{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,25]],"date-time":"2024-06-25T06:10:19Z","timestamp":1719295819275},"reference-count":25,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider elliptic problems with complicated, discontinuous diffusion tensor<jats:inline-formula id=\"j_cmam-2017-0015_ineq_9999_w2aab3b7e2598b1b6b1aab1c14b1b1Aa\"><jats:alternatives><m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><m:msub><m:mi>A<\/m:mi><m:mn>0<\/m:mn><\/m:msub><\/m:math><jats:tex-math>{A_{0}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. One of the standard approaches to numerically treat such problems is to simplify the coefficient by some approximation, say<jats:inline-formula id=\"j_cmam-2017-0015_ineq_9998_w2aab3b7e2598b1b6b1aab1c14b1b3Aa\"><jats:alternatives><m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><m:msub><m:mi>A<\/m:mi><m:mi>\u03b5<\/m:mi><\/m:msub><\/m:math><jats:tex-math>{A_{\\varepsilon}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and to use standard finite elements. In [19] a combined modelling-discretization strategy has been proposed which estimates the discretization and modelling errors by a posteriori estimates of functional type. This strategy allows to balance these two errors in a problem adapted way. However, the estimate of the modelling error was derived under the assumption that the difference<jats:inline-formula id=\"j_cmam-2017-0015_ineq_9997_w2aab3b7e2598b1b6b1aab1c14b1b7Aa\"><jats:alternatives><m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><m:mrow><m:msub><m:mi>A<\/m:mi><m:mn>0<\/m:mn><\/m:msub><m:mo>-<\/m:mo><m:msub><m:mi>A<\/m:mi><m:mi>\u03b5<\/m:mi><\/m:msub><\/m:mrow><\/m:math><jats:tex-math>{A_{0}-A_{\\varepsilon}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>becomes small with respect to the<jats:inline-formula id=\"j_cmam-2017-0015_ineq_9996_w2aab3b7e2598b1b6b1aab1c14b1b9Aa\"><jats:alternatives><m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><m:msup><m:mi>L<\/m:mi><m:mi>\u221e<\/m:mi><\/m:msup><\/m:math><jats:tex-math>{L^{\\infty}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-norm. This implies in particular that interfaces\/discontinuities separating the smooth parts of<jats:inline-formula id=\"j_cmam-2017-0015_ineq_9995_w2aab3b7e2598b1b6b1aab1c14b1c11Aa\"><jats:alternatives><m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><m:msub><m:mi>A<\/m:mi><m:mn>0<\/m:mn><\/m:msub><\/m:math><jats:tex-math>{A_{0}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>have to be matched exactly by the coefficient<jats:inline-formula id=\"j_cmam-2017-0015_ineq_9994_w2aab3b7e2598b1b6b1aab1c14b1c13Aa\"><jats:alternatives><m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><m:msub><m:mi>A<\/m:mi><m:mi>\u03b5<\/m:mi><\/m:msub><\/m:math><jats:tex-math>{A_{\\varepsilon}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Therefore the efficient application of that theory to problems with complicated or curved interfaces is limited. In this paper, we will present a refined theory, where the difference<jats:inline-formula id=\"j_cmam-2017-0015_ineq_9993_w2aab3b7e2598b1b6b1aab1c14b1c15Aa\"><jats:alternatives><m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><m:mrow><m:msub><m:mi>A<\/m:mi><m:mn>0<\/m:mn><\/m:msub><m:mo>-<\/m:mo><m:msub><m:mi>A<\/m:mi><m:mi>\u03b5<\/m:mi><\/m:msub><\/m:mrow><\/m:math><jats:tex-math>{A_{0}-A_{\\varepsilon}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is measured in the<jats:inline-formula id=\"j_cmam-2017-0015_ineq_9992_w2aab3b7e2598b1b6b1aab1c14b1c17Aa\"><jats:alternatives><m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><m:msup><m:mi>L<\/m:mi><m:mi>q<\/m:mi><\/m:msup><\/m:math><jats:tex-math>{L^{q}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-norm for some appropriate<jats:inline-formula id=\"j_cmam-2017-0015_ineq_9991_w2aab3b7e2598b1b6b1aab1c14b1c19Aa\"><jats:alternatives><m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><m:mrow><m:mi>q<\/m:mi><m:mo>\u2208<\/m:mo><m:mo>]<\/m:mo><m:mn>2<\/m:mn><m:mo>,<\/m:mo><m:mi>\u221e<\/m:mi><m:mo>[<\/m:mo><\/m:mrow><\/m:math><jats:tex-math>{q\\in{]2,\\infty[}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and, hence, the geometric resolution condition is significantly relaxed.<\/jats:p>","DOI":"10.1515\/cmam-2017-0015","type":"journal-article","created":{"date-parts":[[2017,7,3]],"date-time":"2017-07-03T07:55:32Z","timestamp":1499068532000},"page":"515-531","source":"Crossref","is-referenced-by-count":1,"title":["A Posteriori Modelling-Discretization Error Estimate for Elliptic Problems with<i>L<\/i><sup>\u221e<\/sup>-Coefficients"],"prefix":"10.1515","volume":"17","author":[{"given":"Monika","family":"Weymuth","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Mathematik , Universit\u00e4t Z\u00fcrich , Winterthurerstrasse 190, CH-8057 Z\u00fcrich , Switzerland"}]},{"given":"Stefan","family":"Sauter","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Mathematik , Universit\u00e4t Z\u00fcrich , Winterthurerstrasse 190, CH-8057 Z\u00fcrich , Switzerland"}]},{"given":"Sergey","family":"Repin","sequence":"additional","affiliation":[{"name":"Russian Academy of Sciences , Saint Petersburg Department of V.\u2009A. Steklov Institute of Mathematics , Fontanka 27, 191 011 Saint Petersburg , Russia ; and University of Jyv\u00e4skyl\u00e4, P.O. Box 35, FI-40014, Jyv\u00e4skyl\u00e4, Finland"}]}],"member":"374","published-online":{"date-parts":[[2017,6,21]]},"reference":[{"key":"2023033114491588266_j_cmam-2017-0015_ref_001_w2aab3b7e2598b1b6b1ab2b1b1Aa","unstructured":"R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Pure Appl. Math. 140, Elsevier, Amsterdam, 2003."},{"key":"2023033114491588266_j_cmam-2017-0015_ref_002_w2aab3b7e2598b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka, U. Banerjee and J. E. Osborn, Generalized finite element methods \u2013 Main ideas, results and perspective, Int. J. Comput. Methods 1 (2004), no. 1, 67\u2013103.","DOI":"10.1142\/S0219876204000083"},{"key":"2023033114491588266_j_cmam-2017-0015_ref_003_w2aab3b7e2598b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka, G. Caloz and J. E. 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