{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T13:48:19Z","timestamp":1781272099000,"version":"3.54.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,9,13]],"date-time":"2022-09-13T00:00:00Z","timestamp":1663027200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,9,13]],"date-time":"2022-09-13T00:00:00Z","timestamp":1663027200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Umea University"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2022,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected by the boundary occur and these elements must in general by stabilized in some way. Discrete extension operators provides such a stabilization by modification of the finite element space close to the boundary. More, precisely the finite element space is extended from the stable interior elements over the boundary in a stable way which also guarantees optimal approximation properties. Our framework is applicable to all standard nodal based finite elements of various order and regularity. We develop an abstract theory for elliptic problems and associated parabolic time dependent partial differential equations and derive a priori error estimates. We finally apply this to some examples of partial differential equations of different order including the interface problems, the biharmonic operator and the sixth order triharmonic operator.<\/jats:p>","DOI":"10.1007\/s00211-022-01313-z","type":"journal-article","created":{"date-parts":[[2022,9,13]],"date-time":"2022-09-13T16:02:47Z","timestamp":1663084967000},"page":"331-369","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":31,"title":["CutFEM based on extended finite element spaces"],"prefix":"10.1007","volume":"152","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":[[2022,9,13]]},"reference":[{"issue":"6","key":"1313_CR1","doi-asserted-by":"publisher","first-page":"B1541","DOI":"10.1137\/18M1185624","volume":"40","author":"S Badia","year":"2018","unstructured":"Badia, S., Martin, A.F., Verdugo, F.: Mixed aggregated finite element methods for the unfitted discretization of the Stokes problem. SIAM J. Sci. Comput. 40(6), B1541\u2013B1576 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"1313_CR2","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1016\/j.cma.2018.03.022","volume":"336","author":"S Badia","year":"2018","unstructured":"Badia, S., Verdugo, F., Mart\u00edn, A.F.: The aggregated unfitted finite element method for elliptic problems. Comput. Methods Appl. Mech. Eng. 336, 533\u2013553 (2018)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"41\u201344","key":"1313_CR3","doi-asserted-by":"publisher","first-page":"3352","DOI":"10.1016\/j.cma.2009.06.017","volume":"198","author":"R Becker","year":"2009","unstructured":"Becker, R., Burman, E., Hansbo, P.: A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Eng. 198(41\u201344), 3352\u20133360 (2009)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1313_CR4","volume-title":"The Mathematical Theory of Finite Element Methods, Volume\u00a015 of Texts in Applied Mathematics.","author":"SC Brenner","year":"2002","unstructured":"Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Volume\u00a015 of Texts in Applied Mathematics., 2nd edn. Springer, New York (2002)","edition":"2"},{"issue":"21\u201322","key":"1313_CR5","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1016\/j.crma.2010.10.006","volume":"348","author":"E Burman","year":"2010","unstructured":"Burman, E.: Ghost penalty. C. R. Math. Acad. Sci. Paris 348(21\u201322), 1217\u20131220 (2010)","journal-title":"C. R. Math. Acad. Sci. Paris"},{"issue":"7","key":"1313_CR6","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":"3","key":"1313_CR7","doi-asserted-by":"publisher","first-page":"1525","DOI":"10.1137\/17M1154266","volume":"56","author":"E Burman","year":"2018","unstructured":"Burman, E., Ern, A.: An unfitted hybrid high-order method for elliptic interface problems. SIAM J. Numer. Anal. 56(3), 1525\u20131546 (2018)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"1313_CR8","doi-asserted-by":"publisher","first-page":"328","DOI":"10.1016\/j.apnum.2011.01.008","volume":"62","author":"E Burman","year":"2012","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)","journal-title":"Appl. Numer. Math."},{"issue":"3","key":"1313_CR9","doi-asserted-by":"publisher","first-page":"859","DOI":"10.1051\/m2an\/2013123","volume":"48","author":"E Burman","year":"2014","unstructured":"Burman, E., Hansbo, P.: Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes\u2019 problem. ESAIM Math. Model. Numer. Anal. 48(3), 859\u2013874 (2014)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"issue":"310","key":"1313_CR10","doi-asserted-by":"publisher","first-page":"633","DOI":"10.1090\/mcom\/3240","volume":"87","author":"E Burman","year":"2018","unstructured":"Burman, E., Hansbo, P., Larson, M.G.: A cut finite element method with boundary value correction. Math. Comput. 