{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T18:54:22Z","timestamp":1776884062435,"version":"3.51.2"},"reference-count":44,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations.\nThe scheme is built to preserve at the discrete level even on severely distorted meshes the energy\/energy dissipation relation.\nThis relation is of paramount importance to capture the long-time behavior of the problem in an accurate way.\nTo enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized.\nWe establish the existence of positive solutions to the scheme. Based on compactness arguments,\nthe convergence of the approximate solution towards a weak solution is established.\nFinally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters\ntend to 0 and when time goes to infinity.<\/jats:p>","DOI":"10.1515\/cmam-2017-0043","type":"journal-article","created":{"date-parts":[[2017,11,12]],"date-time":"2017-11-12T22:16:29Z","timestamp":1510524989000},"page":"407-432","source":"Crossref","is-referenced-by-count":22,"title":["Numerical Analysis of a Nonlinear Free-Energy Diminishing Discrete Duality Finite Volume Scheme for Convection Diffusion Equations"],"prefix":"10.1515","volume":"18","author":[{"given":"Cl\u00e9ment","family":"Canc\u00e8s","sequence":"first","affiliation":[{"name":"Inria , 40 av. Halley, F-59650 Villeneuve d\u2019Ascq , France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Claire","family":"Chainais-Hillairet","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Lille , CNRS , UMR 8524 \u2013 Laboratoire Paul Painlev\u00e9, F-59000 Lille , France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stella","family":"Krell","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Nice , CNRS , UMR 7351 \u2013 Laboratoire J.-A. Dieudonn\u00e9, F-06100 Nice , France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,11,12]]},"reference":[{"key":"2025051309580701310_j_cmam-2017-0043_ref_001_w2aab3b7b1b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"L.  Ambrosio, N.  Gigli and G.  Savar\u00e9,\nGradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed.,\nLecture Notes in Math.,\nBirkh\u00e4user, Basel, 2008.","DOI":"10.1016\/S1874-5717(07)80004-1"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_002_w2aab3b7b1b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"B.  Andreianov, M.  Bendahmane and K. H.  Karlsen,\nDiscrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations,\nJ. Hyperbolic Differ. Equ. 7 (2010), no. 1, 1\u201367.","DOI":"10.1142\/S0219891610002062"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_003_w2aab3b7b1b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"B.  Andreianov, F.  Boyer and F.  Hubert,\nDiscrete duality finite volume schemes for Leray\u2013Lions-type elliptic problems on general 2D meshes,\nNumer. Methods Partial Differential Equations 23 (2007), no. 1, 145\u2013195.","DOI":"10.1002\/num.20170"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_004_w2aab3b7b1b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"B.  Andreianov, C.  Canc\u00e8s and A.  Moussa,\nA nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs,\nJ. Funct. Anal. 273 (2017), no. 12, 3633\u20133670.","DOI":"10.1016\/j.jfa.2017.08.010"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_005_w2aab3b7b1b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"A.  Arnold, J. A.  Carrillo, L.  Desvillettes, J.  Dolbeault, A.  J\u00fcngel, C.  Lederman, P. A.  Markowich, G.  Toscani and C.  Villani,\nEntropies and equilibria of many-particle systems: An essay on recent research,\nMonatsh. Math. 142 (2004), no. 1\u20132, 35\u201343.","DOI":"10.1007\/s00605-004-0239-2"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_006_w2aab3b7b1b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"A.  Arnold, P.  Markowich, G.  Toscani and A.  Unterreiter,\nOn convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker\u2013Planck type equations,\nComm. Partial Differential Equations 26 (2001), no. 1\u20132, 43\u2013100.","DOI":"10.1081\/PDE-100002246"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_007_w2aab3b7b1b1b6b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"M.  Bessemoulin-Chatard and C.  