{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T04:00:02Z","timestamp":1781668802124,"version":"3.54.5"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2020,10,31]],"date-time":"2020-10-31T00:00:00Z","timestamp":1604102400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,10,31]],"date-time":"2020-10-31T00:00:00Z","timestamp":1604102400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"J\u00f6nk\u00f6ping University"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2020,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We develop a robust cut finite element method for a model of diffusion in fractured media consisting of a bulk domain with embedded cracks. The crack has its own pressure field and can cut through the bulk mesh in a very general fashion. Starting from a common background bulk mesh, that covers the domain, finite element spaces are constructed for the interface and bulk subdomains leading to efficient computations of the coupling terms. The crack pressure field also uses the bulk mesh for its representation. The interface conditions are a generalized form of conditions of Robin type previously considered in the literature which allows the modeling of a range of flow regimes across the fracture. The method is robust in the following way: (1) Stability of the formulation in the full range of parameter choices; and (2) Not sensitive to the location of the interface in the background mesh. We derive an optimal order a priori error estimate and present illustrating numerical examples.<\/jats:p>","DOI":"10.1007\/s00211-020-01157-5","type":"journal-article","created":{"date-parts":[[2020,10,31]],"date-time":"2020-10-31T00:07:02Z","timestamp":1604102822000},"page":"783-818","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["A cut finite element method for a model of pressure in fractured media"],"prefix":"10.1007","volume":"146","author":[{"given":"Erik","family":"Burman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter","family":"Hansbo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mats G.","family":"Larson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,10,31]]},"reference":[{"key":"1157_CR1","unstructured":"Angot, P., Boyer, F., Hubert, F.: Asymptotic and numerical modelling of flows in fractured porous media. M2AN Math. Model. Numer. Anal. 43(2), 239\u2013275 (2009)"},{"issue":"1","key":"1157_CR2","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(1), A109\u2013A138 (2019)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"1157_CR3","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(4), 2203\u20132233 (2018)","journal-title":"SIAM J. Numer. Anal."},{"issue":"7","key":"1157_CR4","doi-asserted-by":"publisher","first-page":"472","DOI":"10.1002\/nme.4823","volume":"104","author":"E Burman","year":"2015","unstructured":"Burman, E., Claus, S., Hansbo, P., Larson, M.G., Massing, A.: CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Methods Eng. 104(7), 472\u2013501 (2015)","journal-title":"Int. J. Numer. Methods Eng."},{"issue":"5","key":"1157_CR5","doi-asserted-by":"publisher","first-page":"A3354","DOI":"10.1137\/18M1223836","volume":"41","author":"E Burman","year":"2019","unstructured":"Burman, E., Elfverson, D., Hansbo, P., Larson, M.G., Larsson, K.: Hybridized CutFEM for elliptic interface problems. SIAM J. Sci. Comput. 41(5), A3354\u2013A3380 (2019)","journal-title":"SIAM J. Sci. Comput."},{"key":"1157_CR6","doi-asserted-by":"crossref","unstructured":"Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62(4), 328\u2013341 (2012)","DOI":"10.1016\/j.apnum.2011.01.008"},{"key":"1157_CR7","doi-asserted-by":"publisher","first-page":"188","DOI":"10.1016\/j.cma.2014.10.044","volume":"285","author":"E Burman","year":"2015","unstructured":"Burman, E., Hansbo, P., Larson, M.G.: A stabilized cut finite element method for partial differential equations on surfaces: the Laplace\u2013Beltrami operator. Comput. Methods Appl. Mech. Eng. 285, 188\u2013207 (2015)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1","key":"1157_CR8","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/s10596-018-9792-y","volume":"23","author":"E Burman","year":"2019","unstructured":"Burman, E., Hansbo, P., Larson, M.G.: A simple finite element method for elliptic bulk problems with embedded surfaces. Comput. Geosci. 23(1), 189\u2013199 (2019)","journal-title":"Comput. Geosci."},{"key":"1157_CR9","doi-asserted-by":"publisher","first-page":"726","DOI":"10.1016\/j.compfluid.2018.07.022","volume":"179","author":"E Burman","year":"2019","unstructured":"Burman, E., Hansbo, P., Larson, M.G., Larsson, K.: Cut finite elements for convection in fractured domains. Comput. Fluids 179, 726\u2013734 (2019)","journal-title":"Comput. Fluids"},{"issue":"1","key":"1157_CR10","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1007\/s00211-018-0990-2","volume":"141","author":"E Burman","year":"2019","unstructured":"Burman, E., Hansbo, P., Larson, M.G., Larsson, K., Massing, A.: Finite element approximation of the Laplace\u2013Beltrami operator on a surface with boundary. Numer. Math. 141(1), 141\u2013172 (2019)","journal-title":"Numer. Math."