{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T02:47:31Z","timestamp":1782096451263,"version":"3.54.5"},"reference-count":61,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T00:00:00Z","timestamp":1777248000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T00:00:00Z","timestamp":1777248000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Strategic Priority Research Program of the Chinese Academy of Sciences","award":["XDB0640000"],"award-info":[{"award-number":["XDB0640000"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12571395"],"award-info":[{"award-number":["12571395"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12288201"],"award-info":[{"award-number":["12288201"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012492","name":"Youth Innovation Promotion Association","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100012492","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":[[2026,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Discrete fracture models with reduced-dimensional treatment of conductive and blocking fractures are widely used to simulate fluid flow in fractured porous media. Among these, numerical methods based on interface models are intensively studied, where the fractures are treated as co-dimension one manifolds with governing equations and interface conditions in a bulk matrix. In this paper, we propose a simple yet effective treatment for modeling the fractures on fitted grids in the interior penalty discontinuous Galerkin (IPDG) methods without introducing any additional degrees of freedom or equations on the interfaces. We conduct stability and\n                    <jats:italic>hp<\/jats:italic>\n                    -analysis for the proposed IPDG method, deriving optimal a priori error bounds concerning mesh size (\n                    <jats:italic>h<\/jats:italic>\n                    ) and sub-optimal bounds for polynomial degree (\n                    <jats:italic>k<\/jats:italic>\n                    ) in both the energy norm and the\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$L^2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    norm. Numerical experiments involving published benchmarks validate our theoretical analysis and demonstrate the method\u2019s robust performance. Furthermore, we extend our method to two-phase flows and use numerical tests to confirm the algorithm\u2019s validity.\n                  <\/jats:p>","DOI":"10.1007\/s10915-026-03285-w","type":"journal-article","created":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T16:43:43Z","timestamp":1777308223000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Interior Penalty Discontinuous Galerkin Method for an Interface Model of Flow in Fractured Porous Media"],"prefix":"10.1007","volume":"107","author":[{"given":"Yong","family":"Liu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7298-004X","authenticated-orcid":false,"given":"Ziyao","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,27]]},"reference":[{"key":"3285_CR1","doi-asserted-by":"publisher","first-page":"462","DOI":"10.1016\/j.jcp.2014.12.047","volume":"284","author":"R Ahmed","year":"2015","unstructured":"Ahmed, R., Edwards, M.G., Lamine, S., Huisman, B.A., Pal, M.: Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model. J. Comput. Phys. 284, 462\u2013489 (2015)","journal-title":"J. Comput. Phys."},{"key":"3285_CR2","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1051\/m2an\/2008052","volume":"43","author":"P Angot","year":"2009","unstructured":"Angot, P., Boyer, F., Hubert, F.: Asymptotic and numerical modelling of flows in fractured porous media. ESAIM: Mathematical Modelling and Numerical Analysis 43, 239\u2013275 (2009)","journal-title":"ESAIM: Mathematical Modelling and Numerical Analysis"},{"key":"3285_CR3","doi-asserted-by":"publisher","first-page":"A109","DOI":"10.1137\/17M1138194","volume":"41","author":"PF Antonietti","year":"2019","unstructured":"Antonietti, P.F., Facciol\u00e0, C., Russo, A., Verani, M.: Discontinuous Galerkin approximation of flows in fractured porous media on polytopic grids. SIAM J. Sci. Comput. 41, A109\u2013A138 (2019)","journal-title":"SIAM J. Sci. Comput."