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Cut finite element methods

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations, boundary value problems

Published online by Cambridge University Press:  01 July 2025

Erik Burman ,
Peter Hansbo ,
Mats G. Larson  and
Sara Zahedi Open the ORCID record for Sara Zahedi [Opens in a new window]
Erik Burman
Affiliation:
Department of Mathematics, University College London, London, WC1E 6BT, UK E-mail: e.burman@ucl.ac.uk
Peter Hansbo
Affiliation:
Department of Mechanical Engineering, Jönköping University, 551 11 Jönköping, Sweden E-mail: peter.hansbo@ju.se
Mats G. Larson
Affiliation:
Department of Mathematics and Mathematical Statistics, Umeå University, 90187 Umeå, Sweden E-mail: mats.larson@umu.se
Sara Zahedi
Affiliation:
Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden E-mail: sara.zahedi@math.kth.se
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Abstract

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Cut finite element methods (CutFEM) extend the standard finite element method to unfitted meshes, enabling the accurate resolution of domain boundaries and interfaces without requiring the mesh to conform to them. This approach preserves the key properties and accuracy of the standard method while addressing challenges posed by complex geometries and moving interfaces.

In recent years, CutFEM has gained significant attention for its ability to discretize partial differential equations in domains with intricate geometries. This paper provides a comprehensive review of the core concepts and key developments in CutFEM, beginning with its formulation for common model problems and the presentation of fundamental analytical results, including error estimates and condition number estimates for the resulting algebraic systems. Stabilization techniques for cut elements, which ensure numerical robustness, are also explored. Finally, extensions to methods involving Lagrange multipliers and applications to time-dependent problems are discussed.

MSC classification

Primary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N12: Stability and convergence of numerical methods 65N15: Error bounds 65N85: Fictitious domain methods
Secondary: 65M12: Stability and convergence of numerical methods 65M85: Fictitious domain methods

Information

Type
Research Article
Information
Acta Numerica , Volume 34 , July 2025 , pp. 1 - 121
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2025. Published by Cambridge University Press

References

Ager, C., Schott, B., Winter, M. and Wall, W. A. (2019), A Nitsche-based cut finite element method for the coupling of incompressible fluid flow with poroelasticity, Comput. Methods Appl. Mech. Engrg 351, 253–280.10.1016/j.cma.2019.03.015CrossRefGoogle Scholar
Ahlkrona, J. and Elfverson, D. (2021), A cut finite element method for non-Newtonian free surface flows in 2D: Application to glacier modelling, J. Comput. Phys. X 11, art. 100090.Google Scholar
Annavarapu, C., Hautefeuille, M. and Dolbow, J. E. (2012), A robust Nitsche’s formulation for interface problems, Comput. Methods Appl. Mech. Engrg 225/228, 44–54.10.1016/j.cma.2012.03.008CrossRefGoogle Scholar
Anselmann, M. and Bause, M. (2022), Cut finite element methods and ghost stabilization techniques for space–time discretizations of the Navier–Stokes equations, Int. J. Numer. Methods Fluids 94, 775–802.10.1002/fld.5074CrossRefGoogle Scholar
Aulisa, E. and Loftin, J. (2023), Exact subdomain and embedded interface polynomial integration in finite elements with planar cuts, Numer. Algorithms 94, 315–350.10.1007/s11075-023-01502-3CrossRefGoogle Scholar
Babuška, I. (1972/73), The finite element method with Lagrangian multipliers, Numer. Math. 20, 179–192.10.1007/BF01436561CrossRefGoogle Scholar
Babuška, I. (1973), The finite element method with penalty, Math. Comp. 27, 221–228.10.1090/S0025-5718-1973-0351118-5CrossRefGoogle Scholar
