{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T17:18:44Z","timestamp":1719854324320},"reference-count":21,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,4,1]]},"abstract":"Abstract<\/jats:title>\n The well-posedness and the a priori and a posteriori error analysis of the lowest-order Raviart\u2013Thomas\nmixed finite element method (MFEM) has been established for non-selfadjoint\nindefinite second-order linear elliptic problems recently in an article by Carstensen, Dond, Nataraj\nand Pani (Numer. Math.<\/jats:italic>, 2016). The associated adaptive mesh-refinement strategy faces\nthe difficulty of the flux error\ncontrol in \n \n \n \n H<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n div<\/m:mi>\n ,<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {H({\\operatorname{div}},\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and so involves a data-approximation error \n \n \n \n \u2225<\/m:mo>\n \n f<\/m:mi>\n -<\/m:mo>\n \n \n \u03a0<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n \u2062<\/m:mo>\n f<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n \u2225<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n {\\lVert f-\\Pi_{0}f\\rVert}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in the \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {L^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\nnorm of the right-hand side f<\/jats:italic> and its piecewise constant approximation \n \n \n \n \n \u03a0<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n \u2062<\/m:mo>\n f<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n {\\Pi_{0}f}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. The separate marking strategy has recently been suggested\nwith a split of a D\u00f6rfler marking for the remaining error estimator and an optimal data approximation strategy\nfor the appropriate treatment of \n \n \n \n \n \u2225<\/m:mo>\n \n f<\/m:mi>\n -<\/m:mo>\n \n \n \u03a0<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n \u2062<\/m:mo>\n f<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n \u2225<\/m:mo>\n <\/m:mrow>\n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:msub>\n <\/m:math>\n \n {\\|f-\\Pi_{0}f\\|_{L^{2}(\\Omega)}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nThe resulting strategy presented in this paper\nutilizes the abstract algorithm and convergence analysis of\nCarstensen and Rabus (SINUM<\/jats:italic>, 2017) and generalizes it to general second-order\nelliptic linear PDEs. The argument for the treatment of the piecewise constant displacement approximation\n\n \n \n \n u<\/m:mi>\n RT<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n {u_{{\\mathrm{RT}}}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is its supercloseness to the piecewise constant approximation \n \n \n \n \n \u03a0<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n \u2062<\/m:mo>\n u<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n {\\Pi_{0}u}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of the exact displacement u<\/jats:italic>. The overall convergence analysis then indeed follows the axioms of adaptivity for separate marking.\nSome results on mixed and nonconforming\nfinite element approximations on the multiply connected polygonal 2D Lipschitz domain\nare of general interest.<\/jats:p>","DOI":"10.1515\/cmam-2019-0034","type":"journal-article","created":{"date-parts":[[2019,3,23]],"date-time":"2019-03-23T09:21:26Z","timestamp":1553332886000},"page":"233-250","source":"Crossref","is-referenced-by-count":4,"title":["Quasi-Optimality of Adaptive Mixed FEMs for Non-selfadjoint Indefinite Second-Order Linear Elliptic Problems"],"prefix":"10.1515","volume":"19","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Mathematik , Humboldt-Universit\u00e4t zu Berlin , 10099 Berlin , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Asha K.","family":"Dond","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Indian Institute of Science , Bangalore , India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hella","family":"Rabus","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Mathematik , Humboldt-Universit\u00e4t zu Berlin , 10099 Berlin , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2019,3,23]]},"reference":[{"key":"2023033110021988430_j_cmam-2019-0034_ref_001_w2aab3b7e4230b1b6b1ab2b1b1Aa","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":"2023033110021988430_j_cmam-2019-0034_ref_002_w2aab3b7e4230b1b6b1ab2b1b2Aa","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":"2023033110021988430_j_cmam-2019-0034_ref_003_w2aab3b7e4230b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"D. Boffi, F. Brezzi and M. Fortin,\nMixed Finite Element Methods and Applications,\nSpringer Ser. Comput. Math. 44,\nSpringer, Heidelberg, 2013.","DOI":"10.1007\/978-3-642-36519-5"},{"key":"2023033110021988430_j_cmam-2019-0034_ref_004_w2aab3b7e4230b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"C. Carstensen, A. K. Dond, N. Nataraj and A. K. Pani,\nError analysis of nonconforming and mixed FEMs for second-order linear non-selfadjoint and indefinite elliptic problems,\nNumer. Math. 133 (2016), no. 3, 557\u2013597.","DOI":"10.1007\/s00211-015-0755-0"},{"key":"2023033110021988430_j_cmam-2019-0034_ref_005_w2aab3b7e4230b1b6b1ab2b1b5Aa","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":"2023033110021988430_j_cmam-2019-0034_ref_006_w2aab3b7e4230b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"C. Carstensen and R. H. W. Hoppe,\nError reduction and convergence for an adaptive mixed finite element method,\nMath. Comp. 75 (2006), no. 255, 1033\u20131042.","DOI":"10.1090\/S0025-5718-06-01829-1"},{"key":"2023033110021988430_j_cmam-2019-0034_ref_007_w2aab3b7e4230b1b6b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"C. Carstensen and H. Rabus,\nAn optimal adaptive mixed finite element method,\nMath. Comp. 80 (2011), no. 274, 649\u2013667.","DOI":"10.1090\/S0025-5718-2010-02397-X"},{"key":"2023033110021988430_j_cmam-2019-0034_ref_008_w2aab3b7e4230b1b6b1ab2b1b8Aa","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":"2023033110021988430_j_cmam-2019-0034_ref_009_w2aab3b7e4230b1b6b1ab2b1b9Aa","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":"2023033110021988430_j_cmam-2019-0034_ref_010_w2aab3b7e4230b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"H. Chen, X. Xu and R. H. W. Hoppe,\nConvergence and quasi-optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems,\nNumer. Math. 116 (2010), no. 3, 383\u2013419.","DOI":"10.1007\/s00211-010-0307-6"},{"key":"2023033110021988430_j_cmam-2019-0034_ref_011_w2aab3b7e4230b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"L. Chen, M. Holst and J. Xu,\nConvergence and optimality of adaptive mixed finite element methods,\nMath. Comp. 78 (2009), no. 265, 35\u201353.","DOI":"10.1090\/S0025-5718-08-02155-8"},{"key":"2023033110021988430_j_cmam-2019-0034_ref_012_w2aab3b7e4230b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"M. Dauge,\nElliptic Boundary Value Problems on Corner Domains. Smoothness and Asymptotics of Solutions,\nLecture Notes in Math. 1341,\nSpringer, Berlin, 1988.","DOI":"10.1007\/BFb0086682"},{"key":"2023033110021988430_j_cmam-2019-0034_ref_013_w2aab3b7e4230b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"A. K. Dond, N. 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