{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T22:52:12Z","timestamp":1772837532893,"version":"3.50.1"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2016,12,9]],"date-time":"2016-12-09T00:00:00Z","timestamp":1481241600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2017,7]]},"DOI":"10.1007\/s00211-016-0851-9","type":"journal-article","created":{"date-parts":[[2016,12,9]],"date-time":"2016-12-09T14:17:01Z","timestamp":1481293021000},"page":"805-839","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["A residual-based a posteriori error estimator for single-phase Darcy flow in fractured porous media"],"prefix":"10.1007","volume":"136","author":[{"given":"Huangxin","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shuyu","family":"Sun","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2016,12,9]]},"reference":[{"key":"851_CR1","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s00211-002-0436-7","volume":"96","author":"Y Achdou","year":"2003","unstructured":"Achdou, Y., Bernardi, C., Coquel, F.: A priori and a posteriori analysis of finite volume discretizations of Darcy\u2019s equations. Numer. Math. 96, 17\u201342 (2003)","journal-title":"Numer. Math."},{"key":"851_CR2","volume-title":"Sobolev Spaces","author":"R Adams","year":"1975","unstructured":"Adams, R.: Sobolev Spaces. Academic Press, New York (1975)"},{"key":"851_CR3","doi-asserted-by":"crossref","DOI":"10.1002\/9781118032824","volume-title":"A Posteriori Error Estimation in Finite Element Analysis","author":"M Ainsworth","year":"2000","unstructured":"Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000)"},{"key":"851_CR4","first-page":"189","volume":"30","author":"M Ainsworth","year":"2007","unstructured":"Ainsworth, M.: A posteriori error estimation for lowest order Raviart\u2013Thomas mixed finite elements. SIAM J. Numer. Anal. 30, 189\u2013204 (2007)","journal-title":"SIAM J. Numer. Anal."},{"key":"851_CR5","doi-asserted-by":"crossref","unstructured":"Alboin, C., Jaffr\u00e9, J., Roberts, J.E., Serres, C.: Modeling fractures as interfaces for flow and transport in porous media. In: Fluid Flow and Transport in Porous Media: Mathematical and Numerical Treatment (South Hadley, MA, 2001). Contemporary Mathematics, vol. 295, pp. 13\u201324 (2002)","DOI":"10.1090\/conm\/295\/04999"},{"key":"851_CR6","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1137\/080727646","volume":"48","author":"I Babu\u0161ka","year":"2010","unstructured":"Babu\u0161ka, I., Gatica, G.N.: A residual-based a posteriori error estimator for the Stokes\u2013Darcy coupled problem. SIAM J. Numer. Anal. 48, 498\u2013523 (2010)","journal-title":"SIAM J. Numer. Anal."},{"key":"851_CR7","doi-asserted-by":"crossref","first-page":"1286","DOI":"10.1016\/0021-8928(60)90107-6","volume":"24","author":"G Barenblatt","year":"1960","unstructured":"Barenblatt, G., Zheltov, Y., Kochina, I.: Basic concepts in the theory of seepage of homogeneous fluids in fissured rocks. J. Appl. Math. Mech. 24, 1286\u20131303 (1960)","journal-title":"J. Appl. Math. Mech."},{"key":"851_CR8","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1002\/fld.1650040404","volume":"4","author":"R Baca","year":"1984","unstructured":"Baca, R., Arnett, R., Langford, D.: Modeling fluid flow in fractured porous rock masses by finite element techniques. Int. J. Num. Methods Fluids 4, 337\u2013348 (1984)","journal-title":"Int. J. Num. Methods Fluids"},{"key":"851_CR9","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1007\/s00211-011-0410-3","volume":"120","author":"TP Barrios","year":"2012","unstructured":"Barrios, T.P., Bustinza, R.: An a posteriori error analysis of an augmented discontinuous Galerkin formulation for Darcy flow. Numer. Math. 120, 231\u2013269 (2012)","journal-title":"Numer. Math."},{"key":"851_CR10","doi-asserted-by":"crossref","first-page":"2431","DOI":"10.1137\/S0036142994264079","volume":"33","author":"D Braess","year":"1996","unstructured":"Braess, D., Verf\u00fcrth, R.: A posteriori error estimators for the Raviart\u2013Thomas element. SIAM J. Numer. Anal. 33, 2431\u20132444 (1996)","journal-title":"SIAM J. Numer. Anal."