Badia, S., Caicedo, M. A., Martín, A. F. and Principe, J. (2021), A robust and scalable unfitted adaptive finite element framework for nonlinear solid mechanics, Comput. Methods Appl. Mech. Engrg 386, art. 114093.10.1016/j.cma.2021.114093CrossRefGoogle Scholar
Badia, S., Martorell, P. A. and Verdugo, F. (2024), Space–time unfitted finite elements on moving explicit geometry representations, Comput. Methods Appl. Mech. Engrg 428, art. 117091.10.1016/j.cma.2024.117091CrossRefGoogle Scholar
Badia, S., Verdugo, F. and Martín, A. F. (2018), The aggregated unfitted finite element method for elliptic problems, Comput. Methods Appl. Mech. Engrg 336, 533–553.10.1016/j.cma.2018.03.022CrossRefGoogle Scholar
Barbosa, H. J. C. and Hughes, T. J. R. (1991), The finite element method with Lagrange multipliers on the boundary: Circumventing the Babuška–Brezzi condition, Comput. Methods Appl. Mech. Engrg 85, 109–128.10.1016/0045-7825(91)90125-PCrossRefGoogle Scholar
Barbosa, H. J. C. and Hughes, T. J. R. (1992), Circumventing the Babuška–Brezzi condition in mixed finite element approximations of elliptic variational inequalities, Comput. Methods Appl. Mech. Engrg 97, 193–210.10.1016/0045-7825(92)90163-ECrossRefGoogle Scholar
Barrau, N., Becker, R., Dubach, E. and Luce, R. (2012), A robust variant of NXFEM for the interface problem, C. R . Math. Acad. Sci. Paris 350, 789–792.10.1016/j.crma.2012.09.018CrossRefGoogle Scholar
Barrenechea, G. R. and Chouly, F. (2012), A local projection stabilized method for fictitious domains, Appl. Math. Lett. 25, 2071–2076.10.1016/j.aml.2012.04.020CrossRefGoogle Scholar
Barrett, J. W. and Elliott, C. M. (1986), Finite element approximation of the Dirichlet problem using the boundary penalty method, Numer . Math. 49, 343–366.Google Scholar
Bastian, P. and Engwer, C. (2009), An unfitted finite element method using discontinuous Galerkin, Int. J. Numer. Methods Engrg 79, 1557–1576.10.1002/nme.2631CrossRefGoogle Scholar
Becker, R., Burman, E. and Hansbo, P. (2009), A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity, Comput. Methods Appl. Mech. Engrg 198, 3352–3360.10.1016/j.cma.2009.06.017CrossRefGoogle Scholar
Becker, R., Hansbo, P. and Stenberg, R. (2003), A finite element method for domain decomposition with non-matching grids, ESAIM Math. Model. Numer. Anal. 37, 209–225.10.1051/m2an:2003023CrossRefGoogle Scholar
Berger, M. (2017), Cut cells: Meshes and solvers, in Handbook of Numerical Methods for Hyperbolic Problems, Vol. 18 of Handbook of Numerical Analysis, Elsevier, pp. 1–22.10.1016/bs.hna.2016.10.008CrossRefGoogle Scholar
Bernland, A., Wadbro, E. and Berggren, M. (2018), Acoustic shape optimization using cut finite elements, Int. J. Numer. Methods Engrg 113, 432–449.10.1002/nme.5621CrossRefGoogle Scholar
Berre, N., Rognes, M. E. and Massing, A. (2024), Cut finite element discretizations of cell-by-cell EMI electrophysiology models, SIAM J. Sci. Comput. 46, B527–B553.10.1137/23M1580632CrossRefGoogle Scholar
Bertoluzza, S. (2003), Analysis of a stabilized three-fields domain decomposition method, Numer. Math. 93, 611–634.10.1007/s002110100340CrossRefGoogle Scholar
Bertoluzza, S. (2016), Analysis of a mesh-dependent stabilization for the three fields domain decomposition method, Numer. Math. 133, 1–36.10.1007/s00211-015-0742-5CrossRefGoogle Scholar
Bertoluzza, S. and Burman, E. (2023), An abstract framework for heterogeneous coupling: stability, approximation and applications. Available at arXiv:2312.11733.Google Scholar
Boiveau, T. and Burman, E. (2015), A penalty-free Nitsche method for the weak imposition of boundary conditions in compressible and incompressible elasticity, IMA J. Numer. Anal. 36, 770–795.10.1093/imanum/drv042CrossRefGoogle Scholar
Boiveau, T., Burman, E. and Claus, S. (2017), Penalty-free Nitsche method for interface problems, in Geometrically Unfitted Finite Element Methods and Applications, Vol. 121 of Lecture Notes in Computational Science and Engineering, Springer, pp. 183–210.10.1007/978-3-319-71431-8_6CrossRefGoogle Scholar