},{"key":"3285_CR4","doi-asserted-by":"publisher","first-page":"340","DOI":"10.3934\/mine.2020017","volume":"2","author":"PF Antonietti","year":"2020","unstructured":"Antonietti, P.F., Facciol\u00e0, C., Verani, M.: Unified analysis of discontinuous Galerkin approximations of flows in fractured porous media on polygonal and polyhedral grids. Mathematics in Engineering 2, 340\u2013385 (2020)","journal-title":"Mathematics in Engineering"},{"key":"3285_CR5","doi-asserted-by":"publisher","first-page":"809","DOI":"10.1051\/m2an\/2015087","volume":"50","author":"PF Antonietti","year":"2016","unstructured":"Antonietti, P.F., Formaggia, L., Scotti, A., Verani, M., Verzott, N.: Mimetic finite difference approximation of flows in fractured porous media. ESAIM: Mathematical Modelling and Numerical Analysis 50, 809\u2013832 (2016)","journal-title":"ESAIM: Mathematical Modelling and Numerical Analysis"},{"key":"3285_CR6","doi-asserted-by":"publisher","first-page":"1749","DOI":"10.1137\/S0036142901384162","volume":"39","author":"DN Arnold","year":"2002","unstructured":"Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749\u20131779 (2002)","journal-title":"SIAM J. Numer. Anal."},{"key":"3285_CR7","doi-asserted-by":"publisher","first-page":"B1082","DOI":"10.1137\/18M1224751","volume":"41","author":"A Arrar\u00e1s","year":"2019","unstructured":"Arrar\u00e1s, A., Gaspar, F.J., Portero, L., Rodrigo, C.: Mixed-dimensional geometric multigrid methods for single-phase flow in fractured porous media. SIAM J. Sci. Comput. 41, B1082\u2013B1114 (2019)","journal-title":"SIAM J. Sci. Comput."},{"key":"3285_CR8","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1051\/m2an\/1987210201991","volume":"21","author":"I Babuska","year":"1987","unstructured":"Babuska, I., Suri, M.: The h-p version of the finite element method with quasiuniform meshes. RARIRO- Model. Math. Anal. Numer. 21, 199\u2013238 (1987)","journal-title":"RARIRO- Model. Math. Anal. Numer."},{"key":"3285_CR9","doi-asserted-by":"publisher","first-page":"2203","DOI":"10.1137\/17M1139102","volume":"56","author":"WM Boon","year":"2018","unstructured":"Boon, W.M., Nordbotten, J.M., Yotov, I.: Robust discretization of flow in fractured porous media. SIAM J. Numer. Anal. 56, 2203\u20132233 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"3285_CR10","doi-asserted-by":"publisher","first-page":"2675","DOI":"10.1090\/mcom\/3322","volume":"87","author":"A Cangiani","year":"2018","unstructured":"Cangiani, A., Georgoulis, E., Sabawi, Y.: Adaptive discontinuous Galerkin methods for elliptic interface problems. Math. Comput. 87, 2675\u20132707 (2018)","journal-title":"Math. Comput."},{"key":"3285_CR11","doi-asserted-by":"publisher","first-page":"A1063","DOI":"10.1137\/17M1119500","volume":"40","author":"F Chave","year":"2018","unstructured":"Chave, F., Di Pietro, D.A., Formaggia, L.: A hybrid high-order method for Darcy flows in fractured porous media. SIAM J. Sci. Comput. 40, A1063\u2013A1094 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"3285_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112244","volume":"488","author":"S Chen","year":"2023","unstructured":"Chen, S.: Discontinuous Galerkin method for hybrid-dimensional fracture models of two-phase flow. J. Comput. Phys. 488, 112244 (2023)","journal-title":"J. Comput. Phys."},{"key":"3285_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112384","volume":"491","author":"Z Chen","year":"2023","unstructured":"Chen, Z., Liu, Y.: An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation. J. Comput. Phys. 491, 112384 (2023)","journal-title":"J. Comput. Phys."},{"key":"3285_CR14","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1016\/j.apnum.2024.08.012","volume":"206","author":"Z Chen","year":"2024","unstructured":"Chen, Z., Liu, Y.: An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation, Part II. Piecewise-smooth interfaces, Applied Numerical Mathematics 206, 247\u2013268 (2024)","journal-title":"Piecewise-smooth interfaces, Applied Numerical Mathematics"},{"key":"3285_CR15","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1016\/j.jcp.2018.11.003","volume":"378","author":"N Chuenjarern","year":"2019","unstructured":"Chuenjarern, N., Xu, Z., Yang, Y.: High-order bound-preserving discontinuous Galerkin methods for compressible miscible displacements in porous media on triangular meshes. J. Comput. Phys. 378, 110\u2013128 (2019)","journal-title":"J. Comput. Phys."},{"key":"3285_CR16","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1006\/jcph.1998.5892","volume":"141","author":"B Cockburn","year":"1998","unstructured":"Cockburn, B., Shu, C.