Chainais-Hillairet,\nExponential decay of a finite volume scheme to the thermal equilibrium for drift\u2013diffusion systems,\nJ. Numer. Math. 25 (2017), no. 3, 147\u2013168.","DOI":"10.1515\/jnma-2016-0007"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_008_w2aab3b7b1b1b6b1ab2b1b8Aa","doi-asserted-by":"crossref","unstructured":"M.  Bessemoulin-Chatard, C.  Chainais-Hillairet and F.  Filbet,\nOn discrete functional inequalities for some finite volume schemes,\nIMA J. Numer. Anal. 35 (2015), no. 3, 1125\u20131149.","DOI":"10.1093\/imanum\/dru032"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_009_w2aab3b7b1b1b6b1ab2b1b9Aa","doi-asserted-by":"crossref","unstructured":"D.  Blanchard and A.  Porretta,\nStefan problems with nonlinear diffusion and convection,\nJ. Differential Equations 210 (2005), no. 2, 383\u2013428.","DOI":"10.1016\/j.jde.2004.06.012"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_010_w2aab3b7b1b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"F.  Bolley, I.  Gentil and A.  Guillin,\nConvergence to equilibrium in Wasserstein distance for Fokker\u2013Planck equations,\nJ. Funct. Anal. 263 (2012), no. 8, 2430\u20132457.","DOI":"10.1016\/j.jfa.2012.07.007"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_011_w2aab3b7b1b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"F.  Bolley, I.  Gentil and A.  Guillin,\nUniform convergence to equilibrium for granular media,\nArch. Ration. Mech. Anal. 208 (2013), no. 2, 429\u2013445.","DOI":"10.1007\/s00205-012-0599-z"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_012_w2aab3b7b1b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"C.  Canc\u00e8s, C.  Chainais-Hillairet and S.  Krell,\nA nonlinear discrete duality finite volume scheme for convection-diffusion equations,\nFinite Volumes for Complex Applications VIII\u2014Methods and Theoretical Aspects,\nSpringer Proc. Math. Stat. 199,\nSpringer, Cham (2017), 439\u2013447.","DOI":"10.1007\/978-3-319-57397-7_37"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_013_w2aab3b7b1b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"C.  Canc\u00e8s and C.  Guichard,\nConvergence of a nonlinear entropy diminishing control volume finite element scheme for solving anisotropic degenerate parabolic equations,\nMath. Comp. 85 (2016), no. 298, 549\u2013580.","DOI":"10.1090\/mcom\/2997"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_014_w2aab3b7b1b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"C.  Canc\u00e8s and C.  Guichard,\nNumerical analysis of a robust free energy diminishing finite volume scheme for parabolic equations with gradient structure,\nFound. Comput. Math. (2016), 10.1007\/s10208-016-9328-6.","DOI":"10.1007\/s10208-016-9328-6"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_015_w2aab3b7b1b1b6b1ab2b1c15Aa","doi-asserted-by":"crossref","unstructured":"J. A.  Carrillo and G.  Toscani,\nExponential convergence toward equilibrium for homogeneous Fokker\u2013Planck-type equations,\nMath. Methods Appl. Sci. 21 (1998), no. 13, 1269\u20131286.","DOI":"10.1002\/(SICI)1099-1476(19980910)21:13<1269::AID-MMA995>3.3.CO;2-F"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_016_w2aab3b7b1b1b6b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"J. A.  Carrillo and G.  Toscani,\nAsymptotic L1L^{1}-decay of solutions of the porous medium equation to self-similarity,\nIndiana Univ. Math. J. 49 (2000), no. 1, 113\u2013142.","DOI":"10.1512\/iumj.2000.49.1756"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_017_w2aab3b7b1b1b6b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"C.  Chainais-Hillairet, A.  J\u00fcngel and S.  Schuchnigg,\nEntropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities,\nESAIM Math. Model. Numer. Anal. 50 (2016), no. 1, 135\u2013162.","DOI":"10.1051\/m2an\/2015031"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_018_w2aab3b7b1b1b6b1ab2b1c18Aa","doi-asserted-by":"crossref","unstructured":"C.  Chainais-Hillairet, S.  Krell and A.  Mouton,\nConvergence analysis of a DDFV scheme for a system describing miscible fluid flows in porous media,\nNumer. Methods Partial Differential Equations 31 (2015), no. 3, 723\u2013760.","DOI":"10.1002\/num.21913"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_019_w2aab3b7b1b1b6b1ab2b1c19Aa","doi-asserted-by":"crossref","unstructured":"M.  