{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T13:42:36Z","timestamp":1773409356452,"version":"3.50.1"},"reference-count":33,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This paper deals with the efficient implementation of the finite element method with continuous piecewise linear functions (P1-FEM) in <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi mathvariant=\"double-struck\">R<\/m:mi>\n                              <m:mi>d<\/m:mi>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0239_ineq_0001.png\"\/>\n                        <jats:tex-math>\\mathbb{R}^{d}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> (<jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>d<\/m:mi>\n                              <m:mo>\u2208<\/m:mo>\n                              <m:mi mathvariant=\"double-struck\">N<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0239_ineq_0002.png\"\/>\n                        <jats:tex-math>d\\in\\mathbb{N}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>).\nAlthough at present there does not seem to be a very high practical demand for finite element methods that use higher-dimensional simplicial partitions, there are some advantages in studying the efficient implementation of the method independent of the dimension.\nFor instance, it provides additional insights into necessary data structures and the complexity of implementations.\nThroughout, the focus is on an efficient realization using <jats:sc>Matlab<\/jats:sc> built-in functions and vectorization.\nThe fast and vectorized <jats:sc>Matlab<\/jats:sc> function can be easily implemented in many other vector languages and is provided in Julia, too.\nThe complete implementation of the adaptive FEM is given, including assembling stiffness matrix, building load vector, error estimation, and adaptive mesh-refinement.\nNumerical experiments underline the efficiency of our freely available code which is observed to be of a slightly more than linear complexity with respect to the number of elements when memory limits are not exceeded.<\/jats:p>","DOI":"10.1515\/cmam-2022-0239","type":"journal-article","created":{"date-parts":[[2024,1,24]],"date-time":"2024-01-24T20:41:01Z","timestamp":1706128861000},"page":"283-324","source":"Crossref","is-referenced-by-count":2,"title":["Efficient P1-FEM for Any Space Dimension in Matlab"],"prefix":"10.1515","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-9385-5708","authenticated-orcid":false,"given":"Stefanie","family":"Feifel","sequence":"first","affiliation":[{"name":"Institute of Numerical Mathematics , 9189 Ulm University , Helmholtzstra\u00dfe 20, 89069 Ulm , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4316-1512","authenticated-orcid":false,"given":"Stefan A.","family":"Funken","sequence":"additional","affiliation":[{"name":"Institute of Numerical Mathematics , 9189 Ulm University , Helmholtzstra\u00dfe 20, 89069 Ulm , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2024,1,25]]},"reference":[{"key":"2025062712372308878_j_cmam-2022-0239_ref_001","doi-asserted-by":"crossref","unstructured":"J. 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Funken,\nP1afemnd - efficient implementation of adaptive p1-fem in arbitrary dimensions,\nsoftware download at https:\/\/github.com\/sbeuteruulm\/P1AFEMnD."},{"key":"2025062712372308878_j_cmam-2022-0239_ref_005","doi-asserted-by":"crossref","unstructured":"J. Bezanson, A. Edelman, S. Karpinski and V. B. Shah,\nJulia: A fresh approach to numerical computing,\nSIAM Rev. 59 (2017), no. 1, 65\u201398.","DOI":"10.1137\/141000671"},{"key":"2025062712372308878_j_cmam-2022-0239_ref_006","doi-asserted-by":"crossref","unstructured":"A. Bliss and F. E. Su,\nLower bounds for simplicial covers and triangulations of cubes,\nDiscrete Comput. Geom. 33 (2005), no. 4, 669\u2013686.","DOI":"10.1007\/s00454-004-1128-0"},{"key":"2025062712372308878_j_cmam-2022-0239_ref_007","doi-asserted-by":"crossref","unstructured":"J. Brandts, S. Korotov and M. 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Hughes,\nMinimum-cardinality triangulations of the \ud835\udc51-cube for \n                  \n                     \n                        \n                           d\n                           =\n                           5\n                        \n                     \n                     \n                     d=5\n                  \n                and \n                  \n                     \n                        \n                           d\n                           =\n                           6\n                        \n                     \n                     \n                     d=6\n                  \n               ,\nDiscrete Math. 118 (1993), no. 1\u20133, 75\u2013118.","DOI":"10.1016\/0012-365X(93)90055-X"},{"key":"2025062712372308878_j_cmam-2022-0239_ref_019","doi-asserted-by":"crossref","unstructured":"R. B. Hughes and M. R. 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