-W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199\u2013224 (1998)","journal-title":"J. Comput. Phys."},{"key":"3285_CR17","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1051\/m2an\/2011148","volume":"46","author":"C D\u2019Angelo","year":"2012","unstructured":"D\u2019Angelo, C., Scotti, A.: A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. ESAIM: Mathematical Modelling and Numerical Analysis 46, 465\u2013489 (2012)","journal-title":"ESAIM: Mathematical Modelling and Numerical Analysis"},{"key":"3285_CR18","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1016\/j.cam.2008.08.026","volume":"225","author":"Y Epshteyn","year":"2009","unstructured":"Epshteyn, Y., Rivi\u00e8re, B.: Analysis of hp discontinuous Galerkin methods for incompressible two-phase flow. J. Comput. Appl. Math. 225, 487\u2013509 (2009)","journal-title":"J. Comput. Appl. Math."},{"key":"3285_CR19","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/j.advwatres.2017.10.036","volume":"111","author":"B Flemisch","year":"2018","unstructured":"Flemisch, B., Berre, I., Boon, W., Fumagalli, A., Schwenck, N., Scotti, A., Stefansson, I., Tatomir, A.: Benchmarks for single-phase flow in fractured porous media. Adv. Water Resour. 111, 239\u2013258 (2018)","journal-title":"Adv. Water Resour."},{"key":"3285_CR20","first-page":"47","volume-title":"A review of the XFEM-based approximation of flow in fractured porous media, Advances in Discretization Methods: Discontinuities","author":"B Flemisch","year":"2016","unstructured":"Flemisch, B., Fumagalli, A., Scotti, A.: A review of the XFEM-based approximation of flow in fractured porous media, Advances in Discretization Methods: Discontinuities, pp. 47\u201376. Fictitious Domain Methods, Virtual Elements (2016)"},{"key":"3285_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.advwatres.2023.104390","volume":"173","author":"G Fu","year":"2023","unstructured":"Fu, G., Yang, Y.: A hybridizable discontinuous Galerkin method on unfitted meshes for single-phase Darcy flow in fractured porous media. Adv. Water Resour. 173, 104390 (2023)","journal-title":"Adv. Water Resour."},{"key":"3285_CR22","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1016\/j.advwatres.2017.10.031","volume":"110","author":"D Gl\u00e4ser","year":"2017","unstructured":"Gl\u00e4ser, D., Helmig, R., Flemisch, B., Class, H.: A discrete fracture model for two-phase flow in fractured porous media. Adv. Water Resour. 110, 335\u2013348 (2017)","journal-title":"Adv. Water Resour."},{"key":"3285_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110715","volume":"448","author":"D Gl\u00e4ser","year":"2022","unstructured":"Gl\u00e4ser, D., Schneider, M., Flemisch, B., Helmig, R.: Comparison of cell-and vertex-centered finite-volume schemes for flow in fractured porous media. J. Comput. Phys. 448, 110715 (2022)","journal-title":"J. Comput. Phys."},{"key":"3285_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.advwatres.2021.103951","volume":"153","author":"H Guo","year":"2021","unstructured":"Guo, H., Feng, W., Xu, Z., Yang, Y.: Conservative numerical methods for the reinterpreted discrete fracture model on non-conforming meshes and their applications in contaminant transportation in fractured porous media. Adv. Water Resour. 153, 103951 (2021)","journal-title":"Adv. Water Resour."},{"key":"3285_CR25","doi-asserted-by":"publisher","first-page":"19","DOI":"10.2118\/90276-PA","volume":"11","author":"H Hoteit","year":"2006","unstructured":"Hoteit, H., Firoozabadi, A.: Compositional modeling by the combined discontinuous Galerkin and mixed methods. SPE J. 11, 19\u201334 (2006a)","journal-title":"SPE J."},{"key":"3285_CR26","doi-asserted-by":"publisher","first-page":"341","DOI":"10.2118\/90277-PA","volume":"11","author":"H Hoteit","year":"2006","unstructured":"Hoteit, H., Firoozabadi, A.: Compositional modeling of discrete-fractured media without transfer functions by the discontinuous Galerkin and mixed methods. SPE J. 11, 341\u2013352 (2006b)","journal-title":"SPE J."},{"key":"3285_CR27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109301","volume":"409","author":"H Huang","year":"2020","unstructured":"Huang, H., Li, J., Yan, J.: High order symmetric direct discontinuous Galerkin method for elliptic interface problems with fitted mesh. J. Comput. Phys. 409, 109301 (2020)","journal-title":"J. Comput. Phys."