{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,9]],"date-time":"2025-11-09T07:45:21Z","timestamp":1762674321929},"reference-count":28,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson\u2013Boltzmann equation (PBE).\nWe prove mathematical correctness of the problem, study a suitable class of approximations, and deduce guaranteed and fully computable bounds of approximation errors.\nThe latter goal is achieved by means of the approach suggested in [19] for convex variational problems.\nMoreover, we establish the error identity, which defines the error measure natural for the considered class of problems and show that it yields computable majorants and minorants of the global error as well as indicators of local errors that provide efficient adaptation of meshes.\nTheoretical results are confirmed by a collection of numerical tests that includes problems on 2D and 3D Lipschitz domains.<\/jats:p>","DOI":"10.1515\/cmam-2018-0252","type":"journal-article","created":{"date-parts":[[2019,6,14]],"date-time":"2019-06-14T09:10:56Z","timestamp":1560503456000},"page":"293-319","source":"Crossref","is-referenced-by-count":3,"title":["Reliable Numerical Solution of a Class of Nonlinear Elliptic Problems Generated by the Poisson\u2013Boltzmann Equation"],"prefix":"10.1515","volume":"20","author":[{"given":"Johannes","family":"Kraus","sequence":"first","affiliation":[{"name":"Faculty of Mathematics , University of Duisburg-Essen , Thea-Leymann-Str. 9, 45127 Essen , Germany"}]},{"given":"Svetoslav","family":"Nakov","sequence":"additional","affiliation":[{"name":"RICAM , Austrian Academy of Sciences , Altenbergerstra\u00dfe 69, 4040 Linz , Austria"}]},{"given":"Sergey I.","family":"Repin","sequence":"additional","affiliation":[{"name":"University of Jyv\u00e4skyl\u00e4 , Jyv\u00e4skyl\u00e4n , Finland ; and V.\u2009A. Steklov Institute of Mathematics in St. Petersburg, Russia"}]}],"member":"374","published-online":{"date-parts":[[2019,6,13]]},"reference":[{"key":"2023033110443943080_j_cmam-2018-0252_ref_001_w2aab3b7e2473b1b6b1ab2ab1Aa","unstructured":"D.  Braess and J.  Sch\u00f6berl,\nEquilibrated residual error estimator for Maxwell\u2019s equations,\nRICAM report 2006-19, Austrian Academy of Sciences, 2006."},{"key":"2023033110443943080_j_cmam-2018-0252_ref_002_w2aab3b7e2473b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"H.  Br\u00e9zis,\nFunctional Analysis, Sobolev Spaces and Partial Differential Equations,\nSpringer, New York, 2011.","DOI":"10.1007\/978-0-387-70914-7"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_003_w2aab3b7e2473b1b6b1ab2ab3Aa","unstructured":"H.  Br\u00e9zis and F. E.  Browder,\nSur une propri\u00e9t\u00e9 des espaces de Sobolev,\nC. R. Acad. Sci. Paris S\u00e9r. A-B 287 (1978), no. 3, A113\u2013A115."},{"key":"2023033110443943080_j_cmam-2018-0252_ref_004_w2aab3b7e2473b1b6b1ab2ab4Aa","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":"2023033110443943080_j_cmam-2018-0252_ref_005_w2aab3b7e2473b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"L.  Chen, M. J.  Holst and J.  Xu,\nThe finite element approximation of the nonlinear Poisson\u2013Boltzmann equation,\nSIAM J. Numer. Anal. 45 (2007), no. 6, 2298\u20132320.","DOI":"10.1137\/060675514"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_006_w2aab3b7e2473b1b6b1ab2ab6Aa","unstructured":"H.  Childs, E.  Brugger, B.  Whitlock, J.  Meredith, S.  Ahern, D.  Pugmire, K.  Biagas, M.  Miller, C.  Harrison, G. H.  Weber, H.  Krishnan, T.  Fogal, A.  Sanderson, C.  Garth, E.  Wes Bethel, D.  Camp, O.  R\u00fcbel, M.  Durant, J. M.  Favre and P.  Navr\u00e1til,\nVisIt: An end-user tool for visualizing and analyzing very large data,\nHigh Performance Visualization\u2013Enabling Extreme-Scale Scientific Insight,\nCRC Press, Boca Raton (2012), 357\u2013372."