{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:47:26Z","timestamp":1747198046437,"version":"3.40.5"},"reference-count":19,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["CRC 1283"],"award-info":[{"award-number":["CRC 1283"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>It is an open question if the threshold condition <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\u03b8<\/m:mi>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:msub>\n                                 <m:mi>\u03b8<\/m:mi>\n                                 <m:mo>\u22c6<\/m:mo>\n                              <\/m:msub>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2020-0041_ineq_0001.png\"\/>\n                        <jats:tex-math>\\theta&lt;\\theta_{\\star}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> for the D\u00f6rfler marking parameter is necessary to obtain optimal algebraic rates of adaptive finite element methods.\nWe present a (non-PDE) example fitting into the common abstract convergence framework (axioms of adaptivity) which allows for convergence with exponential rates.\nHowever, for D\u00f6rfler marking <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\u03b8<\/m:mi>\n                              <m:mo>&gt;<\/m:mo>\n                              <m:msub>\n                                 <m:mi>\u03b8<\/m:mi>\n                                 <m:mo>\u22c6<\/m:mo>\n                              <\/m:msub>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2020-0041_ineq_0002.png\"\/>\n                        <jats:tex-math>\\theta&gt;\\theta_{\\star}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, the algebraic convergence rate can be made arbitrarily small.<\/jats:p>","DOI":"10.1515\/cmam-2020-0041","type":"journal-article","created":{"date-parts":[[2020,10,13]],"date-time":"2020-10-13T19:46:47Z","timestamp":1602618407000},"page":"557-567","source":"Crossref","is-referenced-by-count":2,"title":["On the Threshold Condition for D\u00f6rfler Marking"],"prefix":"10.1515","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0523-3079","authenticated-orcid":false,"given":"Lars","family":"Diening","sequence":"first","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik , Universit\u00e4t Bielefeld , 33615 Bielefeld , Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2923-4428","authenticated-orcid":false,"given":"Christian","family":"Kreuzer","sequence":"additional","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik , TU Dortmund , 44221 Dortmund , Germany"}]}],"member":"374","published-online":{"date-parts":[[2020,10,9]]},"reference":[{"key":"2023033111373274042_j_cmam-2020-0041_ref_001","doi-asserted-by":"crossref","unstructured":"L. Belenki, L. Diening and C. Kreuzer,\nOptimality of an adaptive finite element method for the \ud835\udc5d-Laplacian equation,\nIMA J. Numer. Anal. 32 (2012), no. 2, 484\u2013510.","DOI":"10.1093\/imanum\/drr016"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_002","doi-asserted-by":"crossref","unstructured":"P. Binev, W. Dahmen and R. DeVore,\nAdaptive finite element methods with convergence rates,\nNumer. Math. 97 (2004), no. 2, 219\u2013268.","DOI":"10.1007\/s00211-003-0492-7"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_003","doi-asserted-by":"crossref","unstructured":"P. Binev and R. DeVore,\nFast computation in adaptive tree approximation,\nNumer. Math. 97 (2004), no. 2, 193\u2013217.","DOI":"10.1007\/s00211-003-0493-6"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_004","doi-asserted-by":"crossref","unstructured":"A. Bonito and R. H. Nochetto,\nQuasi-optimal convergence rate of an adaptive discontinuous Galerkin method,\nSIAM J. Numer. Anal. 48 (2010), no. 2, 734\u2013771.","DOI":"10.1137\/08072838X"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_005","doi-asserted-by":"crossref","unstructured":"C. Carstensen, M. Feischl, M. Page and D. Praetorius,\nAxioms of adaptivity,\nComput. Math. Appl. 67 (2014), no. 6, 1195\u20131253.","DOI":"10.1016\/j.camwa.2013.12.003"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_006","doi-asserted-by":"crossref","unstructured":"C. Carstensen, D. Peterseim and H. Rabus,\nOptimal adaptive nonconforming FEM for the Stokes problem,\nNumer. Math. 123 (2013), no. 2, 291\u2013308.","DOI":"10.1007\/s00211-012-0490-8"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_007","doi-asserted-by":"crossref","unstructured":"C. Carstensen and H. Rabus,\nAxioms of adaptivity with separate marking for data resolution,\nSIAM J. Numer. Anal. 55 (2017), no. 6, 2644\u20132665.","DOI":"10.1137\/16M1068050"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_008","doi-asserted-by":"crossref","unstructured":"J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert,\nQuasi-optimal convergence rate for an adaptive finite element method,\nSIAM J. Numer. Anal. 46 (2008), no. 5, 2524\u20132550.","DOI":"10.1137\/07069047X"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_009","doi-asserted-by":"crossref","unstructured":"J. M. Casc\u00f3n and R. H. Nochetto,\nQuasioptimal cardinality of AFEM driven by nonresidual estimators,\nIMA J. Numer. Anal. 32 (2012), no. 1, 1\u201329.","DOI":"10.1093\/imanum\/drr014"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_010","doi-asserted-by":"crossref","unstructured":"L. Diening and C. Kreuzer,\nLinear convergence of an adaptive finite element method for the \ud835\udc5d-Laplacian equation,\nSIAM J. Numer. Anal. 46 (2008), no. 2, 614\u2013638.","DOI":"10.1137\/070681508"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_011","doi-asserted-by":"crossref","unstructured":"L. Diening, C. Kreuzer and R. Stevenson,\nInstance optimality of the adaptive maximum strategy,\nFound. Comput. Math. 16 (2016), no. 1, 33\u201368.","DOI":"10.1007\/s10208-014-9236-6"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_012","doi-asserted-by":"crossref","unstructured":"M. Feischl, T. F\u00fchrer and D. Praetorius,\nAdaptive FEM with optimal convergence rates for a certain class of nonsymmetric and possibly nonlinear problems,\nSIAM J. Numer. Anal. 52 (2014), no. 2, 601\u2013625.","DOI":"10.1137\/120897225"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_013","doi-asserted-by":"crossref","unstructured":"M. Feischl, M. Karkulik, J. M. Melenk and D. Praetorius,\nQuasi-optimal convergence rate for an adaptive boundary element method,\nSIAM J. Numer. Anal. 51 (2013), no. 2, 1327\u20131348.","DOI":"10.1137\/110842569"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_014","doi-asserted-by":"crossref","unstructured":"T. Gantumur,\nAdaptive boundary element methods with convergence rates,\nNumer. Math. 124 (2013), no. 3, 471\u2013516.","DOI":"10.1007\/s00211-013-0524-x"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_015","doi-asserted-by":"crossref","unstructured":"C. Kreuzer and M. Schedensack,\nInstance optimal Crouzeix\u2013Raviart adaptive finite element methods for the Poisson and Stokes problems,\nIMA J. Numer. Anal. 36 (2016), no. 2, 593\u2013617.","DOI":"10.1093\/imanum\/drv019"},{"key":"2023033111373274042_j_cmam-2020-0041_ref_016","doi-asserted-by":"crossref","unstructured":"C. Kreuzer and K. G. Siebert,\nDecay rates of adaptive finite elements with D\u00f6rfler marking,\nNumer. 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