{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:18:27Z","timestamp":1776845907634,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"249","license":[{"start":{"date-parts":[[2005,3,23]],"date-time":"2005-03-23T00:00:00Z","timestamp":1111536000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    There has been a long-standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem. This is related to knowing whether the discrete Green\u2019s function is positive for triangular meshes allowing sufficiently good approximation of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    functions. We study this question for the Poisson problem in two dimensions discretized via the Galerkin method with continuous piecewise linears. We give examples which show that in general the answer is negative, and furthermore we extend the number of cases where it is known to be positive. Our techniques utilize some new results about discrete Green\u2019s functions that are of independent interest.\n                  <\/p>","DOI":"10.1090\/s0025-5718-04-01651-5","type":"journal-article","created":{"date-parts":[[2004,9,20]],"date-time":"2004-09-20T09:57:53Z","timestamp":1095674273000},"page":"1-23","source":"Crossref","is-referenced-by-count":80,"title":["Failure of the discrete maximum principle for an elliptic finite element problem"],"prefix":"10.1090","volume":"74","author":[{"given":"Andrei","family":"Dr\u0103g\u0103nescu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Todd","family":"Dupont","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"L.","family":"Scott","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2004,3,23]]},"reference":[{"key":"1","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3658-8","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0387954511","edition":"2"},{"key":"2","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/0045-7825(73)90019-4","article-title":"Maximum principle and uniform convergence for the finite element method","volume":"2","author":"Ciarlet, P. G.","year":"1973","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"Michael S. Floater, One-to-one piecewise linear mappings over triangulations, Math. Comp. 72 (2003), no. 242, 685\u2013696 (electronic).","DOI":"10.1090\/S0025-5718-02-01466-7"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/BF02243548","article-title":"Some remarks on the discrete maximum-principle for finite elements of higher order","volume":"27","author":"H\u00f6hn, W.","year":"1981","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"issue":"233","key":"5","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1090\/S0025-5718-00-01270-9","article-title":"Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle","volume":"70","author":"Korotov, Sergey","year":"2001","journal-title":"Math. 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