Boiveau, T., Burman, E., Claus, S. and Larson, M. (2018), Fictitious domain method with boundary value correction using penalty-free Nitsche method, J. Numer. Math. 26, 77–95.Google Scholar
Bordas, S. P. A., Burman, E., Larson, M. G. and Olshanskii, M. A., eds (2017), Geometrically Unfitted Finite Element Methods and Applications, Vol. 121 of Lecture Notes in Computational Science and Engineering, Springer.10.1007/978-3-319-71431-8CrossRefGoogle Scholar
Bramble, J. H., Dupont, T. and Thomée, V. (1972), Projection methods for Dirichlet’s problem in approximating polygonal domains with boundary-value corrections, Math. Comp. 26, 869–879.Google Scholar
Brenner, S. and Scott, R. (2008), The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, third edition, Springer.10.1007/978-0-387-75934-0CrossRefGoogle Scholar
Bretin, E., Chapelat, J., Outtier, P.-Y. and Renard, Y. (2022), Shape optimization of a linearly elastic rolling structure under unilateral contact using Nitsche’s method and cut finite elements, Comput. Mech. 70, 205–224.10.1007/s00466-022-02164-zCrossRefGoogle Scholar
Brezzi, F. (1974), On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8, 129–151.Google Scholar
Brezzi, F. and Fortin, M. (2001), A minimal stabilisation procedure for mixed finite element methods, Numer. Math. 89, 457–491.10.1007/PL00005475CrossRefGoogle Scholar
Brezzi, F. and Marini, L. D. (1994), A three-field domain decomposition method, in Domain Decomposition Methods in Science and Engineering, Vol. 157 of Contemporary Mathematics, American Mathematical Society, pp. 27–34.10.1090/conm/157/01402CrossRefGoogle Scholar
Buffa, A., Puppi, R. and Vázquez, R. (2020), A minimal stabilization procedure for isogeometric methods on trimmed geometries, SIAM J. Numer. Anal. 58, 2711–2735.10.1137/19M1244718CrossRefGoogle Scholar
Bui, H. P., Tomar, S. and Bordas, S. P. A. (2019), Corotational cut finite element method for real-time surgical simulation: Application to needle insertion simulation, Comput. Methods Appl. Mech. Engrg 345, 183–211.10.1016/j.cma.2018.10.023CrossRefGoogle Scholar
Burman, E. (2010), Ghost penalty, C. R . Math. Acad. Sci. Paris 348, 1217–1220.10.1016/j.crma.2010.10.006CrossRefGoogle Scholar
Burman, E. (2012), A penalty-free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions, SIAM J. Numer. Anal. 50, 1959–1981.10.1137/10081784XCrossRefGoogle Scholar
Burman, E. (2014), Projection stabilization of Lagrange multipliers for the imposition of constraints on interfaces and boundaries, Numer. Methods Partial Differential Equations 30, 567–592.10.1002/num.21829CrossRefGoogle Scholar
Burman, E. and Ern, A. (2018), An unfitted hybrid high-order method for elliptic interface problems, SIAM J. Numer. Anal. 56, 1525–1546.10.1137/17M1154266CrossRefGoogle Scholar
Burman, E. and Fernández, M. A. (2014), An unfitted Nitsche method for incompressible fluid–structure interaction using overlapping meshes, Comput. Methods Appl. Mech. Engrg 279, 497–514.10.1016/j.cma.2014.07.007CrossRefGoogle Scholar
Burman, E. and Hansbo, P. (2010a), Fictitious domain finite element methods using cut elements, I: A stabilized Lagrange multiplier method, Comput . Methods Appl. Mech. Engrg 199, 2680–2686.10.1016/j.cma.2010.05.011CrossRefGoogle Scholar
Burman, E. and Hansbo, P. (2010b), Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems, IMA J. Numer. Anal. 30, 870–885.10.1093/imanum/drn081CrossRefGoogle Scholar
Burman, E. and Hansbo, P. (2012), Fictitious domain finite element methods using cut elements, II: A stabilized Nitsche method, Appl. Numer. Math. 62, 328–341.10.1016/j.apnum.2011.01.008CrossRefGoogle Scholar
Burman, E. and Hansbo, P. (2014), Fictitious domain methods using cut elements, III: A stabilized Nitsche method for Stokes’ problem, ESAIM Math. Model. Numer. Anal. 48, 859–874.10.1051/m2an/2013123CrossRefGoogle Scholar
Burman, E. and Hansbo, P. (2017), Deriving robust unfitted finite element methods from augmented Lagrangian formulations, in Geometrically Unfitted Finite Element Methods and Applications, Vol. 121 of Lecture Notes in Computational Science and Engineering, Springer, pp. 1–24.10.1007/978-3-319-71431-8_1CrossRefGoogle Scholar