},{"key":"2023033110443943080_j_cmam-2018-0252_ref_007_w2aab3b7e2473b1b6b1ab2ab7Aa","unstructured":"C.  Dobrzynski,\nMMG3D: User guide,\nTechnical Report RT-0422, INRIA, 2012."},{"key":"2023033110443943080_j_cmam-2018-0252_ref_008_w2aab3b7e2473b1b6b1ab2ab8Aa","unstructured":"I.  Ekeland and R.  Temam,\nConvex Analysis and Variational Problems,\nNorth-Holland, Amsterdam, 1976."},{"key":"2023033110443943080_j_cmam-2018-0252_ref_009_w2aab3b7e2473b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"M.  Feischl, D.  Praetorius and K. G.  van der Zee,\nAn abstract analysis of optimal goal-oriented adaptivity,\nSIAM J. Numer. Anal. 54 (2016), no. 3, 1423\u20131448.","DOI":"10.1137\/15M1021982"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_010_w2aab3b7e2473b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"F.  Fogolari, A.  Brigo and H.  Molinari,\nThe Poisson\u2013Boltzmann equation for biomolecular electrostatics: A tool for structural biology,\nJ. Mol. Recognit. 15 (2002), 377\u2013392.","DOI":"10.1002\/jmr.577"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_011_w2aab3b7e2473b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"F.  Fogolari, P.  Zuccato, G.  Esposito and P.  Viglino,\nBiomolecular electrostatics with the linearized Poisson\u2013Boltzmann equation,\nBiophys. J. 76 (1999), 1\u201316.","DOI":"10.1016\/S0006-3495(99)77173-0"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_012_w2aab3b7e2473b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"D.  Gilbarg and N. S.  Trudinger,\nElliptic Partial Differential Equations of Second Order,\nClassics Math.,\nSpringer, Berlin, 2001.","DOI":"10.1007\/978-3-642-61798-0"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_013_w2aab3b7e2473b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"F.  Hecht,\nNew development in freefem++,\nJ. Numer. Math. 20 (2012), no. 3\u20134, 251\u2013265.","DOI":"10.1515\/jnum-2012-0013"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_014_w2aab3b7e2473b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"M.  Holst, J. A.  McCammon, Z.  Yu, Y. C.  Zhou and Y.  Zhu,\nAdaptive finite element modeling techniques for the Poisson\u2013Boltzmann equation,\nCommun. Comput. Phys. 11 (2012), no. 1, 179\u2013214.","DOI":"10.4208\/cicp.081009.130611a"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_015_w2aab3b7e2473b1b6b1ab2ac15Aa","doi-asserted-by":"crossref","unstructured":"B.  Kawohl and M.  Lucia,\nBest constants in some exponential Sobolev inequalities,\nIndiana Univ. Math. J. 57 (2008), no. 4, 1907\u20131927.","DOI":"10.1512\/iumj.2008.57.3307"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_016_w2aab3b7e2473b1b6b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"D.  Kinderlehrer and G.  Stampacchia,\nAn Introduction to Variational Inequalities and Their Applications,\nClass. Appl. Math. 31,\nSociety for Industrial and Applied Mathematics (SIAM), Philadelphia, 2000.","DOI":"10.1137\/1.9780898719451"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_017_w2aab3b7e2473b1b6b1ab2ac17Aa","unstructured":"P.  Neittaanm\u00e4ki and S.  Repin,\nReliable Methods for Computer Simulation. Error Control and a Posteriori Estimates,\nStud. Math. Appl. 33,\nElsevier Science, Amsterdam, 2004."},{"key":"2023033110443943080_j_cmam-2018-0252_ref_018_w2aab3b7e2473b1b6b1ab2ac18Aa","doi-asserted-by":"crossref","unstructured":"H.  Oberoi and N.  Allewell,\nMultigrid solution of the nonlinear Poisson\u2013Boltzmann equation and calculation of titration curves,\nBiophys. J. 65 (1993), 48\u201355.","DOI":"10.1016\/S0006-3495(93)81032-4"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_019_w2aab3b7e2473b1b6b1ab2ac19Aa","doi-asserted-by":"crossref","unstructured":"S. I.  