{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T13:20:24Z","timestamp":1775654424577,"version":"3.50.1"},"reference-count":53,"publisher":"Association for Computing Machinery (ACM)","issue":"1","funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"crossref","award":["EP\/W026260\/1"],"award-info":[{"award-number":["EP\/W026260\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2026,3,31]]},"abstract":"<jats:p>\n                    Many classical and modern finite element spaces are derived by dividing each computational cell into finer pieces. Such\n                    <jats:italic toggle=\"yes\">macroelements<\/jats:italic>\n                    frequently enable the enforcement of mathematically desirable properties such as divergence-free conditions or\n                    <jats:inline-formula content-type=\"math\/tex\">\n                      <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(C^{1}\\)<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    continuity in a simpler or more efficient manner than elements without the subdivision. Although a few modern software projects provide one-off support for particular macroelements, a general approach facilitating broad-based support has, until now, been lacking. In this work, we describe a major addition to the FIAT project to support a wide range of different macroelements. These enhancements have been integrated into the Firedrake code stack. We provide numerical evaluation of the new macroelement facility.\n                  <\/jats:p>","DOI":"10.1145\/3797879","type":"journal-article","created":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T16:07:50Z","timestamp":1770998870000},"page":"1-28","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["FIAT: Enabling Classical and Modern Macroelements"],"prefix":"10.1145","volume":"52","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3824-0080","authenticated-orcid":false,"given":"Pablo D.","family":"Brubeck","sequence":"first","affiliation":[{"name":"Mathematical Institute, University of Oxford, Oxford, United Kingdom of Great Britain and Northern Ireland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-9663-8394","authenticated-orcid":false,"given":"Robert C.","family":"Kirby","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Baylor University, Waco, Texas, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2026,4,8]]},"reference":[{"key":"e_1_3_1_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/11082539X"},{"key":"e_1_3_1_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-015-0557-x"},{"key":"e_1_3_1_4_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-08-02071-1"},{"key":"e_1_3_1_5_1","doi-asserted-by":"publisher","DOI":"10.1142\/s021820251350067x"},{"key":"e_1_3_1_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF03167064"},{"key":"e_1_3_1_7_1","first-page":"28","article-title":"Quadratic velocity\/linear pressure Stokes elements","volume":"7","author":"Arnold Douglas N.","year":"1992","unstructured":"Douglas N. Arnold and Jinshui Qin. 1992. Quadratic velocity\/linear pressure Stokes elements. Advances in Computer Methods for Partial Differential Equations 7 (1992), 28\u201334.","journal-title":"Advances in Computer Methods for Partial Differential Equations"},{"key":"e_1_3_1_8_1","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100348"},{"key":"e_1_3_1_9_1","doi-asserted-by":"publisher","DOI":"10.1142\/s0218202503002507"},{"key":"e_1_3_1_10_1","doi-asserted-by":"publisher","DOI":"10.5802\/smai-jcm.91"},{"key":"e_1_3_1_11_1","doi-asserted-by":"publisher","DOI":"10.1145\/1268776.1268779"},{"key":"e_1_3_1_12_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1985-0771031-7"},{"key":"e_1_3_1_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2024.02.005"},{"key":"e_1_3_1_14_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0"},{"key":"e_1_3_1_15_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-004-4135-7"},{"key":"e_1_3_1_16_1","article-title":"FIAT: Improving performance and accuracy for high-order finite elements","author":"Brubeck Pablo D.","year":"2024","unstructured":"Pablo D. Brubeck, Robert C. Kirby, Fabian Laakmann, and Lawrence Mitchell. 2024. FIAT: Improving performance and accuracy for high-order finite elements. ACM Transactions on Mathematical Software 51 (2024), 1\u201317.","journal-title":"ACM Transactions on Mathematical Software"},{"key":"e_1_3_1_17_1","doi-asserted-by":"publisher","DOI":"10.1137\/100802906"},{"key":"e_1_3_1_18_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719208"},{"key":"e_1_3_1_19_1","first-page":"515","volume-title":"Proceedings of the 1st Conference on Matrix Methods in Structural Mechanics","author":"Clough Ray W.","year":"1965","unstructured":"Ray W. Clough and J. L. Tocher. 1965. Finite element stiffness matrices for analysis of plate bending. In Proceedings of the 1st Conference on Matrix Methods in Structural Mechanics, 515\u2013546."},{"key":"e_1_3_1_20_1","doi-asserted-by":"crossref","unstructured":"Jay Gopalakrishnan Johnny Guzman and Jeonghun J. Lee. 2024. The Johnson-Mercier elasticity element in any dimensions. arXiv:2403.13189. Retrieved from https:\/\/arxiv.org\/abs\/2403.13189","DOI":"10.1515\/jnma-2025-0020"},{"key":"e_1_3_1_21_1","doi-asserted-by":"publisher","DOI":"10.1137\/19M1248960"},{"key":"e_1_3_1_22_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.114983"},{"issue":"5","key":"e_1_3_1_23_1","first-page":"2826","article-title":"Inf-sup stable finite elements on barycentric refinements producing divergence\u2013free approximations in arbitrary dimensions","volume":"56","author":"Guzm\u00e1n Johnny","year":"2018","unstructured":"Johnny Guzm\u00e1n and Michael Neilan. 2018. Inf-sup stable finite elements on barycentric refinements producing divergence\u2013free approximations in arbitrary dimensions. 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