Burman, E. and Preuss, J. (2023), Unique continuation for an elliptic interface problem using unfitted isoparametric finite elements. Available at arXiv:2307.05210.Google Scholar
Burman, E. and Zunino, P. (2012), Numerical approximation of large contrast problems with the unfitted Nitsche method, in Frontiers in Numerical Analysis, Vol. 85 of Lecture Notes in Computational Science and Engineering, Springer, pp. 227–282.Google Scholar
Burman, E., Cicuttin, M., Delay, G. and Ern, A. (2021a), An unfitted hybrid high-order method with cell agglomeration for elliptic interface problems, SIAM J. Sci. Comput. 43, A859–A882.10.1137/19M1285901CrossRefGoogle Scholar
Burman, E., Claus, S. and Massing, A. (2015a), A stabilized cut finite element method for the three field Stokes problem, SIAM J. Sci. Comput. 37, A1705–A1726.10.1137/140983574CrossRefGoogle Scholar
Burman, E., Claus, S., Hansbo, P., Larson, M. G. and Massing, A. (2015b), CutFEM: Discretizing geometry and partial differential equations, Int. J. Numer. Methods Engrg 104, 472–501.10.1002/nme.4823CrossRefGoogle Scholar
Burman, E., Delay, G. and Ern, A. (2021b), An unfitted hybrid high-order method for the Stokes interface problem, IMA J. Numer. Anal. 41, 2362–2387.10.1093/imanum/draa059CrossRefGoogle Scholar
Burman, E., Duran, O. and Ern, A. (2022a), Unfitted hybrid high-order methods for the wave equation, Comput. Methods Appl. Mech. Engrg 389, art. 114366.10.1016/j.cma.2021.114366CrossRefGoogle Scholar
Burman, E., Elfverson, D., Hansbo, P., Larson, M. G. and Larsson, K. (2017a), A cut finite element method for the Bernoulli free boundary value problem, Comput. Methods Appl. Mech. Engrg 317, 598–618.10.1016/j.cma.2016.12.021CrossRefGoogle Scholar
Burman, E., Elfverson, D., Hansbo, P., Larson, M. G. and Larsson, K. (2019a), Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions, Comput. Methods Appl. Mech. Engrg 350, 462–479.10.1016/j.cma.2019.03.016CrossRefGoogle Scholar
Burman, E., Elfverson, D., Hansbo, P., Larson, M. G. and Larsson, K. (2019b), Hybridized CutFEM for elliptic interface problems, SIAM J. Sci. Comput. 41, A3354–A3380.10.1137/18M1223836CrossRefGoogle Scholar
Burman, E., Fernández, M. A. and Gerosa, F. M. (2023a), Convergence analysis of an unfitted mesh semi-implicit coupling scheme for incompressible fluid–structure interaction, Vietnam J. Math. 51, 37–69.10.1007/s10013-022-00589-wCrossRefGoogle Scholar
Burman, E., Frei, S. and Massing, A. (2022b), Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains, Numer. Math. 150, 423–478.10.1007/s00211-021-01264-xCrossRefGoogle Scholar
Burman, E., Guzmán, J., Sánchez, M. A. and Sarkis, M. (2018a), Robust flux error estimation of an unfitted Nitsche method for high-contrast interface problems, IMA J. Numer. Anal. 38, 646–668.10.1093/imanum/drx017CrossRefGoogle Scholar
Burman, E., Hansbo, P. and Larson, M. (2024a), Cut finite element method for divergence-free approximation of incompressible flow: A Lagrange multiplier approach, SIAM J. Numer. Anal. 62, 893–918.10.1137/22M1542933CrossRefGoogle Scholar
Burman, E., Hansbo, P. and Larson, M. G. (2015c), A stabilized cut finite element method for partial differential equations on surfaces: The Laplace–Beltrami operator, Comput. Methods Appl. Mech. Engrg 285, 188–207.10.1016/j.cma.2014.10.044CrossRefGoogle Scholar
Burman, E., Hansbo, P. and Larson, M. G. (2018b), A cut finite element method with boundary value correction, Math. Comp. 87, 633–657.10.1090/mcom/3240CrossRefGoogle Scholar
Burman, E., Hansbo, P. and Larson, M. G. (2020a), Cut Bogner–Fox–Schmit elements for plates, Adv. Model. Simul. Eng. Sci. 7, art. 27.10.1186/s40323-020-00164-3CrossRefGoogle Scholar
Burman, E., Hansbo, P. and Larson, M. G. (2020b), A cut finite element method for a model of pressure in fractured media, Numer. Math. 146, 783–818.10.1007/s00211-020-01157-5CrossRefGoogle Scholar