},{"key":"851_CR11","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and Hybrid Finite Element Methods","author":"F Brezzi","year":"1991","unstructured":"Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)"},{"key":"851_CR12","volume-title":"FFW Documentation","author":"A Byfut","year":"2007","unstructured":"Byfut, A., Gedicke, J., G\u00fcnther, D., Reininghaus, J., Wiedemann, S., et al.: FFW Documentation. Humboldt University of Berlin, Germany (2007)"},{"key":"851_CR13","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1137\/080722631","volume":"48","author":"Z Cai","year":"2010","unstructured":"Cai, Z., Zhang, S.: Recovery-based error estimators for interface problems: mixed and nonconforming finite elements. SIAM J. Numer. Anal. 48, 30\u201352 (2010)","journal-title":"SIAM J. Numer. Anal."},{"key":"851_CR14","doi-asserted-by":"crossref","first-page":"1350056","DOI":"10.1142\/S0129183113500563","volume":"24","author":"J Cai","year":"2013","unstructured":"Cai, J., Sun, S.: Fractal analysis of fracture increasing spontaneous imbibition in porous media saturated with gas. Int. J. Mod. Phys. C 24, 1350056 (2013)","journal-title":"Int. J. Mod. Phys. C"},{"key":"851_CR15","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1090\/S0025-5718-97-00837-5","volume":"66","author":"C Carstensen","year":"1997","unstructured":"Carstensen, C.: A posteriori error estimate for the mixed finite element method. Math. Comput. 66, 465\u2013476 (1997)","journal-title":"Math. Comput."},{"key":"851_CR16","doi-asserted-by":"publisher","DOI":"10.1002\/2015WR018450","author":"H Chen","year":"2016","unstructured":"Chen, H., Salama, A., Sun, S.: Adaptive mixed finite element methods for Darcy flow in fractured porous media. Water Resour. Res. (2016). doi: 10.1002\/2015WR018450","journal-title":"Water Resour. Res."},{"key":"851_CR17","doi-asserted-by":"crossref","first-page":"2565","DOI":"10.1016\/j.cma.2003.12.059","volume":"193","author":"C Dawson","year":"2004","unstructured":"Dawson, C., Sun, S., Wheeler, M.F.: Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Eng. 193, 2565\u20132580 (2004)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"851_CR18","doi-asserted-by":"crossref","unstructured":"Demkowicz, L.: Polynomial Exact Sequences and Projection-Based Interpolation with Application to Maxwell Equations. Lecture Notes in Mathematics. Springer, Berlin (2008)","DOI":"10.1007\/978-3-540-78319-0_3"},{"key":"851_CR19","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1615\/JPorMedia.v14.i3.30","volume":"14","author":"C Dong","year":"2011","unstructured":"Dong, C., Sun, S., Taylor, G.A.: Numerical modeling of contaminant transport in fractured porous media using mixed finite element and finite volume methods. J. Porous Media. 14, 219\u2013242 (2011)","journal-title":"J. Porous Media."},{"key":"851_CR20","doi-asserted-by":"crossref","first-page":"929","DOI":"10.1137\/070708354","volume":"47","author":"VJ Ervin","year":"2009","unstructured":"Ervin, V.J., Jenkins, E.W., Sun, S.: Coupled generalized non-linear stokes flow with flow through a porous medium. SIAM J. Numer. Anal. 47, 929\u2013952 (2009)","journal-title":"SIAM J. Numer. Anal."},{"key":"851_CR21","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1051\/m2an\/2013132","volume":"48","author":"L Formaggia","year":"2014","unstructured":"Formaggia, L., Fumagalli, A., Scotti, A., Ruffo, P.: A reduced model for Darcy\u2019s problem in networks of fractures. ESAIM Math. Model. Numer. Anal. 48, 1089\u20131116 (2014)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"851_CR22","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/s10596-007-9062-x","volume":"12","author":"N Frih","year":"2008","unstructured":"Frih, N., Roberts, J.E., Saada, A.: Modeling fractures as interfaces: a model for Forchheimer fractures. Comput. Geosci. 