{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T18:48:07Z","timestamp":1775760487161,"version":"3.50.1"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,11,1]],"date-time":"2020-11-01T00:00:00Z","timestamp":1604188800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,11,1]],"date-time":"2020-11-01T00:00:00Z","timestamp":1604188800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"CONICYT-Chile","award":["PAI77190084"],"award-info":[{"award-number":["PAI77190084"]}]},{"name":"CONICYT-Chile","award":["AFB170001"],"award-info":[{"award-number":["AFB170001"]}]},{"name":"Universidad del Bio-Bio","award":["151408 GI\/VC"],"award-info":[{"award-number":["151408 GI\/VC"]}]},{"name":"CONICYT-Chile","award":["11121347"],"award-info":[{"award-number":["11121347"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2020,11]]},"DOI":"10.1007\/s10915-020-01305-x","type":"journal-article","created":{"date-parts":[[2020,11,4]],"date-time":"2020-11-04T08:02:53Z","timestamp":1604476973000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["A Fully-Mixed Formulation for the Steady Double-Diffusive Convection System Based upon Brinkman\u2013Forchheimer Equations"],"prefix":"10.1007","volume":"85","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7811-759X","authenticated-orcid":false,"given":"Sergio","family":"Caucao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gabriel N.","family":"Gatica","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ricardo","family":"Oyarz\u00faa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nestor","family":"S\u00e1nchez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,11,4]]},"reference":[{"issue":"40","key":"1305_CR1","doi-asserted-by":"publisher","first-page":"2938","DOI":"10.1016\/j.physleta.2018.06.030","volume":"382","author":"AK Alzahrani","year":"2018","unstructured":"Alzahrani, A.K.: Importance of Darcy\u2013Forchheimer porous medium in $$3$$D convective flow of carbon nanotubes. Phys. Lett. A 382(40), 2938\u20132943 (2018)","journal-title":"Phys. Lett. A"},{"issue":"1","key":"1305_CR2","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1007\/s13137-019-0119-5","volume":"10","author":"I Ambartsumyan","year":"2019","unstructured":"Ambartsumyan, I., Khattatov, E., Nguyen, T., Yotov, I.: Flow and transport in fractured poroelastic media. GEM Int. J. Geomath. 10(1), 34 (2019)","journal-title":"GEM Int. J. Geomath."},{"issue":"5","key":"1305_CR3","doi-asserted-by":"publisher","first-page":"1189","DOI":"10.1016\/j.apt.2018.02.010","volume":"29","author":"MM Bhatti","year":"2018","unstructured":"Bhatti, M.M., Zeeshan, A., Ellahi, R., Shit, G.C.: Mathematical modeling of heat and mass transfer effects on MHD peristaltic propulsion of two-phase flow through a Darcy\u2013Brinkman\u2013Forchheimer porous medium. Adv. Powder Technol. 29(5), 1189\u20131197 (2018)","journal-title":"Adv. Powder Technol."},{"key":"1305_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics","author":"F Brezzi","year":"1991","unstructured":"Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)"},{"key":"1305_CR5","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/BF02120313","volume":"1","author":"HC Brinkman","year":"1949","unstructured":"Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1, 27\u201334 (1949)","journal-title":"Appl. Sci. Res."},{"issue":"3","key":"1305_CR6","doi-asserted-by":"publisher","first-page":"1318","DOI":"10.1137\/18M1196108","volume":"57","author":"R B\u00fcrger","year":"2019","unstructured":"B\u00fcrger, R., M\u00e9ndez, P.E., Ruiz-Baier, R.: On $${\\mathbf{H}}(\\div )$$-conforming methods for double-diffusion equations in porous media. SIAM J. Numer. Anal. 57(3), 1318\u20131343 (2019)","journal-title":"SIAM J. Numer. Anal."},{"key":"1305_CR7","unstructured":"Cama\u00f1o, J., Garc\u00eda, C., Oyarz\u00faa, R.: Analysis of a conservative mixed-FEM for the stationary Navier\u2013Stokes problem. Preprint 2018-25, Centro de Investigaci\u00f3n en Ingenier\u00eda Matem\u00e1tica ($$\\text{CI}^2$$MA), Universidad de Concepci\u00f3n, Chile (2018)"},{"issue":"2","key":"1305_CR8","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.1137\/15M1013146","volume":"54","author":"J Cama\u00f1o","year":"2016","unstructured":"Cama\u00f1o, J., Gatica, G.N., Oyarz\u00faa, R., Tierra, G.: An augmented mixed finite element method for the Navier\u2013Stokes equations with variable viscosity. SIAM J. Numer. Anal. 54(2), 1069\u20131092 (2016)","journal-title":"SIAM J. Numer. Anal."