Burman, E., Hansbo, P. and Larson, M. G. (2022c), CutFEM based on extended finite element spaces, Numer. Math. 152, 331–369.10.1007/s00211-022-01313-zCrossRefGoogle Scholar
Burman, E., Hansbo, P. and Larson, M. G. (2022d), Explicit time stepping for the wave equation using CutFEM with discrete extension, SIAM J. Sci. Comput. 44, A1254–A1289.10.1137/20M137937XCrossRefGoogle Scholar
Burman, E., Hansbo, P. and Larson, M. G. (2022e), On the design of locking free ghost penalty stabilization and the relation to CutFEM with discrete extension. Available at arXiv:2205.01340.Google Scholar
Burman, E., Hansbo, P. and Larson, M. G. (2024b), Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions, Math. Comp. 93, 35–54.10.1090/mcom/3875CrossRefGoogle Scholar
Burman, E., Hansbo, P., Larson, M. G. and Larsson, K. (2023b), Extension operators for trimmed spline spaces, Comput. Methods Appl. Mech. Engrg 403, art. 115707.10.1016/j.cma.2022.115707CrossRefGoogle Scholar
Burman, E., Hansbo, P., Larson, M. G. and Massing, A. (2017b), A cut discontinuous Galerkin method for the Laplace–Beltrami operator, IMA J. Numer. Anal. 37, 138–169.10.1093/imanum/drv068CrossRefGoogle Scholar
Burman, E., Hansbo, P., Larson, M. G. and Massing, A. (2018c), Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions, ESAIM Math. Model. Numer. Anal. 52, 2247–2282.10.1051/m2an/2018038CrossRefGoogle Scholar
Burman, E., Hansbo, P., Larson, M. G. and Samvin, D. (2019c), A cut finite element method for elliptic bulk problems with embedded surfaces, GEM Int. J. Geomath. 10, art. 10.10.1007/s13137-019-0120-zCrossRefGoogle Scholar
Burman, E., Hansbo, P., Larson, M. G. and Zahedi, S. (2016a), Cut finite element methods for coupled bulk–surface problems, Numer. Math. 133, 203–231.10.1007/s00211-015-0744-3CrossRefGoogle Scholar
Burman, E., Hansbo, P., Larson, M. G., Massing, A. and Zahedi, S. (2016b), Full gradient stabilized cut finite element methods for surface partial differential equations, Comput. Methods Appl. Mech. Engrg 310, 278–296.10.1016/j.cma.2016.06.033CrossRefGoogle Scholar
Burman, E., He, C. and Larson, M. G. (2021c), Comparison of shape derivatives using CutFEM for ill-posed Bernoulli free boundary problem, J. Sci. Comput. 88, art. 35.10.1007/s10915-021-01544-6CrossRefGoogle Scholar
Burman, E., He, C. and Larson, M. G. (2022f), A posteriori error estimates with boundary correction for a cut finite element method, IMA J. Numer. Anal. 42, 333–362.10.1093/imanum/draa085CrossRefGoogle Scholar
Cáceres, E., Guzmán, J. and Olshanskii, M. (2020), New stability estimates for an unfitted finite element method for two-phase Stokes problem, SIAM J. Numer. Anal. 58, 2165–2192.10.1137/19M1266897CrossRefGoogle Scholar
Cai, Y., Chen, J. and Wang, N. (2021), A Nitsche extended finite element method for the biharmonic interface problem, Comput. Methods Appl. Mech. Engrg 382, art. 113880.10.1016/j.cma.2021.113880CrossRefGoogle Scholar
Cangiani, A., Dong, Z. and Georgoulis, E. H. (2021), hp-version discontinuous Galerkin methods on essentially arbitrarily-shaped elements, Math. Comp. 91, 1–35.10.1090/mcom/3667CrossRefGoogle Scholar
Capatina, D. and He, C. (2021), Flux recovery for cut finite element method and its application in a posteriori error estimation, ESAIM Math. Model. Numer. Anal. 55, 2759–2784.10.1051/m2an/2021071CrossRefGoogle Scholar
Cenanovic, M., Hansbo, P. and Larson, M. G. (2016), Cut finite element modeling of linear membranes, Comput. Methods Appl. Mech. Engrg 310, 98–111.10.1016/j.cma.2016.05.018CrossRefGoogle Scholar
Chen, Z. and Liu, Y. (2023), An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation, J. Comput. Phys. 491, art. 112384.10.1016/j.jcp.2023.112384CrossRefGoogle Scholar
Chen, Z. and Liu, Y. (2024), An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation, Part II: Piecewise-smooth interfaces, Appl. Numer. Math. 206, 247–268.10.1016/j.apnum.2024.08.012CrossRefGoogle Scholar
