00:00:00Z","timestamp":1600905600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["2016\/19648-9","2017\/11428-2","2018\/07643-8"],"award-info":[{"award-number":["2016\/19648-9","2017\/11428-2","2018\/07643-8"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002322","name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior","doi-asserted-by":"publisher","award":["PROEX-9740044\/D"],"award-info":[{"award-number":["PROEX-9740044\/D"]}],"id":[{"id":"10.13039\/501100002322","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the finite element method (FEM) on unstructured grids. Piecewise linear finite element spaces on triangles are used for the space discretization, whereas the time discretization is based on the backward-Euler and the Crank\u2013Nicolson methods. The quadrature rules for discretizing the Volterra integral term are chosen so as to be consistent with the time-stepping schemes; a more efficient version of the implementation that uses a vectorization technique in the assembly process is also presented. The compactness of the approach is demonstrated using the software Matrix Laboratory (MATLAB). The efficiency is demonstrated via a numerical example on an L-shaped domain, for which a comparison is possible against the commercially available finite element software COMSOL Multiphysics. Moreover, further consideration indicates that COMSOL Multiphysics cannot be directly applied to 2D PIDEs containing more complex kernels in the Volterra integral term, whereas our method can. Consequently, the subroutines we present constitute a valuable open and validated resource for solving more general 2D PIDEs.<\/jats:p>","DOI":"10.3390\/a13100242","type":"journal-article","created":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T08:33:03Z","timestamp":1600936383000},"page":"242","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Compact FEM Implementation for Parabolic Integro-Differential Equations in 2D"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0084-9584","authenticated-orcid":false,"given":"Gujji Murali Mohan","family":"Reddy","sequence":"first","affiliation":[{"name":"Department of Mathematics, Birla Institute of Technology &amp; Science, Pilani, Hyderabad Campus, Telangana 500078, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alan B.","family":"Seitenfuss","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Institute of Mathematical and Computer Sciences, University of S\u00e3o Paulo at S\u00e3o Carlos, P.O. Box 668, S\u00e3o Carlos 13560-970, S\u00e3o Paulo, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8783-200X","authenticated-orcid":false,"given":"D\u00e9bora de Oliveira","family":"Medeiros","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Institute of Mathematical and Computer Sciences, University of S\u00e3o Paulo at S\u00e3o Carlos, P.O. Box 668, S\u00e3o Carlos 13560-970, S\u00e3o Paulo, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luca","family":"Meacci","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Institute of Mathematical and Computer Sciences, University of S\u00e3o Paulo at S\u00e3o Carlos, P.O. Box 668, S\u00e3o Carlos 13560-970, S\u00e3o Paulo, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Milton","family":"Assun\u00e7\u00e3o","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Institute of Mathematical and Computer Sciences, University of S\u00e3o Paulo at S\u00e3o Carlos, P.O. Box 668, S\u00e3o Carlos 13560-970, S\u00e3o Paulo, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8318-1251","authenticated-orcid":false,"given":"Michael","family":"Vynnycky","sequence":"additional","affiliation":[{"name":"Division of Processes, Department of Materials Science and Technology, KTH Royal Institute of Technology, Brinellv\u00e4gen 23, 100 44 Stockholm, Sweden"},{"name":"Department of Mathematics and Statistics, University of Limerick, Limerick V94 T9PX, Ireland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/BF00281373","article-title":"A general theory of heat conduction with finite wave speeds","volume":"31","author":"Gurtin","year":"1968","journal-title":"Arch. Ration. Mech. 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