{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,26]],"date-time":"2023-06-26T09:26:38Z","timestamp":1687771598320},"reference-count":23,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper we give a new proof of the <jats:inline-formula id=\"j_cmam-2016-0036_ineq_9999_w2aab3b7e3893b1b6b1aab1c13b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>L<\/m:mi>\n                              <m:mi>\u221e<\/m:mi>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:tex-math>${L^{\\infty}}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-error estimate for the finite element approximation of the\nLaplace\u2013Beltrami equation with an additional lower order term on a surface.\nWhile the proof available in the literature uses the method of perturbed bilinear forms from Schatz and Wahlbin, we adapt Scott\u2019s proof from an Euclidean setting to the surface case. Furthermore, in contrast to the literature we use an approximative Green\u2019s function on the surface instead of an exact Green\u2019s function which is obtained by lifting an Euclidean Green\u2019s function locally from the tangent plane\nto the surface.<\/jats:p>","DOI":"10.1515\/cmam-2016-0036","type":"journal-article","created":{"date-parts":[[2016,11,25]],"date-time":"2016-11-25T09:50:12Z","timestamp":1480067412000},"page":"51-64","source":"Crossref","is-referenced-by-count":1,"title":["Approximative Green\u2019s Functions on Surfaces and Pointwise Error Estimates for the Finite Element Method"],"prefix":"10.1515","volume":"17","author":[{"given":"Heiko","family":"Kr\u00f6ner","sequence":"first","affiliation":[{"name":"Fachbereich Mathematik, Universit\u00e4t Hamburg, Bundesstr. 55, 20146 Hamburg, Germany"}]}],"member":"374","published-online":{"date-parts":[[2016,11,25]]},"reference":[{"key":"2023033115185131377_j_cmam-2016-0036_ref_001_w2aab3b7e3893b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"Bertalmio M., Cheng L.-T., Osher S. and Sapiro G.,\nVariational problems and partial differential equations on implicit surfaces,\nJ. Comput. Phys. 174 (2001), no. 2, 759\u2013780.","DOI":"10.1006\/jcph.2001.6937"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_002_w2aab3b7e3893b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"Brenner S. C. and Scott L. R.,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math.,\nSpringer, Berlin, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_003_w2aab3b7e3893b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"Dedner A., Madhavan P. and Stinner B.,\nAnalysis of the discontinuous Galerkin method for elliptic problems on surfaces,\nIMA J. Numer. Anal. 33 (2013), no. 3, 952\u2013973.","DOI":"10.1093\/imanum\/drs033"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_004_w2aab3b7e3893b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"Demlow A.,\nHigher-order finite element methods and pointwise error estimates for elliptic problems on surfaces,\nSIAM J. Numer. Anal. 47 (2009), no. 2, 805\u2013827.","DOI":"10.1137\/070708135"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_005_w2aab3b7e3893b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"Dziuk G.,\nFinite elements for the Beltrami operator on arbitrary surfaces,\nPartial Differential Equations and Calculus of Variations,\nLecture Notes in Math. 1357,\nSpringer, Berlin (1988), 142\u2013155.","DOI":"10.1007\/BFb0082865"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_006_w2aab3b7e3893b1b6b1ab2b1b6Aa","unstructured":"Dziuk G. and Elliott C. M.,\nFinite elements on evolving surfaces,\nIMA J. Numer. Anal. 25 (2007), 385\u2013407."},{"key":"2023033115185131377_j_cmam-2016-0036_ref_007_w2aab3b7e3893b1b6b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"Dziuk G. and Elliott C. M.,\nFinite element methods for surface PDEs,\nActa Numer. 22 (2013), 289\u2013396.","DOI":"10.1017\/S0962492913000056"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_008_w2aab3b7e3893b1b6b1ab2b1b8Aa","doi-asserted-by":"crossref","unstructured":"Dziuk G. and Elliott C. M.,\nL2${L^{2}}$ estimates for the evolving surface finite element method,\nSIAM J. Numer. Anal. 50 (2013), 2677\u20132694.","DOI":"10.1137\/110828642"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_009_w2aab3b7e3893b1b6b1ab2b1b9Aa","doi-asserted-by":"crossref","unstructured":"Dziuk G., Elliott C. M. and Heine C.