Chatard,\nAsymptotic behavior of the Scharfetter\u2013Gummel scheme for the drift-diffusion model,\nFinite Volumes for Complex Applications VI. Problems & Perspectives. Vols. 1\u20132,\nSpringer Proc. Math. 4,\nSpringer, Heidelberg (2011), 235\u2013243.","DOI":"10.1007\/978-3-642-20671-9_25"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_020_w2aab3b7b1b1b6b1ab2b1c20Aa","doi-asserted-by":"crossref","unstructured":"Y.  Coudi\u00e8re, J.-P.  Vila and P.  Villedieu,\nConvergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem,\nM2AN Math. Model. Numer. Anal. 33 (1999), no. 3, 493\u2013516.","DOI":"10.1051\/m2an:1999149"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_021_w2aab3b7b1b1b6b1ab2b1c21Aa","doi-asserted-by":"crossref","unstructured":"K.  Deimling,\nNonlinear Functional Analysis,\nSpringer, Berlin, 1985.","DOI":"10.1007\/978-3-662-00547-7"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_022_w2aab3b7b1b1b6b1ab2b1c22Aa","unstructured":"C.  Dellacherie and P.-A.  Meyer,\nProbabilities and Potential,\nNorth-Holland Math. Stud. 29,\nNorth-Holland Publishing, Amsterdam, 1978."},{"key":"2025051309580701310_j_cmam-2017-0043_ref_023_w2aab3b7b1b1b6b1ab2b1c23Aa","doi-asserted-by":"crossref","unstructured":"L.  Desvillettes and K.  Fellner,\nExponential decay toward equilibrium via entropy methods for reaction-diffusion equations,\nJ. Math. Anal. Appl. 319 (2006), no. 1, 157\u2013176.","DOI":"10.1016\/j.jmaa.2005.07.003"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_024_w2aab3b7b1b1b6b1ab2b1c24Aa","doi-asserted-by":"crossref","unstructured":"L.  Desvillettes and K.  Fellner,\nDuality and entropy methods for reversible reaction-diffusion equations with degenerate diffusion,\nMath. Methods Appl. Sci. 38 (2015), no. 16, 3432\u20133443.","DOI":"10.1002\/mma.3407"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_025_w2aab3b7b1b1b6b1ab2b1c25Aa","doi-asserted-by":"crossref","unstructured":"K.  Domelevo and P.  Omnes,\nA finite volume method for the Laplace equation on almost arbitrary two-dimensional grids,\nM2AN Math. Model. Numer. Anal. 39 (2005), no. 6, 1203\u20131249.","DOI":"10.1051\/m2an:2005047"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_026_w2aab3b7b1b1b6b1ab2b1c26Aa","doi-asserted-by":"crossref","unstructured":"J.  Droniou and R.  Eymard,\nUniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations,\nNumer. Math. 132 (2016), no. 4, 721\u2013766.","DOI":"10.1007\/s00211-015-0733-6"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_027_w2aab3b7b1b1b6b1ab2b1c27Aa","unstructured":"J.  Droniou, R.  Eymard, T.  Gallou\u00ebt, C.  Guichard and R.  Herbin,\nThe gradient discretisation method,\npreprint (2016), https:\/\/hal.archives-ouvertes.fr\/hal-01382358."},{"key":"2025051309580701310_j_cmam-2017-0043_ref_028_w2aab3b7b1b1b6b1ab2b1c28Aa","doi-asserted-by":"crossref","unstructured":"R.  Eymard, T.  Gallou\u00ebt, M.  Ghilani and R.  Herbin,\nError estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes,\nIMA J. Numer. Anal. 18 (1998), no. 4, 563\u2013594.","DOI":"10.1093\/imanum\/18.4.563"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_029_w2aab3b7b1b1b6b1ab2b1c29Aa","doi-asserted-by":"crossref","unstructured":"R.  Eymard, T.  Gallou\u00ebt and R.  Herbin,\nFinite volume methods,\nHandbook of Numerical Analysis. Vol. VII,\nNorth-Holland, Amsterdam (2000), 713\u20131020.","DOI":"10.1016\/S1570-8659(00)07005-8"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_030_w2aab3b7b1b1b6b1ab2b1c30Aa","doi-asserted-by":"crossref","unstructured":"F.  Filbet,\nAn asymptotically stable scheme for diffusive coagulation-fragmentation models,\nCommun. Math. Sci. 6 (2008), no. 2, 257\u2013280.","DOI":"10.4310\/CMS.2008.v6.n2.a1"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_031_w2aab3b7b1b1b6b1ab2b1c31Aa","doi-asserted-by":"crossref","unstructured":"H.  Gajewski and K.  G\u00e4rtner,\nOn the discretization of van Roosbroeck\u2019s equations with magnetic field,\nZAMM Z. Angew. Math. Mech. 76 (1996), no. 5, 247\u2013264.","DOI":"10.1002\/zamm.19960760502"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_032_w2aab3b7b1b1b6b1ab2b1c32Aa","doi-asserted-by":"crossref","unstructured":"H.  Gajewski and K.  Gr\u00f6ger,\nOn the basic equations for carrier transport in semiconductors,\nJ. Math. Anal. Appl. 113 (1986), no. 1, 12\u201335.","DOI":"10.1016\/0022-247X(86)90330-6"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_033_w2aab3b7b1b1b6b1ab2b1c33Aa","doi-asserted-by":"crossref","unstructured":"H.  Gajewski and K.  