{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T15:16:35Z","timestamp":1773846995332,"version":"3.50.1"},"reference-count":49,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2004,9,7]],"date-time":"2004-09-07T00:00:00Z","timestamp":1094515200000},"content-version":"vor","delay-in-days":3478,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Comput Chem"],"published-print":{"date-parts":[[1995,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We present a robust and efficient numerical method for solution of the nonlinear Poisson\u2010Boltzmann equation arising in molecular biophysics. The equation is discretized with the box method, and solution of the discrete equations is accomplished with a global inexact\u2010Newton method, combined with linear multilevel techniques we have described in an article appearing previously in this journal. A detailed analysis of the resulting method is presented, with comparisons to other methods that have been proposed in the literature, including the classical nonlinear multigrid method, the nonlinear conjugate gradient method, and nonlinear relaxation methods such as successive overrelaxation. Both theoretical and numerical evidence suggests that this method will converge in the case of molecules for which many of the existing methods will not. In addition, for problems which the other methods are able to solve, numerical experiments show that the new method is substantially more efficient, and the superiority of this method grows with the problem size. The method is easy to implement once a linear multilevel solver is available and can also easily be used in conjunction with linear methods other than multigrid. \u00a9 1995 by John Wiley &amp; Sons, Inc.<\/jats:p>","DOI":"10.1002\/jcc.540160308","type":"journal-article","created":{"date-parts":[[2005,1,2]],"date-time":"2005-01-02T01:08:25Z","timestamp":1104628105000},"page":"337-364","source":"Crossref","is-referenced-by-count":208,"title":["Numerical solution of the nonlinear Poisson\u2013Boltzmann equation: Developing more robust and efficient methods"],"prefix":"10.1002","volume":"16","author":[{"given":"Michael J.","family":"Holst","sequence":"first","affiliation":[]},{"given":"Faisal","family":"Saied","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2004,9,7]]},"reference":[{"key":"e_1_2_1_2_2","first-page":"185","volume":"24","author":"Debye P.","year":"1923","journal-title":"Physik. Z."},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.4823072"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.bb.19.060190.001505"},{"key":"e_1_2_1_5_2","volume-title":"Physical Chemistry of Macromolecules","author":"Tanford C.","year":"1961"},{"key":"e_1_2_1_6_2","unstructured":"M.Holst Technical report Department of Applied Mathematics and CRPC California Institute of Technology 1994."},{"key":"e_1_2_1_7_2","unstructured":"M.Holst PhD thesis Numerical Computing Group Department of Computer Science University of Illinois at Urbana\u2013Champaign 1993. Also published as technical report UIUCDCS\u2010R\u201003\u20131821."},{"key":"e_1_2_1_8_2","volume-title":"Matrix Iterative Analysis","author":"Varga R. S.","year":"1962"},{"key":"e_1_2_1_9_2","volume-title":"An Analysis of the Finite Element Method","author":"Strang G.","year":"1973"},{"key":"e_1_2_1_10_2","volume-title":"Iterative Solution of Elliptic Systems and Applications to the Neutron Diffusion Equations of Reactor Physics","author":"Wachspress E. L.","year":"1966"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1137\/0724050"},{"key":"e_1_2_1_12_2","unstructured":"T.Kerkhoven SIAM J. Numer. Anal.(1992)."},{"key":"e_1_2_1_13_2","volume-title":"Numerical Analysis: A Second Course","author":"Ortega J. M.","year":"1972"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1002\/jcc.540120405"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1002\/jcc.540140114"},{"key":"e_1_2_1_16_2","first-page":"409","volume":"49","author":"Hestenes M. R.","year":"1952","journal-title":"J. Research of NBS"},{"key":"e_1_2_1_17_2","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/B978-0-12-141050-6.50023-4","volume-title":"Space Matrix Computations","author":"Concus P.","year":"1976"},{"key":"e_1_2_1_18_2","unstructured":"S. F.Ashby T. A.Manteuffel andP. E.Saylor Technical report UCRL\u201098508 Lawrence Livermore National Laboratory March1988. To appear inSIAM J. Numer. Anal."},{"key":"e_1_2_1_19_2","doi-asserted-by":"publisher","DOI":"10.1002\/jcc.540100313"},{"issue":"6","key":"e_1_2_1_20_2","doi-asserted-by":"crossref","first-page":"1174","DOI":"10.1137\/0910071","volume":"10","author":"van der Vorst H. A.","year":"1989","journal-title":"SIAM J. Sci. Statist. Comp."