-J.,\nA h-narrow band finite-element method for elliptic equations on implicit surfaces,\nIMA J. Numer. Anal. 30 (2013), no. 2, 351\u2013376.","DOI":"10.1093\/imanum\/drn049"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_010_w2aab3b7e3893b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"Dziuk G., Lubich C. and Mansour D.,\nRunge\u2013Kutta time discretization of parabolic differential equations on evolving surfaces,\nIMA J. Numer. Anal. 32 (2012), 394\u2013416.","DOI":"10.1093\/imanum\/drr017"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_011_w2aab3b7e3893b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"Haverkamp R.,\nEine Aussage zur L\u221e${L_{\\infty}}$-Stabilit\u00e4t und zur genauen Konvergenzordnung der H01${H^{1}_{0}}$-Projektionen,\nNumer. Math. 44 (1984), 393\u2013405.","DOI":"10.1007\/BF01405570"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_012_w2aab3b7e3893b1b6b1ab2b1c12Aa","unstructured":"Kov\u00e1cs B. and Power C.,\nMaximum norm stability and error estimates for the evolving surface finite element method,\npreprint 2015, https:\/\/arxiv.org\/abs\/1510.00605."},{"key":"2023033115185131377_j_cmam-2016-0036_ref_013_w2aab3b7e3893b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"Lubich C., Mansour D. and Venkataraman C.,\nBackward difference time discretization of parabolic differential equations on evolving surfaces,\nIMA J. Numer. Anal. 33 (2013), no. 4, 1365\u20131385.","DOI":"10.1093\/imanum\/drs044"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_014_w2aab3b7e3893b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"Natterer F.,\n\u00dcber die punkweise Konvergenz finiter Elemente,\nNumer. Math. 25 (1975\/76), 67\u201377.","DOI":"10.1007\/BF01419529"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_015_w2aab3b7e3893b1b6b1ab2b1c15Aa","doi-asserted-by":"crossref","unstructured":"Nitsche J.,\nL\u221e${L^{\\infty}}$ convergence of finite element approximation,\nMathematical Aspects of Finite Element Methods (Rome 1975),\nLecture Notes in Math. 606,\nSpringer, Berlin (1977), 261\u2013274.","DOI":"10.1007\/BFb0064468"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_016_w2aab3b7e3893b1b6b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"Nitsche J. A. and Schatz A. H.,\nInterior estimates for Ritz\u2013Galerkin methods,\nMath. Comp. 28 (1974), 937\u2013958.","DOI":"10.1090\/S0025-5718-1974-0373325-9"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_017_w2aab3b7e3893b1b6b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"Olshanskii M. A. and Reusken A.,\nError analysis of a space-time finite element method for solving PDEs on evolving surfaces,\nSIAM J. Numer. Anal. 52 (2014), no. 4, 2092\u20132120.","DOI":"10.1137\/130936877"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_018_w2aab3b7e3893b1b6b1ab2b1c18Aa","doi-asserted-by":"crossref","unstructured":"Olshanskii M. A., Reusken A. and Grande J.,\nA finite element method for elliptic equations on surfaces,\nSIAM J. Numer. Anal. 47 (2009), no. 5, 3339\u20133358.","DOI":"10.1137\/080717602"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_019_w2aab3b7e3893b1b6b1ab2b1c19Aa","doi-asserted-by":"crossref","unstructured":"Olshanskii M. A., Reusken A. and Grande J.,\nA finite element method for surface PDEs: Matrix properties,\nNumer. Math. 114 (2010), no. 3, 491\u2013520.","DOI":"10.1007\/s00211-009-0260-4"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_020_w2aab3b7e3893b1b6b1ab2b1c20Aa","doi-asserted-by":"crossref","unstructured":"Olshanskii M. A., Reusken A. and Xu X.,\nAn Eulerian space-time finite element method for diffusion problems on evolving surfaces,\nSIAM J. Numer. Anal. 52 (2014), no. 3, 1354\u20131377.","DOI":"10.1137\/130918149"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_021_w2aab3b7e3893b1b6b1ab2b1c21Aa","doi-asserted-by":"crossref","unstructured":"Schatz A. H.,\nPointwise error estimates and asymptotic error expansion inequalities for the finite element method\non irregular grids. I. Global Estimates,\nMath. Comp. 67 (1998), 877\u2013899.","DOI":"10.1090\/S0025-5718-98-00959-4"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_022_w2aab3b7e3893b1b6b1ab2b1c22Aa","doi-asserted-by":"crossref","unstructured":"Schatz A. H. and Wahlbin L. B.,\nInterior maximum-norm estimates for finite element methods. Part II,\nMath. Comp. 64 (1995), 907\u2013928.","DOI":"10.1090\/S0025-5718-1995-1297478-7"},{"key":"2023033115185131377_j_cmam-2016-0036_ref_023_w2aab3b7e3893b1b6b1ab2b1c23Aa","doi-asserted-by":"crossref","unstructured":"Scott R.,\nOptimal L\u221e${L^{\\infty}}$-estimates for the finite element method on irregular meshes,\nMath. Comp. 30 (1976), no. 136, 681\u2013697.","DOI":"10.1090\/S0025-5718-1976-0436617-2"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/17\/1\/article-p51.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0036\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0036\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T22:01:26Z","timestamp":1680300086000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0036\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,11,25]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,9,18]]},"published-print":{"date-parts":[[2017,1,1]]}},"alternative-id":["10.1515\/cmam-2016-0036"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2016-0036","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2016,11,25]]}}}