},{"key":"3285_CR28","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1002\/nme.4382","volume":"93","author":"LNT Huynh","year":"2013","unstructured":"Huynh, L.N.T., Nguyen, N.C., Peraire, J., Khoo, B.C.: A high-order hybridizable discontinuous Galerkin method for elliptic interface problems. Int. J. Numer. Meth. Eng. 93, 183\u2013200 (2013)","journal-title":"Int. J. Numer. Meth. Eng."},{"key":"3285_CR29","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1016\/j.advwatres.2017.09.017","volume":"109","author":"J Jiang","year":"2017","unstructured":"Jiang, J., Younis, R.M.: An improved projection-based embedded discrete fracture model (pEDFM) for multiphase flow in fractured reservoirs. Adv. Water Resour. 109, 267\u2013289 (2017)","journal-title":"Adv. Water Resour."},{"key":"3285_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/j.advwatres.2020.103620","volume":"142","author":"T Kadeethum","year":"2020","unstructured":"Kadeethum, T., Nick, H.M., Lee, S., Ballarin, F.: Flow in porous media with low dimensional fractures by employing enriched Galerkin method. Adv. Water Resour. 142, 103620 (2020)","journal-title":"Adv. Water Resour."},{"key":"3285_CR31","doi-asserted-by":"publisher","first-page":"227","DOI":"10.2118\/88812-PA","volume":"9","author":"M Karimi-Fard","year":"2004","unstructured":"Karimi-Fard, M., Durlofsky, L.J., Aziz, K.: An efficient discrete-fracture model applicable for general-purpose reservoir simulators. SPE J. 9, 227\u2013236 (2004)","journal-title":"SPE J."},{"key":"3285_CR32","doi-asserted-by":"crossref","unstructured":"Karimi-Fard, M., Firoozabadi, A.: Numerical simulation of water injection in 2D fractured media using discrete-fracture model, in: SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers, January, (2001)","DOI":"10.2118\/71615-MS"},{"key":"3285_CR33","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/s10596-018-9779-8","volume":"23","author":"M K\u00f6ppel","year":"2019","unstructured":"K\u00f6ppel, M., Martin, V., Jaffr\u00e9, J., Roberts, J.E.: A Lagrange multiplier method for a discrete fracture model for flow in porous media. Comput. Geosci. 23, 239\u2013253 (2019)","journal-title":"Comput. Geosci."},{"key":"3285_CR34","doi-asserted-by":"publisher","first-page":"3280","DOI":"10.1137\/120898358","volume":"51","author":"J Kou","year":"2013","unstructured":"Kou, J., Sun, S.: Convergence of discontinuous Galerkin methods for incompressible two-phase flow in heterogeneous media. SIAM J. Numer. Anal. 51, 3280\u20133306 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"3285_CR35","doi-asserted-by":"publisher","first-page":"1120","DOI":"10.1002\/aic.690460604","volume":"46","author":"JG Kim","year":"2000","unstructured":"Kim, J.G., Deo, M.D.: Finite element, discrete-fracture model for multiphase flow in porous media. AIChE J. 46, 1120\u20131130 (2000)","journal-title":"AIChE J."},{"key":"3285_CR36","doi-asserted-by":"publisher","first-page":"750","DOI":"10.2118\/103901-PA","volume":"11","author":"L Li","year":"2008","unstructured":"Li, L., Lee, S.H.: Efficient field-scale simulation of black oil in a naturally fractured reservoir through discrete fracture networks and homogenized media. SPE Reservoir evaluation & engineering 11, 750\u2013758 (2008)","journal-title":"SPE Reservoir evaluation & engineering"},{"key":"3285_CR37","doi-asserted-by":"publisher","DOI":"10.1016\/j.advwatres.2021.104039","volume":"156","author":"T Ma","year":"2021","unstructured":"Ma, T., Zhang, K., Shen, W., Guo, C., Xu, H.: Discontinuous and continuous Galerkin methods for compressible single-phase and two-phase flow in fractured porous media. Adv. Water Resour. 156, 104039 (2021)","journal-title":"Adv. Water Resour."},{"key":"3285_CR38","doi-asserted-by":"publisher","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":"3285_CR39","doi-asserted-by":"publisher","first-page":"1871","DOI":"10.1090\/S0025-5718-10-02362-8","volume":"79","author":"JM Melenk","year":"2010","unstructured":"Melenk, J.M., Sauter, S.: Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions. Math. Comput. 79, 1871\u20131914 (2010)","journal-title":"Math. Comput."