12, 91\u2013104 (2008)","journal-title":"Comput. Geosci."},{"key":"851_CR23","unstructured":"Fumagalli, A.: Numerical Modelling of Flows in Fractured Porous Media by the XFEM Method. Ph.D. thesis. Politecnico di Milano (2012)"},{"key":"851_CR24","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/S0167-5648(02)80080-6","volume":"47","author":"S Gebauer","year":"2002","unstructured":"Gebauer, S., Neunh\u00e4userer, L., Kornhuber, R., Ochs, S., Hinkelmann, R., Helmig, R.: Equidimensional modelling of flow and transport processes in fractured porous systems I. Dev. Water Sci. 47, 335\u2013342 (2002)","journal-title":"Dev. Water Sci."},{"key":"851_CR25","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61623-5","volume-title":"Finite Element Methods for Navier\u2013Stokes Equations: Theory and Algorithms","author":"V Girault","year":"1986","unstructured":"Girault, V., Raviart, P.A.: Finite Element Methods for Navier\u2013Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)"},{"key":"851_CR26","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s11464-012-0189-2","volume":"7","author":"J Gopalakrishnan","year":"2012","unstructured":"Gopalakrishnan, J., Qiu, W.: Partial expansion of a Lipschitz domain and some applications. Front. Math. China 7, 249\u2013272 (2012)","journal-title":"Front. Math. China"},{"key":"851_CR27","doi-asserted-by":"crossref","first-page":"W11412","DOI":"10.1029\/2005WR004339","volume":"41","author":"H Hoteit","year":"2005","unstructured":"Hoteit, H., Firoozabadi, A.: Multicomponent fluid flow by discontinuous Galerkin and mixed methods in unfractured and fractured media. Water Resour. Res. 41, W11412 (2005)","journal-title":"Water Resour. Res."},{"key":"851_CR28","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1016\/j.advwatres.2008.02.004","volume":"31","author":"H Hoteit","year":"2008","unstructured":"Hoteit, H., Firoozabadi, A.: An efficient numerical model for incompressible two-phase flow in fractured media. Adv. Water Resour. 31, 891\u2013905 (2008)","journal-title":"Adv. Water Resour."},{"key":"851_CR29","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1090\/S0025-5718-06-01903-X","volume":"76","author":"K Kim","year":"2007","unstructured":"Kim, K.: A posteriori error analysis for locally conservative mixed methods. Math. Comput. 76, 43\u201366 (2007)","journal-title":"Math. Comput."},{"key":"851_CR30","doi-asserted-by":"crossref","first-page":"1659","DOI":"10.1090\/S0025-5718-06-01872-2","volume":"75","author":"C Lovadina","year":"2006","unstructured":"Lovadina, C., Stenberg, R.: Energy norm a posteriori error estimates for mixed finite element methods. Math. Comput. 75, 1659\u20131674 (2006)","journal-title":"Math. Comput."},{"key":"851_CR31","doi-asserted-by":"crossref","first-page":"1667","DOI":"10.1137\/S1064827503429363","volume":"26","author":"V Martin","year":"2005","unstructured":"Martin, V., Jaffr\u00e9, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26, 1667\u20131691 (2005)","journal-title":"SIAM J. Sci. Comput."},{"key":"851_CR32","doi-asserted-by":"crossref","first-page":"W05511","DOI":"10.1029\/2010WR009801","volume":"47","author":"J Moortgat","year":"2011","unstructured":"Moortgat, J., Sun, S., Firoozabadi, A.: Compositional modeling of three-phase flow with gravity using higher-order finite element methods. Water Resour. Res. 47, W05511 (2011)","journal-title":"Water Resour. Res."},{"key":"851_CR33","doi-asserted-by":"crossref","first-page":"588","DOI":"10.1029\/WR018i003p00588","volume":"18","author":"J Noorishad","year":"1982","unstructured":"Noorishad, J., Mehran, M.: An upstream finite element method for solution of transient transport equation in fractured porous media. Water Resour. Res. 18, 588\u2013596 (1982)","journal-title":"Water Resour. Res."},{"key":"851_CR34","doi-asserted-by":"crossref","first-page":"14","DOI":"10.2118\/10509-PA","volume":"25","author":"K Pruess","year":"1985","unstructured":"Pruess, K., Narasimhan, T.: A practical method for modeling fluid and heat flow in fractured porous media. SPE J. 25, 14\u201326 (1985)","journal-title":"SPE J."