87(310), 633\u2013657 (2018)","journal-title":"Math. Comput."},{"key":"1313_CR11","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1186\/s40323-020-00164-3","volume":"7","author":"E Burman","year":"2020","unstructured":"Burman, E., Hansbo, P., Larson, M.G.: Cut Bogner\u2013Fox\u2013Schmit elements for plates. Adv. Model. Simul. Eng. Sci. 7, 27 (2020)","journal-title":"Adv. Model. Simul. Eng. Sci."},{"issue":"3","key":"1313_CR12","doi-asserted-by":"publisher","first-page":"A1254","DOI":"10.1137\/20M137937X","volume":"44","author":"E Burman","year":"2022","unstructured":"Burman, E., Hansbo, P., Larson, M.G.: Explicit time stepping for the wave equation using CutFEM with discrete extension. SIAM J. Sci. Comput. 44(3), A1254\u2013A1289 (2022)","journal-title":"SIAM J. Sci. Comput."},{"key":"1313_CR13","doi-asserted-by":"publisher","first-page":"943","DOI":"10.1016\/j.cma.2014.10.055","volume":"284","author":"H Casquero","year":"2015","unstructured":"Casquero, H., Bona-Casas, C., Gomez, H.: A NURBS-based immersed methodology for fluid-structure interaction. Comput. Methods Appl. Mech. Eng. 284, 943\u2013970 (2015)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1313_CR14","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1016\/j.jcp.2018.07.020","volume":"374","author":"H Casquero","year":"2018","unstructured":"Casquero, H., Zhang, Y.J., Bona-Casas, C., Dalcin, L., Gomez, H.: Non-body-fitted fluid-structure interaction: divergence-conforming B-splines, fully-implicit dynamics, and variational formulation. J. Comput. Phys. 374, 625\u2013653 (2018)","journal-title":"J. Comput. Phys."},{"issue":"R\u20132","key":"1313_CR15","first-page":"77","volume":"9","author":"P Cl\u00e9ment","year":"1975","unstructured":"Cl\u00e9ment, P.: Approximation by finite element functions using local regularization. Rev. Fran\u00e7aise Automat. Informat. Rech. Op\u00e9ra. S\u00e9r. 9(R\u20132), 77\u201384 (1975)","journal-title":"Rev. Fran\u00e7aise Automat. Informat. Rech. Op\u00e9ra. S\u00e9r."},{"issue":"3","key":"1313_CR16","doi-asserted-by":"publisher","first-page":"1039","DOI":"10.1007\/s10915-015-9997-3","volume":"65","author":"M Dauge","year":"2015","unstructured":"Dauge, M., D\u00fcster, A., Rank, E.: Theoretical and numerical investigation of the finite cell method. J. Sci. Comput. 65(3), 1039\u20131064 (2015)","journal-title":"J. Sci. Comput."},{"key":"1313_CR17","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1016\/j.cma.2016.07.006","volume":"316","author":"F de Prenter","year":"2017","unstructured":"de Prenter, F., Verhoosel, C.V., van Zwieten, G.J., van Brummelen, E.H.: Condition number analysis and preconditioning of the finite cell method. Comput. Methods Appl. Mech. Eng. 316, 297\u2013327 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1313_CR18","doi-asserted-by":"crossref","unstructured":"Eisentr\u00e4ger, S., Kiendl, J., Michaloudis, G., Duy, R., Vetyukov, Y.: Stability analysis of plates using cut Bogner-Fox-Schmit elements. Comput. Struct. 270, 106854 (2022)","DOI":"10.1016\/j.compstruc.2022.106854"},{"issue":"7","key":"1313_CR19","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1002\/nme.2863","volume":"83","author":"A Embar","year":"2010","unstructured":"Embar, A., Dolbow, J., Harari, I.: Imposing Dirichlet boundary conditions with Nitsche\u2019s method and spline-based finite elements. Int. J. Numer. Methods Eng. 83(7), 877\u2013898 (2010)","journal-title":"Int. J. Numer. Methods Eng."},{"issue":"1","key":"1313_CR20","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1051\/m2an:2006006","volume":"40","author":"A Ern","year":"2006","unstructured":"Ern, A., Guermond, J.-L.: Evaluation of the condition number in linear systems arising in finite element approximations. M2AN Math. Model. Numer. Anal. 40(1), 29\u201348 (2006)","journal-title":"M2AN Math. Model. Numer. Anal."},{"key":"1313_CR21","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 (2011). (Reprint of the 1985 original [MR0775683]. With a foreword by Susanne C, Brenner)","DOI":"10.1137\/1.9781611972030"},{"issue":"47\u201348","key":"1313_CR22","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":"1313_CR23","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."},{"issue":"3","key":"1313_CR24","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1051\/m2an:2003039","volume":"37","author":"A Hansbo","year":"2003","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)","journal-title":"ESAIM: Math. Model. Numer. Anal."