},{"key":"e_1_2_1_21_2","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1090\/S0025-5718-1977-0431719-X","volume":"31","author":"Brandt A.","year":"1977","journal-title":"Math. Comp."},{"key":"e_1_2_1_22_2","series-title":"Lecture notes in Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0069927","volume-title":"Multigrid Methods: Proceedings of K\u00f6hn\u2010Porz Conference on Multigrid Methods","author":"Hackbusch W.","year":"1982"},{"key":"e_1_2_1_23_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02427-0"},{"key":"e_1_2_1_24_2","doi-asserted-by":"publisher","DOI":"10.1137\/0902035"},{"key":"e_1_2_1_25_2","volume-title":"Elliptic Problem Solvers","author":"Dendy J. E.","year":"1981"},{"key":"e_1_2_1_26_2","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(82)90057-2"},{"issue":"2","key":"e_1_2_1_27_2","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1137\/0908059","volume":"8","author":"Dendy J. E.","year":"1987","journal-title":"SIAM J. Sci. Statist. Comp."},{"issue":"180","key":"e_1_2_1_28_2","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1090\/S0025-5718-1987-0906174-X","volume":"49","author":"Bramble J. H.","year":"1987","journal-title":"Math. Comp."},{"key":"e_1_2_1_29_2","doi-asserted-by":"publisher","DOI":"10.1021\/j100352a033"},{"key":"e_1_2_1_30_2","doi-asserted-by":"publisher","DOI":"10.1002\/jcc.540130911"},{"key":"e_1_2_1_31_2","unstructured":"H.OberoiandN. M.Allewell Biophysical Journal(1993)."},{"key":"e_1_2_1_32_2","volume-title":"Iterative Solution of Nonlinear Equations in Several Variables","author":"Ortega J. M.","year":"1970"},{"key":"e_1_2_1_33_2","unstructured":"A.Nicholls Private communication 1993."},{"key":"e_1_2_1_34_2","doi-asserted-by":"publisher","DOI":"10.1093\/comjnl\/7.2.149"},{"key":"e_1_2_1_35_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01397652"},{"key":"e_1_2_1_36_2","doi-asserted-by":"crossref","unstructured":"W.HackbuschandA.Reusken InRobust Multigrid Methods W.Hackbusch Ed. Braunschweig 1988 Vieweg p.105.","DOI":"10.1007\/978-3-322-86200-6_9"},{"key":"e_1_2_1_37_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01406516"},{"key":"e_1_2_1_38_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01398879"},{"key":"e_1_2_1_39_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01397135"},{"key":"e_1_2_1_40_2","volume-title":"Functional Analysis","author":"Kantorovich L. V.","year":"1982"},{"issue":"126","key":"e_1_2_1_41_2","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1090\/S0025-5718-1974-0343581-1","volume":"28","author":"Dennis J. E.","year":"1974","journal-title":"Math. Comp."},{"key":"e_1_2_1_42_2","doi-asserted-by":"publisher","DOI":"10.1137\/1019005"},{"key":"e_1_2_1_43_2","doi-asserted-by":"publisher","DOI":"10.1137\/0719025"},{"key":"e_1_2_1_44_2","volume-title":"Topics in Functional Analysis and Applications","author":"Kesavan S.","year":"1989"},{"key":"e_1_2_1_45_2","unstructured":"S. C.EisenstatandH. F.Walker technical report Department of Mathematics and Statistics Utah State University 1992."},{"key":"e_1_2_1_46_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01398257"},{"issue":"160","key":"e_1_2_1_47_2","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1090\/S0025-5718-1982-0669639-X","volume":"39","author":"Bank R. E.","year":"1982","journal-title":"Math. Comp."},{"key":"e_1_2_1_48_2","doi-asserted-by":"publisher","DOI":"10.1002\/bip.360280506"},{"key":"e_1_2_1_49_2","volume-title":"Introduction to Nonlinear Differential and Integral Equations","author":"Davis H. T.","year":"1960"},{"key":"e_1_2_1_50_2","doi-asserted-by":"publisher","DOI":"10.1002\/prot.340180304"}],"container-title":["Journal of Computational Chemistry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjcc.540160308","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jcc.540160308","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,27]],"date-time":"2023-10-27T07:00:13Z","timestamp":1698390013000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jcc.540160308"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,3]]},"references-count":49,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1995,3]]}},"alternative-id":["10.1002\/jcc.540160308"],"URL":"https:\/\/doi.org\/10.1002\/jcc.540160308","archive":["Portico"],"relation":{},"ISSN":["0192-8651","1096-987X"],"issn-type":[{"value":"0192-8651","type":"print"},{"value":"1096-987X","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,3]]}}}