{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T16:22:00Z","timestamp":1781194920487,"version":"3.54.1"},"reference-count":28,"publisher":"American Mathematical Society (AMS)","issue":"260","license":[{"start":{"date-parts":[[2008,5,3]],"date-time":"2008-05-03T00:00:00Z","timestamp":1209772800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Given a two dimensional oriented surface equipped with a simplicial mesh, the standard lowest order finite element spaces provide a complex\n \n \n \n \n X<\/mml:mi>\n \n \u2219\n \n <\/mml:mo>\n <\/mml:msup>\n X^\\bullet<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n centered on Raviart-Thomas divergence conforming vector fields. It can be seen as a realization of the simplicial\n cochain<\/italic>\n complex. We construct a new complex\n \n \n \n \n Y<\/mml:mi>\n \n \u2219\n \n <\/mml:mo>\n <\/mml:msup>\n Y^\\bullet<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of finite element spaces on the barycentric refinement of the mesh which can be seen as a realization of the simplicial\n chain<\/italic>\n complex on the original (unrefined) mesh, such that the\n \n \n \n \n \n L<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n \\mathrm {L}^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n duality is non-degenerate on\n \n \n \n \n \n Y<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msup>\n \n \u00d7\n \n <\/mml:mo>\n \n X<\/mml:mi>\n \n 2<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n i<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n Y^i \\times X^{2-i}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for each\n \n \n \n \n i<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n {<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n }<\/mml:mo>\n <\/mml:mrow>\n i\\in \\{0,1,2\\}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In particular\n \n \n \n \n Y<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n Y^1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a space of\n \n \n \n \n c<\/mml:mi>\n u<\/mml:mi>\n r<\/mml:mi>\n l<\/mml:mi>\n <\/mml:mrow>\n \\mathrm {curl}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -conforming vector fields which is\n \n \n \n \n \n L<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n \\mathrm {L}^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n dual to Raviart-Thomas\n \n \n \n div<\/mml:mi>\n \\operatorname {div}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -conforming elements. When interpreted in terms of differential forms, these two complexes provide a finite-dimensional analogue of Hodge duality.\n <\/p>","DOI":"10.1090\/s0025-5718-07-01965-5","type":"journal-article","created":{"date-parts":[[2007,7,26]],"date-time":"2007-07-26T07:46:03Z","timestamp":1185435963000},"page":"1743-1769","source":"Crossref","is-referenced-by-count":242,"title":["A dual finite element complex on the barycentric refinement"],"prefix":"10.1090","volume":"76","author":[{"given":"Annalisa","family":"Buffa","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Snorre","family":"Christiansen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2007,5,3]]},"reference":[{"issue":"226","key":"1","doi-asserted-by":"publisher","first-page":"607","DOI":"10.1090\/S0025-5718-99-01013-3","article-title":"An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations","volume":"68","author":"Alonso, Ana","year":"1999","journal-title":"Math. 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