{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:34:27Z","timestamp":1787337267107,"version":"build-2736575974"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1912653"],"award-info":[{"award-number":["1912653"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2023,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>While solving PDEs with finite element methods, serendipity elements allow us to obtain the same order of accuracy as rectangular tensor-product elements with many fewer degrees of freedom. To realize the possible computational savings, we develop some additive Schwarz methods based on solving local patch problems. Adapting arguments from Pavarino for the tensor-product case, we prove that patch smoothers give conditioning estimates independent of the polynomial degree for a model problem. We also combine this with a low-order global operator to give an optimal two-grid method, with conditioning estimates independent of the mesh size and polynomial degree. The theory holds for serendipity elements in two and three dimensions and can be extended to multigrid algorithms. Numerical experiments using Firedrake and PETSc confirm this theory and demonstrate efficiency relative to standard elements.<\/jats:p>","DOI":"10.1137\/21m1415674","type":"journal-article","created":{"date-parts":[[2023,6,28]],"date-time":"2023-06-28T17:17:07Z","timestamp":1687972627000},"page":"S401-S420","source":"Crossref","is-referenced-by-count":1,"title":["Additive Schwarz Methods for Serendipity Elements"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9612-3069","authenticated-orcid":true,"given":"Jorge","family":"Marchena-Menendez","sequence":"first","affiliation":[{"name":"Department of Mathematics, Baylor University, Waco, TX 76706 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Robert C.","family":"Kirby","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Baylor University, Waco, TX 76706 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,6,27]]},"reference":[{"key":"ref1","doi-asserted-by":"crossref","unstructured":"R. Abu-Labdeh , \nS. MacLachlan , and \nP. E. Farrell , Monolithic Multigrid for Implicit Runge-Kutta Discretizations of Incompressible Fluid flow, preprint, arXiv:2202.07381, 2022.","DOI":"10.1016\/j.jcp.2023.111961"},{"key":"ref2","first-page":"1","author":"Arbogast T.","year":"2022","journal-title":"Numer. Math."},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-011-9087-3"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-014-0631-3"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1007\/PL00005386"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1145\/3490485"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-97-00826-0"},{"key":"ref9","unstructured":"P. E. Farrell , \nM. G. Knepley , \nF. Wechsung , and \nL. Mitchell , PCPATCH: Software for the Topological Construction of Multigrid Relaxation Methods, preprint, arXiv:1912.08516, 2019."},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1137\/18M1219370"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.5802\/smai-jcm.41"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1137\/17M1133208"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1137\/15M1026092"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.09.037"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385709"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1145\/2998441"},{"key":"ref17","doi-asserted-by":"crossref","unstructured":"J. Sch\u00f6berl , Robust Multigrid Methods for Parameter Dependent Problems, Ph.D. thesis, Johannes Kepler Universit\u00e4t Lin, 1999.","DOI":"10.1007\/978-3-642-58734-4_15"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drl046"},{"key":"ref19","volume-title":"Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics","author":"Thom\u00e9e V.","year":"2006"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1137\/0710062"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/21M1415674","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:53:35Z","timestamp":1787334815000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/21M1415674"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,27]]},"references-count":20,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,6,30]]}},"alternative-id":["10.1137\/21M1415674"],"URL":"https:\/\/doi.org\/10.1137\/21m1415674","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,27]]}}}