},{"key":"1305_CR9","doi-asserted-by":"publisher","first-page":"114","DOI":"10.1553\/etna_vol48s114","volume":"48","author":"J Cama\u00f1o","year":"2018","unstructured":"Cama\u00f1o, J., Mu\u00f1oz, C., Oyarz\u00faa, R.: Numerical analysis of a dual-mixed problem in non-standard Banach spaces. Electron. Trans. Numer. Anal. 48, 114\u2013130 (2018)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"1305_CR10","unstructured":"Caucao, S., Gatica, G.N., Ortega, J.P.: A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman\u2013Forchheimer and double-diffusion equations. (in preparation)"},{"issue":"5","key":"1305_CR11","doi-asserted-by":"publisher","first-page":"1947","DOI":"10.1051\/m2an\/2018027","volume":"52","author":"S Caucao","year":"2018","unstructured":"Caucao, S., Gatica, G.N., Oyarz\u00faa, R.: Analysis of an augmented fully-mixed formulation for the coupling of the Stokes and heat equations. ESAIM Math. Model. Numer. Anal. 52(5), 1947\u20131980 (2018)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"1305_CR12","unstructured":"Caucao, S., Oyarz\u00faa, R., Villa-Fuentes, S.: A new mixed-FEM for steady-state natural convection models allowing conservation of momentum and thermal energy. Calcolo (to appear)"},{"issue":"8","key":"1305_CR13","doi-asserted-by":"publisher","first-page":"801","DOI":"10.1016\/j.aml.2005.11.002","volume":"19","author":"AO Celebi","year":"2006","unstructured":"Celebi, A.O., Kalantarov, V.K., Ugurlu, D.: On continuous dependence on coefficients of the Brinkman\u2013Forchheimer equations. Appl. Math. Lett. 19(8), 801\u2013807 (2006)","journal-title":"Appl. Math. Lett."},{"key":"1305_CR14","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611972597","volume-title":"Linear and Nonlinear Functional Analysis with Applications","author":"PG Ciarlet","year":"2013","unstructured":"Ciarlet, P.G.: Linear and Nonlinear Functional Analysis with Applications. Society for Industrial and Applied Mathematics, Philadelphia (2013)"},{"issue":"5","key":"1305_CR15","doi-asserted-by":"publisher","first-page":"1525","DOI":"10.1051\/m2an\/2020007","volume":"54","author":"E Colmenares","year":"2020","unstructured":"Colmenares, E., Gatica, G.N., Moraga, S.: A Banach spaces-based analysis of a new fully-mixed finite element method for the Boussinesq problem. ESAIM Math. Model. Numer. Anal. 54(5), 1525\u20131568 (2020)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"1305_CR16","doi-asserted-by":"publisher","first-page":"125","DOI":"10.5802\/smai-jcm.64","volume":"6","author":"E Colmenares","year":"2020","unstructured":"Colmenares, E., Gatica, G.N., Moraga, S., Ruiz-Baier, R.: A fully-mixed finite element method for the steady state Oberbeck\u2013Boussinesq system. SMAI J. Comput. Math. 6, 125\u2013157 (2020)","journal-title":"SMAI J. Comput. Math."},{"issue":"1","key":"1305_CR17","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/s10092-016-0182-3","volume":"54","author":"E Colmenares","year":"2017","unstructured":"Colmenares, E., Gatica, G.N., Oyarz\u00faa, R.: An augmented fully-mixed finite element method for the stationary Boussinesq problem. Calcolo 54(1), 167\u2013205 (2017)","journal-title":"Calcolo"},{"issue":"1","key":"1305_CR18","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.crma.2015.10.004","volume":"354","author":"E Colmenares","year":"2016","unstructured":"Colmenares, E., Gatica, G.N., Oyarz\u00faa, R.: Fixed point strategies for mixed variational formulations of the stationary Boussinesq problem. C. R. Math. Acad. Sci. Paris 354(1), 57\u201362 (2016)","journal-title":"C. R. Math. Acad. Sci. Paris"},{"key":"1305_CR19","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1145\/992200.992206","volume":"30","author":"TA Davis","year":"2004","unstructured":"Davis, T.A.: Algorithm 832: UMFPACK V43\u2014an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30, 196\u2013199 (2004)","journal-title":"ACM Trans. Math. Softw."},{"key":"1305_CR20","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4355-5","volume-title":"Theory and Practice of Finite Elements. Applied Mathematical Sciences","author":"A Ern","year":"2004","unstructured":"Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)"},{"issue":"1\u20132","key":"1305_CR21","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1016\/j.jconhyd.2009.08.004","volume":"110","author":"J Faulkner","year":"2009","unstructured":"Faulkner, J., Hu, B.X., Kish, S., Hua, F.: Laboratory analog and numerical study of ground water flow and solute transport in a karst aquifer with conduit and matrix domains. J. Contam. Hydrol. 110(1\u20132), 34\u201344 (2009)","journal-title":"J. Contam. Hydrol."},{"key":"1305_CR22","first-page":"1782","volume":"45","author":"P Forchheimer","year":"1901","unstructured":"Forchheimer, P.: Wasserbewegung durch boden. Z. Ver. Deutsch Ing. 45, 1782\u20131788 (1901)","journal-title":"Z. Ver. Deutsch Ing."},{"key":"1305_CR23","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-03695-3","volume-title":"A Simple Introduction to the Mixed Finite Element Method. Theory and Applications. Springer Briefs in Mathematics","author":"GN Gatica","year":"2014","unstructured":"Gatica, G.N.: A Simple Introduction to the Mixed Finite Element Method. Theory and Applications. Springer Briefs in Mathematics. Springer, Cham (2014)"},{"issue":"4","key":"1305_CR24","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1007\/s00211-013-0577-x","volume":"126","author":"GN Gatica","year":"2014","unstructured":"Gatica, G.N., Gatica, L.F., M\u00e1rquez, A.: Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow. Numer. Math. 126(4), 635\u2013677 (2014)","journal-title":"Numer. Math."