{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T03:51:57Z","timestamp":1777607517861,"version":"3.51.4"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T00:00:00Z","timestamp":1663718400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T00:00:00Z","timestamp":1663718400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100006162","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Ci\u00eancia e Tecnologia do Estado de Pernambuco","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100006162","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002322","name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002322","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2022,11]]},"DOI":"10.1007\/s10915-022-01978-6","type":"journal-article","created":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T13:04:02Z","timestamp":1663765442000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear Defect Correction Strategy for the Numerical Solution of Steady State Diffusion Problems in Heterogeneous and Anisotropic Media Satisfying the Discrete Maximum Principle"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3471-4090","authenticated-orcid":false,"given":"T. M.","family":"Cavalcante","sequence":"first","affiliation":[]},{"given":"R. J. M. Lira","family":"Filho","sequence":"additional","affiliation":[]},{"given":"A. C. R.","family":"Souza","sequence":"additional","affiliation":[]},{"given":"D. K. E.","family":"Carvalho","sequence":"additional","affiliation":[]},{"given":"P. R. M.","family":"Lyra","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,9,21]]},"reference":[{"key":"1978_CR1","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s10596-010-9191-5","volume":"15","author":"E Keilegavlen","year":"2011","unstructured":"Keilegavlen, E., Aavatsmark, I.: Monotonicity for MPFA methods on triangular grids. Comput. Geosci. 15, 3\u201316 (2011). https:\/\/doi.org\/10.1007\/s10596-010-9191-5","journal-title":"Comput. Geosci."},{"key":"1978_CR2","doi-asserted-by":"publisher","first-page":"1700","DOI":"10.1137\/S1064827595293582","volume":"19","author":"I Aavatsmark","year":"1998","unstructured":"Aavatsmark, I., Barkve, T., B\u00f8e, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I derivation of the methods. SIAM J. Sci. Comput. 19, 1700\u20131716 (1998). https:\/\/doi.org\/10.1137\/S1064827595293582","journal-title":"SIAM J. Sci. Comput."},{"key":"1978_CR3","doi-asserted-by":"publisher","first-page":"1717","DOI":"10.1137\/S1064827595293594","volume":"19","author":"I Aavatsmark","year":"1998","unstructured":"Aavatsmark, I., Barkve, T., B\u00f8e, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part II. Discussion and numerical results. SIAM J. Sci. Comput. 19, 1717\u20131736 (1998). https:\/\/doi.org\/10.1137\/S1064827595293594","journal-title":"SIAM J. Sci. Comput."},{"key":"1978_CR4","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1023\/A:1011510505406","volume":"2","author":"MG Edwards","year":"1998","unstructured":"Edwards, M.G., Rogers, C.F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2, 259\u2013290 (1998). https:\/\/doi.org\/10.1023\/A:1011510505406","journal-title":"Comput. Geosci."},{"key":"1978_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-85949-6","volume-title":"Computational Galerkin Methods","author":"CAJ Fletcher","year":"1984","unstructured":"Fletcher, C.A.J.: Computational Galerkin Methods. Springer, Berlin (1984). https:\/\/doi.org\/10.1007\/978-3-642-85949-6\u3039"},{"key":"1978_CR6","doi-asserted-by":"publisher","unstructured":"Ciarlet, P.G.: The finite element method for elliptic problems. Society for Industrial and Applied Mathematics; 2002. https:\/\/doi.org\/10.1137\/1.9780898719208.","DOI":"10.1137\/1.9780898719208"},{"key":"1978_CR7","doi-asserted-by":"publisher","unstructured":"Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2-nd order elliptic problems. Gall. I., Magenes E. Math. Asp. Finite Elem. Methods. Lect. Notes Math. vol 606., Springer, Berlin, 1977, pp. 292\u2013315. https:\/\/doi.org\/10.1007\/BFb0064470.","DOI":"10.1007\/BFb0064470"},{"key":"1978_CR8","doi-asserted-by":"publisher","unstructured":"Dur\u00e1n, R.G.: Mixed finite element methods. Boffi D., Gastaldi