},{"key":"1313_CR25","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1016\/j.apnum.2014.06.009","volume":"85","author":"P Hansbo","year":"2014","unstructured":"Hansbo, P., Larson, M.G., Zahedi, S.: A cut finite element method for a Stokes interface problem. Appl. Numer. Math. 85, 90\u2013114 (2014)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"1313_CR26","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1002\/nme.4337","volume":"92","author":"I Harari","year":"2012","unstructured":"Harari, I., Shavelzon, E.: Embedded kinematic boundary conditions for thin plate bending by Nitsche\u2019s approach. Int. J. Numer. Methods Eng. 92(1), 99\u2013114 (2012)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"1313_CR27","doi-asserted-by":"publisher","first-page":"324","DOI":"10.1016\/j.cma.2018.03.034","volume":"337","author":"T Hoang","year":"2018","unstructured":"Hoang, T., Verhoosel, C.V., Auricchio, F., van Brummelen, E.H., Reali, A.: Skeleton-stabilized isogeometric analysis: high-regularity interior-penalty methods for incompressible viscous flow problems. Comput. Methods Appl. Mech. Eng. 337, 324\u2013351 (2018)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1313_CR28","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1016\/j.cma.2017.06.004","volume":"323","author":"P Huang","year":"2017","unstructured":"Huang, P., Wu, H., Xiao, Y.: An unfitted interface penalty finite element method for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 323, 439\u2013460 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"4","key":"1313_CR29","doi-asserted-by":"publisher","first-page":"607","DOI":"10.1007\/s00211-012-0497-1","volume":"123","author":"A Johansson","year":"2013","unstructured":"Johansson, A., Larson, M.G.: A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary. Numer. Math. 123(4), 607\u2013628 (2013)","journal-title":"Numer. Math."},{"key":"1313_CR30","unstructured":"Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications, vol. I. Springer, New York (1972). (Translated from the French by P, p. 181. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band)"},{"issue":"4","key":"1313_CR31","doi-asserted-by":"publisher","first-page":"1059","DOI":"10.1007\/s11831-017-9220-9","volume":"25","author":"B Marussig","year":"2018","unstructured":"Marussig, B., Hughes, T.J.R.: A review of trimming in isogeometric analysis: challenges, data exchange and simulation aspects. Arch. Comput. Methods Eng. 25(4), 1059\u20131127 (2018)","journal-title":"Arch. Comput. Methods Eng."},{"key":"1313_CR32","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1016\/j.cma.2016.07.040","volume":"316","author":"B Marussig","year":"2017","unstructured":"Marussig, B., Zechner, J., Beer, G., Fries, T.-P.: Stable isogeometric analysis of trimmed geometries. Comput. Methods Appl. Mech. Eng. 316, 497\u2013521 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1","key":"1313_CR33","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1007\/s00211-013-0603-z","volume":"128","author":"A Massing","year":"2014","unstructured":"Massing, A., Larson, M.G., Logg, A., Rognes, M.E.: A stabilized Nitsche overlapping mesh method for the Stokes problem. Numer. Math. 128(1), 73\u2013101 (2014)","journal-title":"Numer. Math."},{"issue":"1","key":"1313_CR34","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1007\/s00466-007-0173-y","volume":"41","author":"J Parvizian","year":"2007","unstructured":"Parvizian, J., D\u00fcster, A., Rank, E.: Finite cell method: $$h$$- and $$p$$-extension for embedded domain problems in solid mechanics. Comput. Mech. 41(1), 121\u2013133 (2007)","journal-title":"Comput. Mech."},{"issue":"190","key":"1313_CR35","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. Comput. 54(190), 483\u2013493 (1990)","journal-title":"Math. Comput."},{"key":"1313_CR36","series-title":"Princeton Mathematical Series","volume-title":"Singular Integrals and Differentiability Properties of Functions","author":"EM Stein","year":"1970","unstructured":"Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)"},{"key":"1313_CR37","volume-title":"Galerkin Finite Element Methods for Parabolic Problems, Volume\u00a025 of Springer Series in Computational Mathematics","author":"V Thom\u00e9e","year":"2006","unstructured":"Thom\u00e9e, V.: Galerkin Finite Element Methods for Parabolic Problems, Volume\u00a025 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2006)","edition":"2"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-022-01313-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-022-01313-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-022-01313-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,26]],"date-time":"2022-09-26T03:23:46Z","timestamp":1664162626000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-022-01313-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,13]]},"references-count":37,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["1313"],"URL":"https:\/\/doi.org\/10.1007\/s00211-022-01313-z","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,13]]},"assertion":[{"value":"7 May 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 July 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 July 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 September 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}