Chen, Z., Li, K. and Xiang, X. (2021), An adaptive high-order unfitted finite element method for elliptic interface problems, Numer. Math. 149, 507–548.10.1007/s00211-021-01243-2CrossRefGoogle Scholar
Chen, Z., Li, K., Lyu, M. and Xiang, X. (2024), A high order unfitted finite element method for time-harmonic Maxwell interface problems, Int. J. Numer. Anal. Model. 21, 822–849.10.4208/ijnam2024-1033CrossRefGoogle Scholar
Chernyshenko, A. Y. and Olshanskii, M. A. (2020), An unfitted finite element method for the Darcy problem in a fracture network, J. Comput. Appl. Math. 366, art. 112424.10.1016/j.cam.2019.112424CrossRefGoogle Scholar
Chessa, J. and Belytschko, T. (2004), Arbitrary discontinuities in space–time finite elements by level sets and X-FEM, Int. J. Numer. Methods Engrg 61, 2595–2614.10.1002/nme.1155CrossRefGoogle Scholar
Claus, S. and Kerfriden, P. (2018), A stable and optimally convergent LaTIn-CutFEM algorithm for multiple unilateral contact problems, Int. J. Numer. Methods Engrg 113, 938–966.10.1002/nme.5694CrossRefGoogle Scholar
Claus, S. and Kerfriden, P. (2019), A CutFEM method for two-phase flow problems, Comput. Methods Appl. Mech. Engrg 348, 185–206.10.1016/j.cma.2019.01.009CrossRefGoogle Scholar
Claus, S., Bigot, S. and Kerfriden, P. (2018), CutFEM method for Stefan–Signorini problems with application in pulsed laser ablation, SIAM J. Sci. Comput. 40, B1444–B1469.10.1137/18M1185697CrossRefGoogle Scholar
Claus, S., Kerfriden, P., Moshfeghifar, F., Darkner, S., Erleben, K. and Wong, C. (2021), Contact modeling from images using cut finite element solvers, Adv. Model. Simul. Eng. Sci. 8, art. 13.10.1186/s40323-021-00197-2CrossRefGoogle Scholar
Cockburn, B. and Solano, M. (2012), Solving Dirichlet boundary-value problems on curved domains by extensions from subdomains, SIAM J. Sci. Comput. 34, A497–A519.10.1137/100805200CrossRefGoogle Scholar
Corti, D. C., Delay, G., Fernández, M. A., Vergnet, F. and Vidrascu, M. (2024), Low-order fictitious domain method with enhanced mass conservation for an interface Stokes problem, ESAIM Math. Model. Numer. Anal. 58, 303–333.10.1051/m2an/2023103CrossRefGoogle Scholar
Court, S. (2019), A fictitious domain approach for a mixed finite element method solving the two-phase Stokes problem with surface tension forces, J. Comput. Appl. Math. 359, 30–54.10.1016/j.cam.2019.03.029CrossRefGoogle Scholar
de Prenter, F., Lehrenfeld, C. and Massing, A. (2018), A note on the stability parameter in Nitsche’s method for unfitted boundary value problems, Comput. Math. Appl. 75, 4322–4336.10.1016/j.camwa.2018.03.032CrossRefGoogle Scholar
Dilgen, S. B., Jensen, J. S. and Aage, N. (2021), Shape optimization of the time-harmonic response of vibroacoustic devices using cut elements, Finite Elem. Anal. Des. 196, art. 103608.10.1016/j.finel.2021.103608CrossRefGoogle Scholar
Dryja, M. (2003), On discontinuous Galerkin methods for elliptic problems with discontinuous coefficients, Comput. Methods Appl. Math. 3, 76–85.10.2478/cmam-2003-0007CrossRefGoogle Scholar
Dunn, K., Lui, R. and Sarkis, M. (2021), An unconditionally stable semi-implicit CutFEM for an interaction problem between an elastic membrane and an incompressible fluid, Electron. Trans. Numer. Anal. 54, 296–322.10.1553/etna_vol54s296CrossRefGoogle Scholar
Duprez, M., Lleras, V. and Lozinski, A. (2023), A new ϕ-FEM approach for problems with natural boundary conditions, Numer. Methods Partial Differential Equations 39, 281–303.10.1002/num.22878CrossRefGoogle Scholar
Egger, H. (2009), A class of hybrid mortar finite element methods for interface problems with non-matching meshes. Preprint: AICES-2009-2.Google Scholar
Engwer, C., May, S., Nüßing, A. and Streitbürger, F. (2020), A stabilized DG cut cell method for discretizing the linear transport equation, SIAM J. Sci. Comput. 42, A3677–A3703.10.1137/19M1268318CrossRefGoogle Scholar
Ern, A. and Guermond, J.-L. (2006), Evaluation of the condition number in linear systems arising in finite element approximations, ESAIM Math. Model. Numer. Anal. 40, 29–48.10.1051/m2an:2006006CrossRefGoogle Scholar