L. Mix. Finite Elem. Compat. Cond. Appl. Lect. Notes Math. vol 1939., Springer, Berlin, 2008, pp. 1\u201344. https:\/\/doi.org\/10.1007\/978-3-540-78319-0_1","DOI":"10.1007\/978-3-540-78319-0_1"},{"key":"1978_CR9","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1002\/fld.1948","volume":"61","author":"DKE de Carvalho","year":"2009","unstructured":"de Carvalho, D.K.E., Willmersdorf, R.B., Lyra, P.R.M.: Some results on the accuracy of an edge-based finite volume formulation for the solution of elliptic problems in non-homogeneous and non-isotropic media. Int. J. Numer. Methods Fluids 61, 237\u2013254 (2009). https:\/\/doi.org\/10.1002\/fld.1948","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"1978_CR10","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/0045-7825(73)90019-4","volume":"2","author":"PG Ciarlet","year":"1973","unstructured":"Ciarlet, P.G., Raviart, P.-A.: Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2, 17\u201331 (1973). https:\/\/doi.org\/10.1016\/0045-7825(73)90019-4","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1978_CR11","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1090\/S0025-5718-00-01270-9","volume":"70","author":"S Korotov","year":"2000","unstructured":"Korotov, S., K\u0159\u00ed\u017eek, M., Neittaanm\u00e4ki, P.: Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle. Math. Comput. 70, 107\u2013120 (2000). https:\/\/doi.org\/10.1090\/S0025-5718-00-01270-9","journal-title":"Math. Comput."},{"key":"1978_CR12","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1016\/j.crma.2004.02.010","volume":"338","author":"E Burman","year":"2004","unstructured":"Burman, E., Ern, A.: Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes. Comptes Rendus Math. 338, 641\u2013646 (2004). https:\/\/doi.org\/10.1016\/j.crma.2004.02.010","journal-title":"Comptes Rendus Math."},{"key":"1978_CR13","unstructured":"Le Potier, C.: A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators. Int. J. Finite Vol Inst Math\u00e9matiques Marseille, AMU, pp. 1\u201320 (2009)"},{"key":"1978_CR14","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1007\/s00211-013-0545-5","volume":"125","author":"C Canc\u00e8s","year":"2013","unstructured":"Canc\u00e8s, C., Cathala, M., Le Potier, C.: Monotone corrections for generic cell-centered finite volume approximations of anisotropic diffusion equations. Numer. Math. 125, 387\u2013417 (2013). https:\/\/doi.org\/10.1007\/s00211-013-0545-5","journal-title":"Numer. Math."},{"key":"1978_CR15","unstructured":"Pal, M., Edwards, M.G.: Flux-splitting schemes for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients. Eur. Conf. Comput. Fluid Dyn. (2006)."},{"key":"1978_CR16","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1002\/fld.2258","volume":"66","author":"M Pal","year":"2011","unstructured":"Pal, M., Edwards, M.G.: Non-linear flux-splitting schemes with imposed discrete maximum principle for elliptic equations with highly anisotropic coefficients. Int. J. Numer. Methods Fluids 66, 299\u2013323 (2011). https:\/\/doi.org\/10.1002\/fld.2258","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"1978_CR17","doi-asserted-by":"publisher","first-page":"1701","DOI":"10.1016\/j.jcp.2007.09.021","volume":"227","author":"Q-Y Chen","year":"2008","unstructured":"Chen, Q.-Y., Wan, J., Yang, Y., Mifflin, R.T.: Enriched multi-point flux approximation for general grids. J. Comput. Phys. 227, 1701\u20131721 (2008). https:\/\/doi.org\/10.1016\/j.jcp.2007.09.021","journal-title":"J. Comput. Phys."},{"key":"1978_CR18","doi-asserted-by":"publisher","first-page":"3448","DOI":"10.1016\/j.jcp.2009.01.031","volume":"228","author":"D Kuzmin","year":"2009","unstructured":"Kuzmin, D., Shashkov, M.J., Svyatskiy, D.: A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems. J. Comput. Phys. 228, 3448\u20133463 (2009). https:\/\/doi.org\/10.1016\/j.jcp.2009.01.031","journal-title":"J. Comput. Phys."},{"key":"1978_CR19","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1016\/j.jcp.2018.06.052","volume":"372","author":"S Su","year":"2018","unstructured":"Su, S., Dong, Q., Wu, J.: A decoupled and positivity-preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes. J. Comput. Phys. 372, 773\u2013798 (2018). https:\/\/doi.org\/10.1016\/j.jcp.2018.06.052","journal-title":"J. Comput. Phys."