},{"key":"1157_CR11","doi-asserted-by":"crossref","unstructured":"Burman, E., Hansbo, P., Larson, M.G., Samvin, D.: A cut finite element method for elliptic bulk problems with embedded surfaces. GEM Int. J. Geomath. 10(1), Art. 10, 22 (2019)","DOI":"10.1007\/s13137-019-0120-z"},{"issue":"2","key":"1157_CR12","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1007\/s00211-015-0744-3","volume":"133","author":"E Burman","year":"2016","unstructured":"Burman, E., Hansbo, P., Larson, M.G., Zahedi, S.: Cut finite element methods for coupled bulk-surface problems. Numer. Math. 133(2), 203\u2013231 (2016)","journal-title":"Numer. Math."},{"issue":"10","key":"1157_CR13","doi-asserted-by":"publisher","first-page":"2224","DOI":"10.1080\/00036811.2015.1075007","volume":"95","author":"D Capatina","year":"2016","unstructured":"Capatina, D., Luce, R., El-Otmany, H., Barrau, N.: Nitsche\u2019s extended finite element method for a fracture model in porous media. Appl. Anal. 95(10), 2224\u20132242 (2016)","journal-title":"Appl. Anal."},{"issue":"2","key":"1157_CR14","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(2), A1063\u2013A1094 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"1157_CR15","doi-asserted-by":"crossref","unstructured":"Chernyshenko, A.Y., Olshanskii, M.A.: An unfitted finite element method for the Darcy problem in a fracture network. J. Comput. Appl. Math. 366, 112424, 14 (2020)","DOI":"10.1016\/j.cam.2019.112424"},{"key":"1157_CR16","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1002\/1522-2616(200202)235:1<29::AID-MANA29>3.0.CO;2-6","volume":"235","author":"M Costabel","year":"2002","unstructured":"Costabel, M., Dauge, M.: Crack singularities for general elliptic systems. Math. Nachr. 235, 29\u201349 (2002)","journal-title":"Math. Nachr."},{"issue":"5\u20136","key":"1157_CR17","doi-asserted-by":"publisher","first-page":"869","DOI":"10.1081\/PDE-120021180","volume":"28","author":"M Costabel","year":"2003","unstructured":"Costabel, M., Dauge, M., Duduchava, R.: Asymptotics without logarithmic terms for crack problems. Comm. Partial Differ. Equ. 28(5\u20136), 869\u2013926 (2003)","journal-title":"Comm. Partial Differ. Equ."},{"issue":"1","key":"1157_CR18","doi-asserted-by":"publisher","first-page":"B228","DOI":"10.1137\/16M1098231","volume":"40","author":"A Fumagalli","year":"2018","unstructured":"Fumagalli, A., Keilegavlen, E.: Dual virtual element method for discrete fractures networks. SIAM J. Sci. Comput. 40(1), B228\u2013B258 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"1157_CR19","doi-asserted-by":"crossref","unstructured":"Grisvard, P.: Elliptic problems in nonsmooth domains, volume 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA: Reprint of the 1985 original [MR0775683]. With a foreword by Susanne C, Brenner (2011)","DOI":"10.1137\/1.9781611972030"},{"issue":"5","key":"1157_CR20","doi-asserted-by":"publisher","first-page":"1303","DOI":"10.1051\/m2an\/2015013","volume":"49","author":"S Gross","year":"2015","unstructured":"Gross, S., Olshanskii, M.A., Reusken, A.: A trace finite element method for a class of coupled bulk-interface transport problems. ESAIM Math. Model. Numer. Anal. 49(5), 1303\u20131330 (2015)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"issue":"47\u201348","key":"1157_CR21","doi-asserted-by":"publisher","first-page":"5537","DOI":"10.1016\/S0045-7825(02)00524-8","volume":"191","author":"A Hansbo","year":"2002","unstructured":"Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche\u2019s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191(47\u201348), 5537\u20135552 (2002)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"33\u201335","key":"1157_CR22","doi-asserted-by":"publisher","first-page":"3523","DOI":"10.1016\/j.cma.2003.12.041","volume":"193","author":"A Hansbo","year":"2004","unstructured":"Hansbo, A., Hansbo, P.: A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Eng. 193(33\u201335), 3523\u20133540 (2004)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1157_CR23","unstructured":"Hansbo, A., Hansbo, P., Larson, M.G.: A finite element method on composite grids based on Nitsche\u2019s method. ESAIM: Math. Model. Numer. Anal. 37(3), 495\u2013514 (2003)"},{"key":"1157_CR24","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1016\/j.cma.2017.08.033","volume":"326","author":"P Hansbo","year":"2017","unstructured":"Hansbo, P., Jonsson, T., Larson, M.G., Larsson, K.: A Nitsche method for elliptic problems on composite surfaces. Comput. Methods Appl. Mech. Eng. 326, 505\u2013525 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1157_CR25","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1016\/0022-247X(74)90023-7","volume":"47","author":"PH Hung","year":"1974","unstructured":"Hung, P.H., S\u00e1nchez-Palencia, E.: Ph\u00e9nom\u00e8nes de transmission \u00e0 travers des couches minces de conductivit\u00e9 \u00e9lev\u00e9e. J. Math. Anal. Appl. 47, 284\u2013309 (1974)","journal-title":"J. Math. Anal. Appl."