Fabre, M., Pousin, J. and Renard, Y. (2016), A fictitious domain method for frictionless contact problems in elasticity using Nitsche’s method, SMAI J. Comput. Math. 2, 19–50.10.5802/smai-jcm.8CrossRefGoogle Scholar
Farina, S., Claus, S., Hale, J. S., Skupin, A. and Bordas, S. P. A. (2021), A cut finite element method for spatially resolved energy metabolism models in complex neuro-cell morphologies with minimal remeshing, Adv. Model. Simul. Eng. Sci. 8, art. 5.10.1186/s40323-021-00191-8CrossRefGoogle Scholar
Fernández, M. A. and Landajuela, M. (2020), Splitting schemes and unfitted-mesh methods for the coupling of an incompressible fluid with a thin-walled structure, IMA J. Numer. Anal. 40, 1407–1453.10.1093/imanum/dry098CrossRefGoogle Scholar
Fernández, M. A. and Gerosa, F. M. (2021), An unfitted mesh semi-implicit coupling scheme for fluid–structure interaction with immersed solids, Int. J. Numer. Methods Engrg 122, 5384–5408.10.1002/nme.6449CrossRefGoogle Scholar
Fournié, M. and Lozinski, A. (2017), Stability and optimal convergence of unfitted extended finite element methods with Lagrange multipliers for the Stokes equations, in Geometrically Unfitted Finite Element Methods and Applications, Vol. 121 of Lecture Notes in Computational Science and Engineering, Springer, pp. 143–182.10.1007/978-3-319-71431-8_5CrossRefGoogle Scholar
Frachon, T. and Myrbäck, S. (2024), CutFEM-Library, GitHub repository. Available at https://github.com/CutFEM/CutFEM-Library.Google Scholar
Frachon, T. and Zahedi, S. (2019), A cut finite element method for incompressible two-phase Navier–Stokes flows, J. Comput. Phys. 384, 77–98.10.1016/j.jcp.2019.01.028CrossRefGoogle Scholar
Frachon, T. and Zahedi, S. (2023), A cut finite element method for two-phase flows with insoluble surfactants, J. Comput. Phys. 473, art. 111734.10.1016/j.jcp.2022.111734CrossRefGoogle Scholar
Frachon, T., Hansbo, P., Nilsson, E. and Zahedi, S. (2024a), A divergence preserving cut finite element method for Darcy flow, SIAM J. Sci. Comput. 46, A1793–A1820.10.1137/22M149702XCrossRefGoogle Scholar
Frachon, T., Nilsson, E. and Zahedi, S. (2024b), Divergence-free cut finite element methods for Stokes flow, BIT Numer. Math. 64, art. 39.10.1007/s10543-024-01040-xCrossRefGoogle Scholar
Fries, T.-P. and Belytschko, T. (2010), The extended/generalized finite element method: An overview of the method and its applications, Int. J. Numer. Methods Engrg 84, 253–304.10.1002/nme.2914CrossRefGoogle Scholar
Fries, T.-P. and Kaiser, M. W. (2023), On the simultaneous solution of structural membranes on all level sets within a bulk domain, Comput. Methods Appl. Mech. Engrg 415, art. 116223.10.1016/j.cma.2023.116223CrossRefGoogle Scholar
Fu, P. and Kreiss, G. (2021), High order cut discontinuous Galerkin methods for hyperbolic conservation laws in one space dimension, SIAM J. Sci. Comput. 43, A2404–A2424.10.1137/20M1349060CrossRefGoogle Scholar
Fu, P., Frachon, T., Kreiss, G. and Zahedi, S. (2022), High order discontinuous cut finite element methods for linear hyperbolic conservation laws with an interface, J. Sci. Comput. 90, art. 84.10.1007/s10915-021-01756-wCrossRefGoogle Scholar
Fu, P., Kreiss, G. and Zahedi, S. (2024), A bound preserving cut discontinuous Galerkin method for one dimensional hyperbolic conservation laws, ESAIM Math. Model. Numer. Anal. 58, 1651–1680.10.1051/m2an/2024042CrossRefGoogle Scholar
Garcke, H., Nürnberg, R. and Zhao, Q. (2023), Structure-preserving discretizations of two-phase Navier–Stokes flow using fitted and unfitted approaches, J. Comput. Phys. 489, art. 112276.10.1016/j.jcp.2023.112276CrossRefGoogle Scholar
Garhuom, W. and Düster, A. (2022), Non-negative moment fitting quadrature for cut finite elements and cells undergoing large deformations, Comput. Mech. 70, 1059–1081.10.1007/s00466-022-02203-9CrossRefGoogle Scholar
Georgievskii, Y., Medvedev, E. S. and Stuchebrukhov, A. A. (2002), Proton transport via the membrane surface, Biophys J. 82, 2833–2846.10.1016/S0006-3495(02)75626-9CrossRefGoogle ScholarPubMed
