{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:47:18Z","timestamp":1747198038175,"version":"3.40.5"},"reference-count":13,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,7,1]]},"abstract":"Abstract<\/jats:title>\n We present a numerical approximation method for linear elliptic diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known singular functions as well as of an \n \n \n \n H<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {H^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-regular part. The latter part is expressed in terms of an elliptic problem with regularized Dirichlet boundary conditions, and can be approximated by means of a Nitsche finite element approach. The discrete solution of the original problem is then defined by adding back the singular part of the exact solution to the Nitsche approximation. In this way, the discrete solution can be shown to converge of second order 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>-norm with respect to the mesh size.<\/jats:p>","DOI":"10.1515\/cmam-2017-0057","type":"journal-article","created":{"date-parts":[[2017,12,13]],"date-time":"2017-12-13T07:59:31Z","timestamp":1513151971000},"page":"373-381","source":"Crossref","is-referenced-by-count":1,"title":["A Nitsche Finite Element Approach for Elliptic Problems with Discontinuous Dirichlet Boundary Conditions"],"prefix":"10.1515","volume":"18","author":[{"given":"Ramona","family":"Baumann","sequence":"first","affiliation":[{"name":"Mathematics Institute , University of Bern , Sidlerstrasse 5, CH-3012 Bern , Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1232-0637","authenticated-orcid":false,"given":"Thomas P.","family":"Wihler","sequence":"additional","affiliation":[{"name":"Mathematics Institute , University of Bern , Sidlerstrasse 5, CH-3012 Bern , Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,12,13]]},"reference":[{"key":"2025051309570896828_j_cmam-2017-0057_ref_001_w2aab3b7e2617b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini,\nUnified analysis of discontinuous Galerkin methods for elliptic problems,\nSIAM J. Numer. Anal. 39 (2001), 1749\u20131779.","DOI":"10.1137\/S0036142901384162"},{"key":"2025051309570896828_j_cmam-2017-0057_ref_002_w2aab3b7e2617b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka,\nError bounds for finite element method,\nNumer. Math. 16 (1971), no. 4, 322\u2013333.","DOI":"10.1007\/BF02165003"},{"key":"2025051309570896828_j_cmam-2017-0057_ref_003_w2aab3b7e2617b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka and B. Q. Guo,\nRegularity of the solution of elliptic problems with piecewise analytic data. I. Boundary value problems for linear elliptic equation of second order,\nSIAM J. Math. Anal. 19 (1988), 172\u2013203.","DOI":"10.1137\/0519014"},{"key":"2025051309570896828_j_cmam-2017-0057_ref_004_w2aab3b7e2617b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka and B. Q. Guo,\nRegularity of the solution of elliptic problems with piecewise analytic data. II. The trace spaces and application to the boundary value problems with nonhomogeneous boundary conditions,\nSIAM J. Math. Anal. 20 (1989), no. 4, 763\u2013781.","DOI":"10.1137\/0520054"},{"key":"2025051309570896828_j_cmam-2017-0057_ref_005_w2aab3b7e2617b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka, R. B. Kellogg and J. Pitk\u00e4ranta,\nDirect and inverse error estimates for finite elements with mesh refinements,\nNumer. Math. 33 (1979), no. 4, 447\u2013471.","DOI":"10.1007\/BF01399326"},{"key":"2025051309570896828_j_cmam-2017-0057_ref_006_w2aab3b7e2617b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"M. Dauge,\nElliptic Boundary Value Problems on Corner Domains,\nLecture Notes in Math. 1341,\nSpringer, Berlin, 1988.","DOI":"10.1007\/BFb0086682"},{"key":"2025051309570896828_j_cmam-2017-0057_ref_007_w2aab3b7e2617b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"J. Douglas, Jr., T. Dupont, H. H. Rachford, Jr. and M. F. Wheeler,\nLocal H-1H^{-1} Galerkin procedures for elliptic equations,\nRAIRO Anal. Num\u00e9r. 11 (1977), no. 1, 3\u201312, 111.","DOI":"10.1051\/m2an\/1977110100031"},{"key":"2025051309570896828_j_cmam-2017-0057_ref_008_w2aab3b7e2617b1b6b1ab2ab8Aa","unstructured":"P. Grisvard,\nElliptic Problems in Nonsmooth Domains,\nMonogr. Stud. Math. 24,\nPitman, Boston, 1985."},{"key":"2025051309570896828_j_cmam-2017-0057_ref_009_w2aab3b7e2617b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"P. Houston and T. P. Wihler,\nSecond-order elliptic PDEs with discontinuous boundary data,\nIMA J. Numer. Anal. 32 (2012), no. 1, 48\u201374.","DOI":"10.1093\/imanum\/drq032"},{"key":"2025051309570896828_j_cmam-2017-0057_ref_010_w2aab3b7e2617b1b6b1ab2ac10Aa","unstructured":"J. M. Melenk,\nOn generalized finite element methods,\nPh.D. thesis, University of Maryland, 1995."},{"key":"2025051309570896828_j_cmam-2017-0057_ref_011_w2aab3b7e2617b1b6b1ab2ac11Aa","unstructured":"J. Ne\u010das,\nSur une m\u00e9thode pour r\u00e9soudre les \u00e9quations aux d\u00e9riv\u00e9es partielles du type elliptique, voisine de la variationelle,\nAnn. Sc. Norm. Super. Pisa 16 (1962), 305\u2013326."},{"key":"2025051309570896828_j_cmam-2017-0057_ref_012_w2aab3b7e2617b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"J. Nitsche,\n\u00dcber ein Variationsprinzip zur L\u00f6sung von Dirichlet Problemen bei Verwendung von Teilr\u00e4umen, die keinen Randbedingungen unterworfen sind,\nAbh. Math. Semin. Univ. Hambg. 36 (1971), 9\u201315.","DOI":"10.1007\/BF02995904"},{"key":"2025051309570896828_j_cmam-2017-0057_ref_013_w2aab3b7e2617b1b6b1ab2ac13Aa","unstructured":"C. Schwab,\np- and hp-FEM \u2013 Theory and Application to Solid and Fluid Mechanics,\nOxford University Press, Oxford, 1998."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/18\/3\/article-p373.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0057\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0057\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T09:58:54Z","timestamp":1747130334000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0057\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,13]]},"references-count":13,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,12,13]]},"published-print":{"date-parts":[[2018,7,1]]}},"alternative-id":["10.1515\/cmam-2017-0057"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2017-0057","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"type":"electronic","value":"1609-9389"},{"type":"print","value":"1609-4840"}],"subject":[],"published":{"date-parts":[[2017,12,13]]}}}