Gr\u00f6ger,\nSemiconductor equations for variable mobilities based on Boltzmann statistics or Fermi\u2013Dirac statistics,\nMath. Nachr. 140 (1989), 7\u201336.","DOI":"10.1002\/mana.19891400102"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_034_w2aab3b7b1b1b6b1ab2b1c34Aa","doi-asserted-by":"crossref","unstructured":"T.  Gallou\u00ebt,\nSome discrete functional analysis tools,\nFinite Volumes for Complex Applications VIII\u2014Methods and Theoretical Aspects,\nSpringer Proc. Math. Stat. 199,\nSpringer, Cham (2017), 29\u201341.","DOI":"10.1007\/978-3-319-57397-7_3"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_035_w2aab3b7b1b1b6b1ab2b1c35Aa","doi-asserted-by":"crossref","unstructured":"A.  Glitzky,\nExponential decay of the free energy for discretized electro-reaction-diffusion systems,\nNonlinearity 21 (2008), no. 9, 1989\u20132009.","DOI":"10.1088\/0951-7715\/21\/9\/003"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_036_w2aab3b7b1b1b6b1ab2b1c36Aa","doi-asserted-by":"crossref","unstructured":"A.  Glitzky,\nUniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems,\nMath. Nachr. 284 (2011), no. 17\u201318, 2159\u20132174.","DOI":"10.1002\/mana.200910215"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_037_w2aab3b7b1b1b6b1ab2b1c37Aa","doi-asserted-by":"crossref","unstructured":"R.  Jordan, D.  Kinderlehrer and F.  Otto,\nThe variational formulation of the Fokker\u2013Planck equation,\nSIAM J. Math. Anal. 29 (1998), no. 1, 1\u201317.","DOI":"10.1137\/S0036141096303359"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_038_w2aab3b7b1b1b6b1ab2b1c38Aa","doi-asserted-by":"crossref","unstructured":"A.  J\u00fcngel,\nEntropy Methods for Diffusive Partial Differential Equations,\nSpringer Briefs Math.,\nSpringer, Cham, 2016.","DOI":"10.1007\/978-3-319-34219-1"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_039_w2aab3b7b1b1b6b1ab2b1c39Aa","doi-asserted-by":"crossref","unstructured":"A.  J\u00fcngel and S.  Schuchnigg,\nEntropy-dissipating semi-discrete Runge\u2013Kutta schemes for nonlinear diffusion equations,\nCommun. Math. Sci. 15 (2017), no. 1, 27\u201353.","DOI":"10.4310\/CMS.2017.v15.n1.a2"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_040_w2aab3b7b1b1b6b1ab2b1c40Aa","doi-asserted-by":"crossref","unstructured":"J.  Leray and J.  Schauder,\nTopologie et \u00e9quations fonctionnelles,\nAnn. Sci. \u00c9c. Norm. Sup\u00e9r (3) 51 (1934), 45\u201378.","DOI":"10.24033\/asens.836"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_041_w2aab3b7b1b1b6b1ab2b1c41Aa","doi-asserted-by":"crossref","unstructured":"F.  Otto,\nThe geometry of dissipative evolution equations: The porous medium equation,\nComm. Partial Differential Equations 26 (2001), no. 1\u20132, 101\u2013174.","DOI":"10.1081\/PDE-100002243"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_042_w2aab3b7b1b1b6b1ab2b1c42Aa","doi-asserted-by":"crossref","unstructured":"F.  Santambrogio,\nOptimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling,\nProgr. Nonlinear Differential Equations Appl. 87,\nBirkh\u00e4user, Cham, 2015.","DOI":"10.1007\/978-3-319-20828-2"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_043_w2aab3b7b1b1b6b1ab2b1c43Aa","doi-asserted-by":"crossref","unstructured":"D.  Scharfetter and H.  Gummel,\nLarge signal analysis of a silicon Read diode,\nIEEE Trans. Elec. Dev. 16 (1969), 64\u201377.","DOI":"10.1109\/T-ED.1969.16566"},{"key":"2025051309580701310_j_cmam-2017-0043_ref_044_w2aab3b7b1b1b6b1ab2b1c44Aa","doi-asserted-by":"crossref","unstructured":"G.  Toscani and C.  Villani,\nOn the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds,\nJ. Stat. Phys. 98 (2000), no. 5\u20136, 1279\u20131309.","DOI":"10.1023\/A:1018623930325"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/18\/3\/article-p407.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0043\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0043\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T10:00:35Z","timestamp":1747130435000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0043\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,12]]},"references-count":44,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,11,12]]},"published-print":{"date-parts":[[2018,7,1]]}},"alternative-id":["10.1515\/cmam-2017-0043"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2017-0043","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2017,11,12]]}}}