},{"key":"3285_CR40","doi-asserted-by":"crossref","unstructured":"Monteagudo, J.E.P., Firoozabadi, A.: Control-volume method for numerical simulation of two-phase immiscible flow in two-and three-dimensional discrete-fractured media. Water Resour. Res. 40, (2004)","DOI":"10.1029\/2003WR002996"},{"key":"3285_CR41","doi-asserted-by":"publisher","first-page":"698","DOI":"10.1002\/nme.1783","volume":"69","author":"JEP Monteagudo","year":"2007","unstructured":"Monteagudo, J.E.P., Firoozabadi, A.: Comparison of fully implicit and IMPES formulations for simulation of water injection in fractured and unfractured media. Int. J. Numer. Meth. Eng. 69, 698\u2013728 (2007)","journal-title":"Int. J. Numer. Meth. Eng."},{"key":"3285_CR42","doi-asserted-by":"publisher","first-page":"289","DOI":"10.2118\/154246-PA","volume":"19","author":"A Moinfar","year":"2014","unstructured":"Moinfar, A., Varavei, A., Sepehrnoori, K., Johns, R.T.: Development of an efficient embedded discrete fracture model for 3D compositional reservoir simulation in fractured reservoirs. SPE J. 19, 289\u2013303 (2014)","journal-title":"SPE J."},{"key":"3285_CR43","doi-asserted-by":"publisher","first-page":"1317","DOI":"10.1016\/j.matcom.2021.07.012","volume":"190","author":"I Mozolevski","year":"2021","unstructured":"Mozolevski, I., Murad, M.A., Schuh, L.A.: High order discontinuous Galerkin method for reduced flow models in fractured porous media. Math. Comput. Simul. 190, 1317\u20131341 (2021)","journal-title":"Math. Comput. Simul."},{"key":"3285_CR44","doi-asserted-by":"publisher","first-page":"1020","DOI":"10.1016\/j.advwatres.2005.09.001","volume":"29","author":"V Reichenberger","year":"2006","unstructured":"Reichenberger, V., Jakobs, H., Bastian, P., Helmig, R.: A mixed-dimensional finite volume method for two-phase flow in fractured porous media. Adv. Water Resour. 29, 1020\u20131036 (2006)","journal-title":"Adv. Water Resour."},{"key":"3285_CR45","doi-asserted-by":"crossref","unstructured":"Rivi\u00e8re, B.: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation, Society for Industrial and Applied Mathematics, (2008)","DOI":"10.1137\/1.9780898717440"},{"key":"3285_CR46","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1002\/cnm.464","volume":"18","author":"B Rivi\u00e8re","year":"2002","unstructured":"Rivi\u00e8re, B., Wheeler, M.F.: Discontinuous Galerkin methods for flow and transport problems in porous media. Commun. Numer. Methods Eng. 18, 63\u201368 (2002)","journal-title":"Commun. Numer. Methods Eng."},{"key":"3285_CR47","first-page":"337","volume":"4","author":"B Rivi\u00e8re","year":"2000","unstructured":"Rivi\u00e8re, B., Wheeler, M.F., Bana\u015b, K.: Part II. Discontinuous Galerkin method applied to a single phase flow in porous media, Computational Geosciences 4, 337\u2013349 (2000)","journal-title":"Discontinuous Galerkin method applied to a single phase flow in porous media, Computational Geosciences"},{"key":"3285_CR48","doi-asserted-by":"publisher","first-page":"3784","DOI":"10.1016\/j.jcp.2012.01.023","volume":"231","author":"TH Sandve","year":"2012","unstructured":"Sandve, T.H., Berre, I., Nordbotten, J.M.: An efficient multi-point flux approximation method for discrete fracture-matrix simulations. J. Comput. Phys. 231, 3784\u20133800 (2012)","journal-title":"J. Comput. Phys."},{"key":"3285_CR49","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.cageo.2019.06.014","volume":"132","author":"P Sch\u00e4dle","year":"2019","unstructured":"Sch\u00e4dle, P., Zulian, P., Vogler, D., Bhopalam, S.R., Nestola, M.G., Ebigbo, A., Krause, R., Saar, M.O.: 3D non-conforming mesh model for flow in fractured porous media using Lagrange multipliers. Computers & Geosciences 132, 42\u201355 (2019)","journal-title":"Computers & Geosciences"},{"key":"3285_CR50","volume-title":"p- and hp-Finite Element Methods","author":"Ch Schwab","year":"1998","unstructured":"Schwab, Ch.: p- and hp-Finite Element Methods. Oxford Science Publications, New York (1998)"},{"key":"3285_CR51","doi-asserted-by":"publisher","first-page":"1219","DOI":"10.1007\/s10596-015-9536-1","volume":"19","author":"N Schwenck","year":"2015","unstructured":"Schwenck, N., Flemisch, B., Helmig, R., Wohlmuth, B.I.: Dimensionally reduced flow models in fractured porous media: crossings and boundaries. Comput. Geosci. 19, 1219\u20131230 (2015)","journal-title":"Comput. Geosci."},{"key":"3285_CR52","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1016\/0021-9991(88)90177-5","volume":"77","author":"C-W Shu","year":"1988","unstructured":"Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439\u2013471 (1988)","journal-title":"J. Comput. Phys."