{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T11:25:45Z","timestamp":1791285945823,"version":"4.1.0"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>In this paper we study the approximation of solutions to linear and nonlinear elliptic problems with discontinuous coefficients in the discrete duality finite volume (DDFV) framework. This family of schemes allows very general meshes and inherits the main properties of the continuous problem. In order to take into account the discontinuities and to prevent consistency defect in the scheme, we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs. We first illustrate our approach by the study of the 1D situation. Then, we show how to design our new scheme, called m-DDFV, and we propose its analysis. We also describe an iterative solver, whose convergence is proved, which can be used to solve the nonlinear discrete equations defining the finite volume scheme. Finally, we provide numerical results which confirm that the m-DDFV scheme significantly improves the convergence rate of the usual DDFV method for both linear and nonlinear problems.<\/jats:p>","DOI":"10.1137\/060666196","type":"journal-article","created":{"date-parts":[[2008,9,4]],"date-time":"2008-09-04T19:18:32Z","timestamp":1220555912000},"page":"3032-3070","source":"Crossref","is-referenced-by-count":89,"title":["Finite Volume Method for 2D Linear and Nonlinear Elliptic Problems with Discontinuities"],"prefix":"10.1137","volume":"46","author":[{"given":"Franck","family":"Boyer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Florence","family":"Hubert","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,9,4]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595293582"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595293594"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1002\/num.20170"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901400006"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"F. Boyer and F. Hubert,\n                      Finite volume method for nonlinear transmission problems\n                      , in Domain Decomposition Methods in Science and Engineering XVII, Lecture Notes in Computational Science and Engineering 60, U. Langer, M. Discacciati, D. Keyes, O. Widlund, and W. Zulehner, eds., Heidelberg, 2008, Springer, pp. 443\u2013450. Proceedings of the 17th International Conference on Domain Decomposition Methods held at St. Wolfgang \/ Strobl, Austria, July 3\u20137, 2006.","DOI":"10.1007\/978-3-540-75199-1_56"},{"key":"R6","unstructured":"Y. Coudi\u00e8re, C. Pierre, and R. Turpault,\n                      Solving the fully coupled heart and torso problems of electrocardiology with a 3D discrete duality finite volume method\n                      , submitted, (2006). http:\/\/hal.archives-ouvertes.fr\/hal-00016825\/en\/."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:1999149"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2005047"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2007001"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"R. Eymard, T. Gallou\u00ebt, and R. Herbin,\n                      Finite volume methods\n                      , in Handbook of Numerical Analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713\u20131020.","DOI":"10.1016\/S1570-8659(00)07005-8"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/1990240404571"},{"key":"R12","first-page":"41","volume":"9","author":"Glowinski R.","year":"1975","journal-title":"Rev. Fran\u00e7aise Automat. Informat. Recherche Op\u00e9rationnelle"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"R. Glowinski,\n                      Numerical methods for nonlinear variational problems\n                      , Springer Series in Computational Physics, Springer-Verlag, New York, 1984.","DOI":"10.1007\/978-3-662-12613-4"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2000.6466"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(02)00644-8"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.24033\/bsmf.1617"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/S0362-546X(98)00014-5"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/s002110000157"},{"key":"R19","unstructured":"C. Pierre,\n                      Mod\u00e9lisation et simulation de l'activit\u00e9 \u00e9lectrique du couer dans le thorax, analyse num\u00e9rique et m\u00e9thodes de volumes finis\n                      . Ph.D. thesis, Universit\u00e9 de Nantes, 2005. http:\/\/tel.ccsd.cnrs.fr\/tel-00010705."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385610"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060666196","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:14:45Z","timestamp":1787325285000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060666196"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":20,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060666196"],"URL":"https:\/\/doi.org\/10.1137\/060666196","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}