},{"key":"1978_CR20","doi-asserted-by":"publisher","DOI":"10.1002\/zamm.201900320","author":"F Zhao","year":"2020","unstructured":"Zhao, F., Sheng, Z., Yuan, G.: A monotone combination scheme of diffusion equations on polygonal meshes. ZAMM: J. Appl. Math. Mech.\/Zeitschrift F\u00fcr Angew Math Und Mech (2020). https:\/\/doi.org\/10.1002\/zamm.201900320","journal-title":"ZAMM: J. Appl. Math. Mech.\/Zeitschrift F\u00fcr Angew Math Und Mech"},{"key":"1978_CR21","unstructured":"Herbin, R., Hubert, F.: Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids. Finite Vol. complex Appl. V, pp. 659\u2013692 (2008)."},{"key":"1978_CR22","doi-asserted-by":"publisher","unstructured":"Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D.: Discontinuous Galerkin Methods for Elliptic Problems, pp. 89\u2013101 (2000). https:\/\/doi.org\/10.1007\/978-3-642-59721-3_5.","DOI":"10.1007\/978-3-642-59721-3_5"},{"key":"1978_CR23","doi-asserted-by":"publisher","first-page":"2157","DOI":"10.1002\/fld.2496","volume":"67","author":"Z Gao","year":"2011","unstructured":"Gao, Z., Wu, J.: A linearity-preserving cell-centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes. Int. J. Numer. Methods Fluids 67, 2157\u20132183 (2011). https:\/\/doi.org\/10.1002\/fld.2496","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"1978_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.compfluid.2015.11.013","volume":"127","author":"FRL Contreras","year":"2016","unstructured":"Contreras, F.R.L., Lyra, P.R.M., Souza, M.R.A., Carvalho, D.K.E.: A cell-centered multipoint flux approximation method with a diamond stencil coupled with a higher order finite volume method for the simulation of oil\u2013water displacements in heterogeneous and anisotropic petroleum reservoirs. Comput. Fluids 127, 1\u201316 (2016). https:\/\/doi.org\/10.1016\/j.compfluid.2015.11.013","journal-title":"Comput. Fluids"},{"key":"1978_CR25","doi-asserted-by":"publisher","DOI":"10.1002\/fld.4829","author":"TdeM Cavalcante","year":"2020","unstructured":"Cavalcante, TdeM., Contreras, F.R.L., Lyra, P.R.M., Carvalho, D.K.E.: A multipoint flux approximation with diamond stencil finite volume scheme for the two-dimensional simulation of fluid flows in naturally fractured reservoirs using a hybrid-grid method. Int. J. Numer. Methods Fluids (2020). https:\/\/doi.org\/10.1002\/fld.4829","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"1978_CR26","doi-asserted-by":"publisher","first-page":"106510","DOI":"10.1016\/j.compstruc.2021.106510","volume":"250","author":"RJM Lira Filho","year":"2021","unstructured":"Lira Filho, R.J.M., Santos, S.R., Cavalcante, TdeM., Contreras, F.R.L., Lyra, P.R.M., Carvalho, D.K.E.: A linearity-preserving finite volume scheme with a diamond stencil for the simulation of anisotropic and highly heterogeneous diffusion problems using tetrahedral meshes. Comput Struct 250, 106510 (2021). https:\/\/doi.org\/10.1016\/j.compstruc.2021.106510","journal-title":"Comput Struct"},{"key":"1978_CR27","doi-asserted-by":"publisher","first-page":"A607","DOI":"10.1137\/16M1098000","volume":"40","author":"Z Sheng","year":"2018","unstructured":"Sheng, Z., Yuan, G.: Construction of Nonlinear Weighted Method for Finite Volume Schemes Preserving Maximum Principle. SIAM J Sci Comput 40, A607\u2013A628 (2018). https:\/\/doi.org\/10.1137\/16M1098000","journal-title":"SIAM J Sci Comput"},{"key":"1978_CR28","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1016\/j.apnum.2020.04.014","volume":"156","author":"Z Sheng","year":"2020","unstructured":"Sheng, Z., Yuan, G., Yue, J.: A nonlinear convex combination in the construction of finite volume scheme satisfying maximum principle. Appl. Numer. Math. 156, 125\u2013139 (2020). https:\/\/doi.org\/10.1016\/j.apnum.2020.04.014","journal-title":"Appl. Numer. Math."},{"key":"1978_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1006\/jcph.2000.6418","volume":"160","author":"MG Edwards","year":"2000","unstructured":"Edwards, M.G.: M-matrix flux splitting for general full tensor discretization operators on structured and unstructured grids. J. Comput. Phys. 160, 1\u201328 (2000). https:\/\/doi.org\/10.1006\/jcph.2000.6418","journal-title":"J. Comput. Phys."