},{"key":"3285_CR53","doi-asserted-by":"publisher","first-page":"3145","DOI":"10.1002\/num.23003","volume":"39","author":"Q Tao","year":"2023","unstructured":"Tao, Q., Liu, Y., Jiang, Y., Lu, J.: An oscillation free local discontinuous Galerkin method for nonlinear degenerate parabolic equations. Numerical Methods for Partial Differential Equations 39, 3145\u20133169 (2023)","journal-title":"Numerical Methods for Partial Differential Equations"},{"key":"3285_CR54","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/j.advwatres.2017.05.009","volume":"105","author":"M \u0162ene","year":"2017","unstructured":"\u0162ene, M., Bosma, S.B., Al Kobaisi, M.S., Hajibeygi, H.: Projection-based embedded discrete fracture model (pEDFM). Adv. Water Resour. 105, 205\u2013216 (2017)","journal-title":"Adv. Water Resour."},{"key":"3285_CR55","doi-asserted-by":"publisher","first-page":"1274","DOI":"10.1007\/s10915-018-0673-2","volume":"76","author":"G Wang","year":"2018","unstructured":"Wang, G., He, Y., Yang, J.: Weak Galerkin finite element methods for the simulation of single-phase flow in fractured porous media. J. Sci. Comput. 76, 1274\u20131300 (2018)","journal-title":"J. Sci. Comput."},{"key":"3285_CR56","doi-asserted-by":"publisher","DOI":"10.1016\/j.advwatres.2024.104869","volume":"195","author":"Z Xu","year":"2025","unstructured":"Xu, Z., Gl\u00e4ser, D.: An extension of the box method discrete fracture model (Box-DFM) to include low-permeable barriers with minimal additional degrees of freedom. Adv. Water Resour. 195, 104869 (2025)","journal-title":"Adv. Water Resour."},{"key":"3285_CR57","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111749","volume":"473","author":"Z Xu","year":"2023","unstructured":"Xu, Z., Huang, Z., Yang, Y.: The hybrid-dimensional Darcy\u2019s law: a non-conforming reinterpreted discrete fracture model (RDFM) for single-phase flow in fractured media. J. Comput. Phys. 473, 111749 (2023)","journal-title":"J. Comput. Phys."},{"key":"3285_CR58","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109523","volume":"415","author":"Z Xu","year":"2020","unstructured":"Xu, Z., Yang, Y.: The hybrid dimensional representation of permeability tensor: a reinterpretation of the discrete fracture model and its extension on nonconforming meshes. J. Comput. Phys. 415, 109523 (2020)","journal-title":"J. Comput. Phys."},{"key":"3285_CR59","doi-asserted-by":"publisher","first-page":"3091","DOI":"10.1016\/j.jcp.2009.12.030","volume":"229","author":"X Zhang","year":"2010","unstructured":"Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091\u20133120 (2010)","journal-title":"J. Comput. Phys."},{"key":"3285_CR60","doi-asserted-by":"publisher","first-page":"420","DOI":"10.1016\/j.jcp.2012.12.006","volume":"242","author":"N Zhang","year":"2013","unstructured":"Zhang, N., Yao, J., Huang, Z., Wang, Y.: Accurate multiscale finite element method for numerical simulation of two-phase flow in fractured media using discrete-fracture model. J. Comput. Phys. 242, 420\u2013438 (2013)","journal-title":"J. Comput. Phys."},{"key":"3285_CR61","doi-asserted-by":"crossref","unstructured":"Zhao, J., Rui, H.: A discrete fracture-matrix approach based on Petrov-Galerkin immersed finite element for fractured porous media flow on nonconforming mesh. J. Comput. Phys. 112718 (2023)","DOI":"10.1016\/j.jcp.2023.112718"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03285-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03285-w","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03285-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T02:03:07Z","timestamp":1782093787000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-026-03285-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,27]]},"references-count":61,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["3285"],"URL":"https:\/\/doi.org\/10.1007\/s10915-026-03285-w","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,27]]},"assertion":[{"value":"14 January 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 March 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 March 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 April 2026","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 declare that we have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}}],"article-number":"83"}}