{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T09:14:32Z","timestamp":1776849272601,"version":"3.51.2"},"reference-count":15,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["CA 151\/22-1"],"award-info":[{"award-number":["CA 151\/22-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This paper provides a discrete Poincar\u00e9 inequality in <jats:italic>n<\/jats:italic> space\ndimensions on a simplex <jats:italic>K<\/jats:italic> with explicit constants. This inequality bounds the norm of the piecewise derivative of\nfunctions with integral mean zero on <jats:italic>K<\/jats:italic> and all integrals of jumps zero\nalong all interior sides by its Lebesgue norm times <jats:inline-formula id=\"j_cmam-2017-0044_ineq_9999_w2aab3b7d645b1b6b1aab1c14b1b7Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>C<\/m:mi>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:mi>n<\/m:mi>\n                                 <m:mo stretchy=\"false\">)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mi>diam<\/m:mi>\n                                 <m:mo>\u2061<\/m:mo>\n                                 <m:mrow>\n                                    <m:mo stretchy=\"false\">(<\/m:mo>\n                                    <m:mi>K<\/m:mi>\n                                    <m:mo stretchy=\"false\">)<\/m:mo>\n                                 <\/m:mrow>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2017-0044_eq_1029.png\"\/>\n                        <jats:tex-math>{C(n)\\operatorname{diam}(K)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.\nThe explicit constant <jats:inline-formula id=\"j_cmam-2017-0044_ineq_9998_w2aab3b7d645b1b6b1aab1c14b1b9Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>C<\/m:mi>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:mi>n<\/m:mi>\n                                 <m:mo stretchy=\"false\">)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2017-0044_eq_1031.png\"\/>\n                        <jats:tex-math>{C(n)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> depends only on the dimension <jats:inline-formula id=\"j_cmam-2017-0044_ineq_9997_w2aab3b7d645b1b6b1aab1c14b1c11Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>n<\/m:mi>\n                              <m:mo>=<\/m:mo>\n                              <m:mrow>\n                                 <m:mn>2<\/m:mn>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mn>3<\/m:mn>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2017-0044_eq_1335.png\"\/>\n                        <jats:tex-math>{n=2,3}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>\nin case of an adaptive triangulation with the newest vertex bisection.\nThe second part of this paper proves the stability of an enrichment\noperator, which leads to the stability and approximation of a (discrete)\nquasi-interpolator applied in the proofs of the discrete Friedrichs\ninequality and discrete reliability estimate with explicit bounds on the\nconstants in terms of the minimal angle <jats:inline-formula id=\"j_cmam-2017-0044_ineq_9996_w2aab3b7d645b1b6b1aab1c14b1c13Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>\u03c9<\/m:mi>\n                              <m:mn>0<\/m:mn>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2017-0044_eq_1251.png\"\/>\n                        <jats:tex-math>{\\omega_{0}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> in the triangulation.\nThe analysis allows the bound of two constants <jats:inline-formula id=\"j_cmam-2017-0044_ineq_9995_w2aab3b7d645b1b6b1aab1c14b1c15Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi mathvariant=\"normal\">\u039b<\/m:mi>\n                              <m:mn>1<\/m:mn>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2017-0044_eq_1127.png\"\/>\n                        <jats:tex-math>{\\Lambda_{1}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and\n<jats:inline-formula id=\"j_cmam-2017-0044_ineq_9994_w2aab3b7d645b1b6b1aab1c14b1c17Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi mathvariant=\"normal\">\u039b<\/m:mi>\n                              <m:mn>3<\/m:mn>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2017-0044_eq_1130.png\"\/>\n                        <jats:tex-math>{\\Lambda_{3}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> in the axioms of adaptivity for the practical choice of the\nbulk parameter with guaranteed optimal convergence rates.<\/jats:p>","DOI":"10.1515\/cmam-2017-0044","type":"journal-article","created":{"date-parts":[[2017,11,17]],"date-time":"2017-11-17T22:16:14Z","timestamp":1510956974000},"page":"433-450","source":"Crossref","is-referenced-by-count":14,"title":["Constants in Discrete Poincar\u00e9 and 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