},{"issue":"7","key":"1305_CR25","doi-asserted-by":"publisher","first-page":"2420","DOI":"10.1016\/j.camwa.2017.12.029","volume":"75","author":"LF Gatica","year":"2018","unstructured":"Gatica, L.F., Oyarz\u00faa, R., S\u00e1nchez, N.: A priori and a posteriori error analysis of an augmented mixed-FEM for the Navier\u2013Stokes\u2013Brinkman problem. Comput. Math. Appl. 75(7), 2420\u20132444 (2018)","journal-title":"Comput. Math. Appl."},{"key":"1305_CR26","doi-asserted-by":"crossref","unstructured":"Girault, V., Raviart, P.A.: Finite Element Methods for Navier\u2013Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)","DOI":"10.1007\/978-3-642-61623-5"},{"key":"1305_CR27","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1515\/jnum-2012-0013","volume":"20","author":"F Hecht","year":"2012","unstructured":"Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251\u2013265 (2012)","journal-title":"J. Numer. Math."},{"issue":"1","key":"1305_CR28","doi-asserted-by":"publisher","first-page":"138","DOI":"10.1006\/jmaa.1996.0428","volume":"204","author":"PN Kaloni","year":"1996","unstructured":"Kaloni, P.N., Guo, J.: Steady nonlinear double-diffusive convection in a porous medium based upon the Brinkman\u2013Forchheimer model. J. Math. Anal. Appl. 204(1), 138\u2013155 (1996)","journal-title":"J. Math. Anal. Appl."},{"key":"1305_CR29","volume-title":"Function Spaces. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis","author":"A Kufner","year":"1977","unstructured":"Kufner, A., Jhon, O., Fu\u010d\u00edk, S.: Function Spaces. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff International Publishing, Leyden (1977)"},{"issue":"1","key":"1305_CR30","first-page":"79","volume":"33","author":"M \u00d4tani","year":"2013","unstructured":"\u00d4tani, M., Uchida, S.: Global solvability of some double-diffusive convection system coupled with Brinkman\u2013Forchheimer equations. Lib. Math. (N.S.) 33(1), 79\u2013107 (2013)","journal-title":"Lib. Math. (N.S.)"},{"issue":"4","key":"1305_CR31","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1111\/1467-9590.00116","volume":"102","author":"LE Payne","year":"1999","unstructured":"Payne, L.E., Straughan, B.: Convergence and continuous dependence for the Brinkman\u2013Forchheimer equations. Stud. Appl. Math. 102(4), 419\u2013439 (1999)","journal-title":"Stud. Appl. Math."},{"key":"1305_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-85268-1","volume-title":"Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics","author":"A Quarteroni","year":"1994","unstructured":"Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)"},{"key":"1305_CR33","doi-asserted-by":"crossref","unstructured":"Roberts, J.E., Thomas, J.M.: Mixed and Hybrid Methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. II, Finite Element Methods (Part 1), North-Holland, Amsterdam (1991)","DOI":"10.1016\/S1570-8659(05)80041-9"},{"issue":"1","key":"1305_CR34","first-page":"89","volume":"70","author":"S Safi","year":"2018","unstructured":"Safi, S., Benissaad, S.: Double-diffusive convection in an anisotropic porous layer using the Darcy\u2013Brinkman\u2013Forchheimer formulation. Arch. Mech. (Arch. Mech. Stos.) 70(1), 89\u2013102 (2018)","journal-title":"Arch. Mech. (Arch. Mech. Stos.)"},{"key":"1305_CR35","doi-asserted-by":"publisher","first-page":"670","DOI":"10.1016\/j.ijheatmasstransfer.2017.08.068","volume":"115\u2013B","author":"YJ Zhuang","year":"2017","unstructured":"Zhuang, Y.J., Yu, H.Z., Zhu, Q.Y.: A thermal non-equilibrium model for $$3$$D double diffusive convection of power-law fluids with chemical reaction in the porous medium. Int. J. Heat Mass Transf. 115\u2013B, 670\u2013694 (2017)","journal-title":"Int. J. Heat Mass Transf."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01305-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-020-01305-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01305-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T10:35:58Z","timestamp":1697020558000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-020-01305-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11]]},"references-count":35,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,11]]}},"alternative-id":["1305"],"URL":"https:\/\/doi.org\/10.1007\/s10915-020-01305-x","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11]]},"assertion":[{"value":"14 January 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 September 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 November 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"44"}}