{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T20:34:01Z","timestamp":1778704441332,"version":"3.51.4"},"reference-count":37,"publisher":"American Mathematical Society (AMS)","issue":"354","license":[{"start":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T00:00:00Z","timestamp":1761782400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper we present a new\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H left-parenthesis d i v right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>div<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H(\\operatorname {div})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -conforming unfitted finite element method for the mixed Poisson problem which is robust in the cut configuration and preserves conservation properties of body-fitted finite element methods. The key is to formulate the divergence-constraint on the active mesh, instead of the physical domain, in order to obtain robustness with respect to cut configurations without the need for a stabilization that pollutes the mass balance. This change in the formulation results in a slight inconsistency, but does not affect the accuracy of the flux variable. By applying post-processings for the scalar variable, in virtue of classical local post-processings in body-fitted methods, we retain optimal convergence rates for both variables and even the superconvergence after post-processing of the scalar variable. We present the method and perform a rigorous a priori error analysis of the method and discuss several variants and extensions. Numerical experiments confirm the theoretical results.\n                  <\/p>","DOI":"10.1090\/mcom\/4027","type":"journal-article","created":{"date-parts":[[2024,10,30]],"date-time":"2024-10-30T12:53:06Z","timestamp":1730292786000},"page":"1667-1699","source":"Crossref","is-referenced-by-count":2,"title":["Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem"],"prefix":"10.1090","volume":"94","author":[{"given":"Christoph","family":"Lehrenfeld","sequence":"first","affiliation":[]},{"given":"Tim","family":"van Beeck","sequence":"additional","affiliation":[]},{"given":"Igor","family":"Voulis","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,10,30]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"Paper No. 114232, 23","DOI":"10.1016\/j.cma.2021.114232","article-title":"Linking ghost penalty and aggregated unfitted methods","volume":"388","author":"Badia, Santiago","year":"2022","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1016\/j.cma.2018.03.022","article-title":"The aggregated unfitted finite element method for elliptic problems","volume":"336","author":"Badia, Santiago","year":"2018","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"T. Belytschko, N. Moes, S. Usui, and C. Parimi, Arbitrary discontinuities in finite elements, Internat. J. Numer. Methods Engrg. 50 (2001), 993\u20131013.","DOI":"10.1002\/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M"},{"key":"4","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-36519-5","volume-title":"Mixed finite element methods and applications","volume":"44","author":"Boffi, Daniele","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642365188"},{"issue":"21-22","key":"5","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1016\/j.crma.2010.10.006","article-title":"Ghost penalty","volume":"348","author":"Burman, Erik","year":"2010","journal-title":"C. R. Math. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/1631-073X","issn-type":"print"},{"issue":"7","key":"6","doi-asserted-by":"publisher","first-page":"472","DOI":"10.1002\/nme.4823","article-title":"CutFEM: discretizing geometry and partial differential equations","volume":"104","author":"Burman, Erik","year":"2015","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1002\/num.21829","article-title":"Projection stabilization of Lagrange multipliers for the imposition of constraints on interfaces and boundaries","volume":"30","author":"Burman, Erik","year":"2014","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"893","DOI":"10.1137\/22M1542933","article-title":"Cut finite element method for divergence-free approximation of incompressible flow: a Lagrange multiplier approach","volume":"62","author":"Burman, Erik","year":"2024","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"9","doi-asserted-by":"publisher","first-page":"2247","DOI":"10.1051\/m2an\/2018038","article-title":"Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions","volume":"52","author":"Burman, Erik","year":"2018","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/2822-7840","issn-type":"print"},{"issue":"19","key":"10","doi-asserted-by":"publisher","first-page":"4586","DOI":"10.1002\/nme.7047","article-title":"An extended finite element method for coupled Darcy-Stokes problems","volume":"123","author":"Cao, Pei","year":"2022","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"key":"11","doi-asserted-by":"crossref","unstructured":"C. D\u2019Angelo and A. Scotti, A mixed finite element method for Darcy flow in fractured porous media with non-matching grids, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 2, 465\u2013489.","DOI":"10.1051\/m2an\/2011148"},{"issue":"3","key":"12","doi-asserted-by":"publisher","first-page":"A1793--A1820","DOI":"10.1137\/22M149702X","article-title":"A divergence preserving cut finite element method for Darcy flow","volume":"46","author":"Frachon, Thomas","year":"2024","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1002\/nme.2914","article-title":"The extended\/generalized finite element method: an overview of the method and its applications","volume":"84","author":"Fries, Thomas-Peter","year":"2010","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"key":"14","series-title":"Springer Monographs in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-09620-9","volume-title":"An introduction to the mathematical theory of the Navier-Stokes equations","author":"Galdi, G. P.","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387096193","edition":"2"},{"issue":"1","key":"15","doi-asserted-by":"publisher","first-page":"228","DOI":"10.1137\/16M1102203","article-title":"Analysis of a high-order trace finite element method for PDEs on level set surfaces","volume":"56","author":"Grande, J\u00f6rg","year":"2018","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"16","doi-asserted-by":"publisher","first-page":"466","DOI":"10.1016\/j.cma.2018.12.041","article-title":"A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems","volume":"348","author":"G\u00fcrkan, Ceren","year":"2019","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"313","key":"17","doi-asserted-by":"publisher","first-page":"2091","DOI":"10.1090\/mcom\/3288","article-title":"Inf-sup