Repin,\nA posteriori error estimation for variational problems with uniformly convex functionals,\nMath. Comp. 69 (2000), no. 230, 481\u2013500.","DOI":"10.1090\/S0025-5718-99-01190-4"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_020_w2aab3b7e2473b1b6b1ab2ac20Aa","doi-asserted-by":"crossref","unstructured":"S. I.  Repin,\nOn measures of errors for nonlinear variational problems,\nRussian J. Numer. Anal. Math. Modelling 27 (2012), no. 6, 577\u2013584.","DOI":"10.1515\/rnam-2012-0033"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_021_w2aab3b7e2473b1b6b1ab2ac21Aa","doi-asserted-by":"crossref","unstructured":"S.  Repin and J.  Valdman,\nError identities for variational problems with obstacles,\nZAMM Z. Angew. Math. Mech. 98 (2018), no. 4, 635\u2013658.","DOI":"10.1002\/zamm.201700105"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_022_w2aab3b7e2473b1b6b1ab2ac22Aa","doi-asserted-by":"crossref","unstructured":"I.  Sakalli, J.  Sch\u00f6berl and E. W.  Knapp,\nmfes: A robust molecular finite element solver for electrostatic energy computations,\nJ. Chem. Theory Comput. 10 (2014), 5095\u20135112.","DOI":"10.1021\/ct5005092"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_023_w2aab3b7e2473b1b6b1ab2ac23Aa","doi-asserted-by":"crossref","unstructured":"K.  Sharp and B.  Honig,\nCalculating total electrostatic energies with the nonlinear Poisson\u2013Boltzmann equation,\nJ. Phys. Chem 94 (1990), 7684\u20137692.","DOI":"10.1021\/j100382a068"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_024_w2aab3b7e2473b1b6b1ab2ac24Aa","unstructured":"R. E.  Showalter,\nHilbert Space Methods for Partial Differential Equations,\nPitman, London, 1977."},{"key":"2023033110443943080_j_cmam-2018-0252_ref_025_w2aab3b7e2473b1b6b1ab2ac25Aa","doi-asserted-by":"crossref","unstructured":"H.  Si,\nTetGen, a Delaunay-based quality tetrahedral mesh generator,\nACM Trans. Math. Software 41 (2015), no. 2, Article ID 11.","DOI":"10.1145\/2629697"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_026_w2aab3b7e2473b1b6b1ab2ac26Aa","doi-asserted-by":"crossref","unstructured":"G.  Stampacchia,\nLe probl\u00e8me de Dirichlet pour les \u00e9quations elliptiques du second ordre \u00e0 coefficients discontinus,\nAnn. Inst. Fourier (Grenoble) 15 (1965), no. 1, 189\u2013258.","DOI":"10.5802\/aif.204"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_027_w2aab3b7e2473b1b6b1ab2ac27Aa","doi-asserted-by":"crossref","unstructured":"N. S.  Trudinger,\nOn imbeddings into Orlicz spaces and some applications,\nJ. Math. Mech. 17 (1967), 473\u2013483.","DOI":"10.1512\/iumj.1968.17.17028"},{"key":"2023033110443943080_j_cmam-2018-0252_ref_028_w2aab3b7e2473b1b6b1ab2ac28Aa","unstructured":"A collection of molecular surface meshes,\nhttps:\/\/www.rocq.inria.fr\/gamma\/gamma\/download\/affichage.php?dir=MOLECULE&name=water_mol&last_page=6, Accessed: 2017-08-18."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/20\/2\/article-p293.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0252\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0252\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T12:56:23Z","timestamp":1680267383000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0252\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,13]]},"references-count":28,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,8,14]]},"published-print":{"date-parts":[[2020,4,1]]}},"alternative-id":["10.1515\/cmam-2018-0252"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2018-0252","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2019,6,13]]}}}