{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T05:25:32Z","timestamp":1783056332526,"version":"3.54.6"},"reference-count":14,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Preserving positivity precludes that linear operators onto continuous piecewise affine functions provide near best approximations of gradients. Linear interpolation thus does not capture the approximation properties of positive continuous piecewise affine functions. To remedy, we assign nodal values in a nonlinear fashion such that their global best error is equivalent to a suitable sum of local best errors with positive affine functions. As one of the applications of this equivalence, we consider the linear finite element solution to the elliptic obstacle problem and derive that its error is bounded in terms of these local best errors.<\/jats:p>","DOI":"10.1515\/cmam-2018-0017","type":"journal-article","created":{"date-parts":[[2018,6,30]],"date-time":"2018-06-30T22:16:22Z","timestamp":1530396982000},"page":"295-310","source":"Crossref","is-referenced-by-count":5,"title":["Positivity Preserving Gradient Approximation with Linear Finite Elements"],"prefix":"10.1515","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2152-2911","authenticated-orcid":false,"given":"Andreas","family":"Veeser","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica , Universit\u00e0 degli Studi di Milano , Via Saldini 50, 20133 Milano , Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2018,6,30]]},"reference":[{"key":"2023033110022005526_j_cmam-2018-0017_ref_001_w2aab3b7e1909b1b6b1ab2b1b1Aa","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":"2023033110022005526_j_cmam-2018-0017_ref_002_w2aab3b7e1909b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L. R.  Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2023033110022005526_j_cmam-2018-0017_ref_003_w2aab3b7e1909b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"F.  Brezzi, W. W.  Hager and P.-A.  Raviart,\nError estimates for the finite element solution of variational inequalities,\nNumer. Math. 28 (1977), no. 4, 431\u2013443.","DOI":"10.1007\/BF01404345"},{"key":"2023033110022005526_j_cmam-2018-0017_ref_004_w2aab3b7e1909b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"Z.  Chen and R. H.  Nochetto,\nResidual type a posteriori error estimates for elliptic obstacle problems,\nNumer. Math. 84 (2000), no. 4, 527\u2013548.","DOI":"10.1007\/s002110050009"},{"key":"2023033110022005526_j_cmam-2018-0017_ref_005_w2aab3b7e1909b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"P. G.  Ciarlet,\nThe Finite Element Method for Elliptic Problems,\nStud. Math. Appl. 4,\nNorth-Holland Publishing, Amsterdam, 1978,","DOI":"10.1115\/1.3424474"},{"key":"2023033110022005526_j_cmam-2018-0017_ref_006_w2aab3b7e1909b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"P. G.  Ciarlet,\nBasic error estimates for elliptic problems,\nHandbook of Numerical Analysis. Vol. II,\nNorth-Holland, Amsterdam (1991), 17\u2013352.","DOI":"10.1016\/S1570-8659(05)80039-0"},{"key":"2023033110022005526_j_cmam-2018-0017_ref_007_w2aab3b7e1909b1b6b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"R. S.  Falk,\nError estimates for the approximation of a class of variational inequalities,\nMath. Comput. 28 (1974), 963\u2013971.","DOI":"10.1090\/S0025-5718-1974-0391502-8"},{"key":"2023033110022005526_j_cmam-2018-0017_ref_008_w2aab3b7e1909b1b6b1ab2b1b8Aa","doi-asserted-by":"crossref","unstructured":"F.  Fierro and A.  Veeser,\nA posteriori error estimators for regularized total variation of characteristic functions,\nSIAM J. Numer. Anal. 41 (2003), 2032\u20132055.","DOI":"10.1137\/S0036142902408283"},{"key":"2023033110022005526_j_cmam-2018-0017_ref_009_w2aab3b7e1909b1b6b1ab2b1b9Aa","doi-asserted-by":"crossref","unstructured":"D.  Gilbarg and N. S.  Trudinger,\nElliptic Partial Differential Equations of Second Order,\nClassics in Mathematics, Springer, Berlin, 2001.","DOI":"10.1007\/978-3-642-61798-0"},{"key":"2023033110022005526_j_cmam-2018-0017_ref_010_w2aab3b7e1909b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"R.  Nochetto, K.  Siebert and A.  Veeser,\nFully localized a posteriori error estimators and barrier sets for contact problems,\nSIAM J. Numer. Anal. 42 (2005), 2118\u20132135.","DOI":"10.1137\/S0036142903424404"},{"key":"2023033110022005526_j_cmam-2018-0017_ref_011_w2aab3b7e1909b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"R. H.  Nochetto and L. B.  Wahlbin,\nPositivity preserving finite element approximation,\nMath. Comp. 71 (2002), no. 240, 1405\u20131419.","DOI":"10.1090\/S0025-5718-01-01369-2"},{"key":"2023033110022005526_j_cmam-2018-0017_ref_012_w2aab3b7e1909b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"L. R.  Scott and S.  Zhang,\nFinite element interpolation of nonsmooth functions satisfying boundary conditions,\nMath. Comp. 54 (1990), no. 190, 483\u2013493.","DOI":"10.1090\/S0025-5718-1990-1011446-7"},{"key":"2023033110022005526_j_cmam-2018-0017_ref_013_w2aab3b7e1909b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"A.  Veeser,\nApproximating gradients with continuous piecewise polynomial functions,\nFound. Comput. Math. 16 (2016), no. 3, 723\u2013750.","DOI":"10.1007\/s10208-015-9262-z"},{"key":"2023033110022005526_j_cmam-2018-0017_ref_014_w2aab3b7e1909b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"A.  Veeser and R.  Verf\u00fcrth,\nExplicit upper bounds for dual norms of residuals,\nSIAM J. Numer. Anal. 47 (2009), no. 3, 2387\u20132405.","DOI":"10.1137\/080738283"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/19\/2\/article-p295.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0017\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0017\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T10:45:53Z","timestamp":1680259553000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0017\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,30]]},"references-count":14,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,2,24]]},"published-print":{"date-parts":[[2019,4,1]]}},"alternative-id":["10.1515\/cmam-2018-0017"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2018-0017","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2018,6,30]]}}}