stability of geometrically unfitted Stokes finite elements","volume":"87","author":"Guzm\u00e1n, Johnny","year":"2018","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"47-48","key":"18","doi-asserted-by":"publisher","first-page":"5537","DOI":"10.1016\/S0045-7825(02)00524-8","article-title":"An unfitted finite element method, based on Nitsche\u2019s method, for elliptic interface problems","volume":"191","author":"Hansbo, Anita","year":"2002","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"2","key":"19","doi-asserted-by":"publisher","first-page":"Paper No. 66, 33","DOI":"10.1007\/s10915-022-01839-2","article-title":"An immersed Raviart-Thomas mixed finite element method for elliptic interface problems on unfitted meshes","volume":"91","author":"Ji, Haifeng","year":"2022","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"A1019--A1043","DOI":"10.1137\/15M1011779","article-title":"Analysis of an XFEM discretization for Stokes interface problems","volume":"38","author":"Kirchhart, Matthias","year":"2016","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"21","doi-asserted-by":"crossref","unstructured":"C. Lehrenfeld, F. Heimann, J. Preu\u00df, and H. von Wahl, ngsxfem: add-on to NGSolve for geometrically unfitted finite element discretizations, Journal of Open Source Software, 6 (2021), no. 64, 3237.","DOI":"10.21105\/joss.03237"},{"key":"22","doi-asserted-by":"publisher","first-page":"716","DOI":"10.1016\/j.cma.2015.12.005","article-title":"High order unfitted finite element methods on level set domains using isoparametric mappings","volume":"300","author":"Lehrenfeld, Christoph","year":"2016","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"2","key":"23","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1051\/m2an\/2018068","article-title":"An Eulerian finite element method for PDEs in time-dependent domains","volume":"53","author":"Lehrenfeld, Christoph","year":"2019","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/2822-7840","issn-type":"print"},{"issue":"2","key":"24","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1007\/s00211-016-0801-6","article-title":"Optimal preconditioners for Nitsche-XFEM discretizations of interface problems","volume":"135","author":"Lehrenfeld, Christoph","year":"2017","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"3","key":"25","doi-asserted-by":"publisher","first-page":"1351","DOI":"10.1093\/imanum\/drx041","article-title":"Analysis of a high-order unfitted finite element method for elliptic interface problems","volume":"38","author":"Lehrenfeld, Christoph","year":"2018","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"key":"26","doi-asserted-by":"crossref","unstructured":"C. Lehrenfeld, I. Voulis, and T. van Beeck, Replication data for: analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem, 2023, GRO.data, DOI:10.25625\/LYR2CG.","DOI":"10.1090\/mcom\/4027"},{"issue":"1","key":"27","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1051\/m2an\/2022072","article-title":"A CutFEM divergence-free discretization for the Stokes problem","volume":"57","author":"Liu, Haoran","year":"2023","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/2822-7840","issn-type":"print"},{"issue":"3","key":"28","doi-asserted-by":"publisher","first-page":"604","DOI":"10.1007\/s10915-014-9838-9","article-title":"A stabilized Nitsche fictitious domain method for the Stokes problem","volume":"61","author":"Massing, Andr\u00e9","year":"2014","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"1","key":"29","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1002\/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J","article-title":"A finite element method for crack growth without remeshing","volume":"46","author":"Mo\u00ebs, Nicolas","year":"1999","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"key":"30","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198508885.001.0001","volume-title":"Finite element methods for Maxwell's equations","author":"Monk, Peter","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0198508883"},{"issue":"1","key":"31","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1007\/s00466-007-0173-y","article-title":"Finite cell method: \u210e- and \ud835\udc5d-extension for embedded domain problems in solid mechanics","volume":"41","author":"Parvizian, Jamshid","year":"2007","journal-title":"Comput. Mech.","ISSN":"https:\/\/id.crossref.org\/issn\/0178-7675","issn-type":"print"},{"key":"32","unstructured":"J. Preu\u00df, Higher order unfitted isoparametric space-time FEM on moving domains, Master\u2019s Thesis, NAM, University of G\u00f6ttingen, February 2018."},{"key":"33","unstructured":"R. Puppi, A cut finite element method for the Darcy problem, Preprint,  arXiv:2111.09922, 2021."},{"key":"34","unstructured":"J. Sch\u00f6berl, C++11 implementation of finite elements in NGSolve, Tech. Report, Institute for Analysis and Scientific Computing, TU Wien."},{"key":"35","doi-asserted-by":"crossref","unstructured":"J. Sch\u00f6berl, NETGEN an advancing front 2D\/3D-mesh generator based on abstract rules, Computing and Visualization in Science (1997) 1, no. 1, 41\u201352.","DOI":"10.1007\/s007910050004"},{"issue":"1","key":"36","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1051\/m2an\/1991250101511","article-title":"Postprocessing schemes for some mixed finite elements","volume":"25","author":"Stenberg, Rolf","year":"1991","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"2","key":"37","doi-asserted-by":"publisher","first-page":"Paper No. 40, 16","DOI":"10.1007\/s10915-020-01284-z","article-title":"Robust and local optimal a priori error estimates for interface problems with low regularity: mixed finite element approximations","volume":"84","author":"Zhang, Shun","year":"2020","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-354\/S0025-5718-2024-04027-9\/S0025-5718-2024-04027-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:42:36Z","timestamp":1776836556000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-354\/S0025-5718-2024-04027-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,30]]},"references-count":37,"journal-issue":{"issue":"354","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["S0025-5718-2024-04027-9"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4027","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2024,10,30]]}}}