},{"key":"1978_CR30","doi-asserted-by":"publisher","first-page":"314","DOI":"10.1016\/j.apnum.2020.08.008","volume":"158","author":"H Zhou","year":"2020","unstructured":"Zhou, H., Sheng, Z., Yuan, G.: A finite volume method preserving maximum principle for the diffusion equations with imperfect interface. Appl. Numer. Math. 158, 314\u2013335 (2020). https:\/\/doi.org\/10.1016\/j.apnum.2020.08.008","journal-title":"Appl. Numer. Math."},{"key":"1978_CR31","doi-asserted-by":"publisher","unstructured":"V\u00e9ron, L.: Elliptic Equations Involving Measures, pp. 593\u2013712 (2004). https:\/\/doi.org\/10.1016\/S1874-5733(04)80010-X.","DOI":"10.1016\/S1874-5733(04)80010-X"},{"key":"1978_CR32","doi-asserted-by":"publisher","unstructured":"Borsuk, M., Kondratiev, V.: The Dirichlet problem for elliptic linear divergent equations in a nonsmooth domain, 2006, pp. 165\u2013213. https:\/\/doi.org\/10.1016\/S0924-6509(06)80018-8.","DOI":"10.1016\/S0924-6509(06)80018-8"},{"key":"1978_CR33","doi-asserted-by":"publisher","first-page":"1329","DOI":"10.1002\/num.20320","volume":"24","author":"I Aavatsmark","year":"2008","unstructured":"Aavatsmark, I., Eigestad, G.T., Mallison, B.T., Nordbotten, J.M.: A compact multipoint flux approximation method with improved robustness. Numer. Methods Partial Differ. Equ 24, 1329\u20131360 (2008). https:\/\/doi.org\/10.1002\/num.20320","journal-title":"Numer. Methods Partial Differ. Equ"},{"key":"1978_CR34","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1016\/j.jcp.2014.07.003","volume":"275","author":"O M\u00f8yner","year":"2014","unstructured":"M\u00f8yner, O., Lie, K.-A.: A multiscale two-point flux-approximation method. J. Comput. Phys. 275, 273\u2013293 (2014). https:\/\/doi.org\/10.1016\/j.jcp.2014.07.003","journal-title":"J. Comput. Phys."},{"key":"1978_CR35","doi-asserted-by":"publisher","unstructured":"Eymard, R., Henry, G., Herbin, R., Hubert, F., Kl\u00f6fkorn, R., Manzini, G.: 3D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids, 2011, pp. 895\u2013930. https:\/\/doi.org\/10.1007\/978-3-642-20671-9_89.","DOI":"10.1007\/978-3-642-20671-9_89"},{"key":"1978_CR36","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1002\/fld.3850","volume":"74","author":"LES Queiroz","year":"2014","unstructured":"Queiroz, L.E.S., Souza, M.R.A., Contreras, F.R.L., Lyra, P.R.M., de Carvalho, D.K.E.: On the accuracy of a nonlinear finite volume method for the solution of diffusion problems using different interpolations strategies. Int. J. Numer. Methods Fluids 74, 270\u2013291 (2014). https:\/\/doi.org\/10.1002\/fld.3850","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"1978_CR37","doi-asserted-by":"publisher","DOI":"10.1515\/RJNAMM.2009.014","author":"AA Danilov","year":"2009","unstructured":"Danilov, A.A., Vassilevski, Y.V.: A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes. Russ J. Numer. Anal. Math. Model (2009). https:\/\/doi.org\/10.1515\/RJNAMM.2009.014","journal-title":"Russ J. Numer. Anal. Math. Model"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-022-01978-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-022-01978-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-022-01978-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,22]],"date-time":"2022-10-22T17:29:11Z","timestamp":1666459751000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-022-01978-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,21]]},"references-count":37,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,11]]}},"alternative-id":["1978"],"URL":"https:\/\/doi.org\/10.1007\/s10915-022-01978-6","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,21]]},"assertion":[{"value":"3 November 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 May 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 August 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 September 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article. All data generated or analyzed during this study are included in this published article. Even so, whoever considers that need additional information, feel free to contact the corresponding author.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"42"}}