},{"key":"851_CR35","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1090\/S0025-5718-1990-1011446-7","volume":"54","author":"R Scott","year":"1990","unstructured":"Scott, R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483\u2013493 (1990)","journal-title":"Math. Comput."},{"key":"851_CR36","doi-asserted-by":"crossref","first-page":"389","DOI":"10.4208\/aamm.10-m1108","volume":"4","author":"P Song","year":"2012","unstructured":"Song, P., Sun, S.: Contaminant flow and transport simulation in cracked porous media using locally conservative schemes. Adv. Appl. Math. Mech. 4, 389\u2013421 (2012)","journal-title":"Adv. Appl. Math. Mech."},{"key":"851_CR37","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/s10208-005-0183-0","volume":"2","author":"R Stevenson","year":"2007","unstructured":"Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 2, 245\u2013269 (2007)","journal-title":"Found. Comput. Math."},{"key":"851_CR38","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1137\/S003614290241708X","volume":"43","author":"S Sun","year":"2005","unstructured":"Sun, S., Wheeler, M.F.: Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal. 43, 195\u2013219 (2005)","journal-title":"SIAM J. Numer. Anal."},{"key":"851_CR39","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1007\/s10915-004-4148-2","volume":"22","author":"S Sun","year":"2005","unstructured":"Sun, S., Wheeler, M.F.: L2(H1) norm a posteriori error estimation for discontinuous Galerkin approximations of reactive transport problems. J. Sci. Comput. 22, 501\u2013530 (2005)","journal-title":"J. Sci. Comput."},{"key":"851_CR40","volume-title":"Adaptive Computations: Theory and Algorithms","author":"T Tang","year":"2007","unstructured":"Tang, T., Xu, J.: Adaptive Computations: Theory and Algorithms. Science Press, Beijing (2007)"},{"key":"851_CR41","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/0377-0427(94)90290-9","volume":"50","author":"R Verf\u00fcrth","year":"1994","unstructured":"Verf\u00fcrth, R.: A posteriori error estimates and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50, 67\u201383 (1994)","journal-title":"J. Comput. Appl. Math."},{"key":"851_CR42","volume-title":"A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques","author":"R Verf\u00fcrth","year":"1996","unstructured":"Verf\u00fcrth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester (1996)"},{"key":"851_CR43","doi-asserted-by":"crossref","unstructured":"Warren, J., Root, P.: The behavior of naturally fractured reservoirs. SPE J. 3, 245\u2013255 (1963)","DOI":"10.2118\/426-PA"},{"key":"851_CR44","doi-asserted-by":"crossref","first-page":"327","DOI":"10.2118\/15129-PA","volume":"3","author":"YS Wu","year":"1988","unstructured":"Wu, Y.S., Pruess, K.: A multiple-porosity method for simulation of naturally fractured petroleum reservoirs. SPE Reserv. Eng. 3, 327\u2013336 (1988)","journal-title":"SPE Reserv. Eng."},{"key":"851_CR45","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.advwatres.2014.08.010","volume":"74","author":"A Zidane","year":"2014","unstructured":"Zidane, A., Firoozabadi, A.: An efficient numerical model for multicomponent compressible flow in fractured porous media. Adv. Water Resour. 74, 127\u2013147 (2014)","journal-title":"Adv. Water Resour."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-016-0851-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-016-0851-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-016-0851-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,16]],"date-time":"2019-09-16T07:11:45Z","timestamp":1568617905000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-016-0851-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,9]]},"references-count":45,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,7]]}},"alternative-id":["851"],"URL":"https:\/\/doi.org\/10.1007\/s00211-016-0851-9","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,12,9]]}}}