{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:38:03Z","timestamp":1776793083522,"version":"3.51.2"},"reference-count":31,"publisher":"American Mathematical Society (AMS)","issue":"273","license":[{"start":{"date-parts":[[2011,7,8]],"date-time":"2011-07-08T00:00:00Z","timestamp":1310083200000},"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                    The purpose of this paper is to propose a proof for the Poincar\u00e9-Friedrichs inequality for piecewise\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 on\n                    <italic>anisotropic<\/italic>\n                    meshes. By verifying suitable assumptions involved in the newly proposed proof, we show that the Poincar\u00e9-Friedrichs inequality for piecewise\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 holds independently of the aspect ratio which characterizes the shape-regular condition in finite element analysis. In addition, under the maximum angle condition, we establish the Poincar\u00e9-Friedrichs inequality for the Crouzeix-Raviart nonconforming linear finite element. Counterexamples show that the maximum angle condition is only sufficient.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2010-02296-3","type":"journal-article","created":{"date-parts":[[2010,10,4]],"date-time":"2010-10-04T11:01:54Z","timestamp":1286190114000},"page":"119-140","source":"Crossref","is-referenced-by-count":2,"title":["On the Poincar\u00e9-Friedrichs inequality for piecewise \ud835\udc3b\u00b9 functions in anisotropic discontinuous Galerkin finite element methods"],"prefix":"10.1090","volume":"80","author":[{"given":"Huo-Yuan","family":"Duan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Roger","family":"Tan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2010,7,8]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1016\/S0377-0427(00)00564-1","article-title":"Lagrange and average interpolation over 3D anisotropic elements","volume":"135","author":"Acosta, Gabriel","year":"2001","journal-title":"J. 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