},{"issue":"267","key":"1157_CR26","doi-asserted-by":"publisher","first-page":"1353","DOI":"10.1090\/S0025-5718-08-02183-2","volume":"78","author":"M Juntunen","year":"2009","unstructured":"Juntunen, M., Stenberg, R.: Nitsche\u2019s method for general boundary conditions. Math. Comp. 78(267), 1353\u20131374 (2009)","journal-title":"Math. Comp."},{"issue":"1","key":"1157_CR27","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1007\/s13137-019-0117-7","volume":"10","author":"M K\u00f6ppel","year":"2019","unstructured":"K\u00f6ppel, M., Martin, V., Roberts, J.E.: A stabilized Lagrange multiplier finite-element method for flow in porous media with fractures. GEM Int. J. Geomath. 10(1), 7 (2019)","journal-title":"GEM Int. J. Geomath."},{"issue":"3","key":"1157_CR28","doi-asserted-by":"publisher","first-page":"1702","DOI":"10.1093\/imanum\/drz021","volume":"40","author":"MG Larson","year":"2020","unstructured":"Larson, M.G., Zahedi, S.: Stabilization of high order cut finite element methods on surfaces. IMA J. Numer. Anal. 40(3), 1702\u20131745 (2020)","journal-title":"IMA J. Numer. Anal."},{"issue":"5","key":"1157_CR29","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(5), 1667\u20131691 (2005)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1157_CR30","doi-asserted-by":"publisher","first-page":"320","DOI":"10.1016\/j.jmaa.2009.10.042","volume":"365","author":"F Morales","year":"2010","unstructured":"Morales, F., Showalter, R.E.: The narrow fracture approximation by channeled flow. J. Math. Anal. Appl. 365(1), 320\u2013331 (2010)","journal-title":"J. Math. Anal. Appl."},{"key":"1157_CR31","unstructured":"Ods\u00e6\u00a0ter, L.\u00a0H., Kvamsdal, T., Larson, M.G.: A simple embedded discrete fracture-matrix model for a coupled flow and transport problem in porous media. Comput. Methods Appl. Mech. Eng. 343, 572\u2013601 (2019)"},{"key":"1157_CR32","doi-asserted-by":"crossref","unstructured":"Olshanskii, M.A., Reusken, A.: Trace finite element methods for PDEs on surfaces. In: Geometrically Unfitted Finite Element Methods and Applications, volume 121 of Lecture Notes Computer Science Engineering, pp. 211\u2013258. Springer, Cham (2017)","DOI":"10.1007\/978-3-319-71431-8_7"},{"issue":"190","key":"1157_CR33","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1090\/S0025-5718-1990-1011446-7","volume":"54","author":"LR Scott","year":"1990","unstructured":"Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), 483\u2013493 (1990)","journal-title":"Math. Comp."},{"key":"1157_CR34","unstructured":"Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton (1970)"},{"key":"1157_CR35","doi-asserted-by":"crossref","unstructured":"Taylor, M.E.: Partial Differential Equations I. Basic Theory, volume 115 of Applied Mathematical Sciences, 2nd edn. Springer, New York (2011)","DOI":"10.1007\/978-1-4419-7055-8"},{"issue":"3","key":"1157_CR36","doi-asserted-by":"publisher","first-page":"316","DOI":"10.4208\/jcm.1805-m2017-0151","volume":"37","author":"H Wu","year":"2019","unstructured":"Wu, H., Xiao, Y.: An unfitted $$hp$$-interface penalty finite element method for elliptic interface problems. J. Comput. Math. 37(3), 316\u2013339 (2019)","journal-title":"J. Comput. Math."},{"key":"1157_CR37","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1016\/j.cma.2016.02.032","volume":"304","author":"EL Yedeg","year":"2016","unstructured":"Yedeg, E.L., Wadbro, E., Hansbo, P., Larson, M.G., Berggren, M.: A Nitsche-type method for Helmholtz equation with an embedded acoustically permeable interface. Comput. Methods Appl. Mech. Eng. 304, 479\u2013500 (2016)","journal-title":"Comput. Methods Appl. Mech. Eng."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-020-01157-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-020-01157-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-020-01157-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,11,24]],"date-time":"2020-11-24T15:40:23Z","timestamp":1606232423000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-020-01157-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,31]]},"references-count":37,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,12]]}},"alternative-id":["1157"],"URL":"https:\/\/doi.org\/10.1007\/s00211-020-01157-5","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,31]]},"assertion":[{"value":"2 June 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 October 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 October 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 October 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}