{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:35:04Z","timestamp":1787337304678,"version":"build-2736575974"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>In this paper, we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with Dirichlet\u2010type boundary condition. First we formulate the qualitative properties for the differential equations and shed light on their relations. Then we show how the well\u2010known discretization schemes can be written in the form of a one\u2010step iterative process. We give necessary and sufficient conditions of the main qualitative properties of these iterations. We apply the results to the finite difference and Galerkin finite element solutions of linear parabolic problems. In our main result we show that the nonnegativity preservation property is equivalent to the maximum\u2010minimum principle and they imply the maximum norm contractivity. In one, two, and three dimensions, we list sufficient a priori conditions that ensure the required qualitative properties. Finally, we demonstrate the above results on numerical examples.<\/jats:p>","DOI":"10.1137\/050627241","type":"journal-article","created":{"date-parts":[[2006,12,26]],"date-time":"2006-12-26T11:06:15Z","timestamp":1167131175000},"page":"2313-2336","source":"Crossref","is-referenced-by-count":62,"title":["Discrete Maximum Principle and Adequate Discretizations of Linear Parabolic Problems"],"prefix":"10.1137","volume":"28","author":[{"given":"Istv\u00e1n","family":"Farag\u00f3","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R\u00f3bert","family":"Horv\u00e1th","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,12,21]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"A. Berman and R. J. Plemmons,\n                      Nonnegative Matrices in the Mathematical Sciences\n                      , Academic Press, New York, 1979.","DOI":"10.1016\/B978-0-12-092250-5.50009-6"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479802409687"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827502406695"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/BF01844166"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(73)90019-4"},{"key":"R6","first-page":"38","volume":"6","author":"Farag\u00f3 I.","year":"1996","journal-title":"Pure Math. Appl."},{"key":"R7","unstructured":"I. Farag\u00f3 and R. Horv\u00e1th,\n                      On the nonnegativity conservation of finite element solutions of parabolic problems\n                      , in Proceedings of the Conference on Finite Element Methods: Three\u2010Dimensional Problems, P. Neittaanm\u00e4ki and M. Krizek, eds., GAKUTO Internat. Ser. Math. Sci. Appl. 15, Gakk\u014dtosho, Tokyo, 2001, pp. 76\u201384."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2004.09.001"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01876627"},{"key":"R10","unstructured":"A. Friedmann,\n                      Partial Differential Equations of Parabolic Type\n                      , Prentice\u2010Hall, Englewood Cliffs, NJ, 1964."},{"key":"R11","unstructured":"H. Fujii,\n                      Some remarks on finite element analysis of time\u2010dependent field problems\n                      , in Theory and Practice in Finite Element Structural Analysis, University of Tokyo Press, Tokyo, 1973, pp. 91\u2013106."},{"key":"R12","unstructured":"A. H. Hariton,\n                      Some Qualitative Properties of the Numerical Solution to the Heat Conduction Equation\n                      , Thesis for Cand. of Math. Science, E\u00f6tv\u00f6s Lor\u00e1nd University, Budapest, 1995."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(99)00007-0"},{"key":"R14","first-page":"281","volume":"11","author":"Horv\u00e1th R.","year":"2000","journal-title":"Pure Math. Appl."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(01)00227-9"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(01)00150-7"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-004-0559-0"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-00-01270-9"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/0168-9274(92)90003-V"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"O. A. Lady\u017eenskaja, V. A. Solonnikov, and N. N. Ural\u2019ceva,\n                      Linear and Quasilinear Equations of Parabolic Type\n                      , Transl. Math. Monogr. 23, AMS, Providence RI, 1967 (in Russian).","DOI":"10.1090\/mmono\/023"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396643"},{"key":"R22","first-page":"139","volume":"36","author":"Pfeil T.","year":"1993","journal-title":"Ann. Univ. Sci. Budapest. E\u00f6tv\u00f6s Sect. Math."},{"key":"R23","first-page":"473","volume":"29","author":"Ruas Santos V.","year":"1982","journal-title":"J. Fac. Sci. Univ. Tokyo Sect. IA Math."},{"key":"R24","doi-asserted-by":"crossref","unstructured":"T. Vejchodsky,\n                      On the nonnegativity conservation in semidiscrete parabolic problems\n                      , in Conjugate Gradient Algorithms and Finite Element Methods, M. Krizek et al., eds., Springer\u2010Verlag, Berlin, 2004, pp. 282\u2013295.","DOI":"10.1007\/978-3-642-18560-1_13"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/050627241","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:55:44Z","timestamp":1787334944000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/050627241"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":24,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1137\/050627241"],"URL":"https:\/\/doi.org\/10.1137\/050627241","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,1]]}}}