Giovanardi, B., Formaggia, L., Scotti, A. and Zunino, P. (2017), Unfitted FEM for modelling the interaction of multiple fractures in a poroelastic medium, in Geometrically Unfitted Finite Element Methods and Applications, Vol. 121 of Lecture Notes in Computational Science and Engineering, Springer, pp. 331–352.10.1007/978-3-319-71431-8_11CrossRefGoogle Scholar
Girault, V. and Glowinski, R. (1995), Error analysis of a fictitious domain method applied to a Dirichlet problem, Japan J. Indust. Appl. Math. 12, 487–514.10.1007/BF03167240CrossRefGoogle Scholar
Glowinski, R., Pan, T.-W. and Périaux, J. (1994), A fictitious domain method for Dirichlet problem and applications, Comput. Methods Appl. Mech. Engrg 111, 283–303.10.1016/0045-7825(94)90135-XCrossRefGoogle Scholar
Grande, J. (2014), Eulerian finite element methods for parabolic equations on moving surfaces, SIAM J. Sci. Comput. 36, B248–B271.10.1137/130920095CrossRefGoogle Scholar
Grande, J., Lehrenfeld, C. and Reusken, A. (2018), Analysis of a high-order trace finite element method for PDEs on level set surfaces, SIAM J. Numer. Anal. 56, 228–255.10.1137/16M1102203CrossRefGoogle Scholar
Gross, S. and Reusken, A. (2023), Analysis of optimal preconditioners for CutFEM, Numer. Linear Algebra Appl. 30, art. e2486.10.1002/nla.2486CrossRefGoogle Scholar
Guo, R., Lin, Y. and Zou, J. (2023), Solving two-dimensional H(curl)-elliptic interface systems with optimal convergence on unfitted meshes, European J. Appl. Math. 34, 774–805.10.1017/S0956792522000390CrossRefGoogle Scholar
Gürkan, C. and Massing, A. (2019), A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems, Comput. Methods Appl. Mech. Engrg 348, 466–499.10.1016/j.cma.2018.12.041CrossRefGoogle Scholar
Gürkan, C., Sticko, S. and Massing, A. (2020), Stabilized cut discontinuous Galerkin methods for advection–reaction problems, SIAM J. Sci. Comput. 42, A2620–A2654.10.1137/18M1206461CrossRefGoogle Scholar
Guzmán, J. and Olshanskii, M. (2018), Inf-sup stability of geometrically unfitted Stokes finite elements, Math. Comp. 87, 2091–2112.10.1090/mcom/3288CrossRefGoogle Scholar
Hansbo, A. and Hansbo, P. (2002), An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg 191, 5537–5552.10.1016/S0045-7825(02)00524-8CrossRefGoogle Scholar
Hansbo, A. and Hansbo, P. (2004), A finite element method for the simulation of strong and weak discontinuities in solid mechanics, Comput. Methods Appl. Mech. Engrg 193, 3523–3540.10.1016/j.cma.2003.12.041CrossRefGoogle Scholar
Hansbo, A., Hansbo, P. and Larson, M. G. (2003), A finite element method on composite grids based on Nitsche’s method, ESAIM Math. Model. Numer. Anal. 37, 495–514.10.1051/m2an:2003039CrossRefGoogle Scholar
Hansbo, P. (2005), Nitsche’s method for interface problems in computational mechanics, GAMM-Mitt. 28, 183–206.10.1002/gamm.201490018CrossRefGoogle Scholar
Hansbo, P., Jonsson, T., Larson, M. G. and Larsson, K. (2017a), A Nitsche method for elliptic problems on composite surfaces, Comput. Methods Appl. Mech. Engrg 326, 505–525.10.1016/j.cma.2017.08.033CrossRefGoogle Scholar
Hansbo, P., Larson, M. G. and Larsson, K. (2017b), Cut finite element methods for linear elasticity problems, in Geometrically Unfitted Finite Element Methods and Applications, Vol. 121 of Lecture Notes in Computational Science and Engineering, Springer, pp. 25–63.10.1007/978-3-319-71431-8_2CrossRefGoogle Scholar
Hansbo, P., Larson, M. G. and Larsson, K. (2020), Analysis of finite element methods for vector Laplacians on surfaces, IMA J. Numer. Anal. 40, 1652–1701.10.1093/imanum/drz018CrossRefGoogle Scholar
Hansbo, P., Larson, M. G. and Zahedi, S. (2014), A cut finite element method for a Stokes interface problem, Appl. Numer. Math. 85, 90–114.10.1016/j.apnum.2014.06.009CrossRefGoogle Scholar
Hansbo, P., Larson, M. G. and Zahedi, S. (2015a), Characteristic cut finite element methods for convection–diffusion problems on time dependent surfaces, Comput. Methods Appl. Mech. Engrg 293, 431–461.10.1016/j.cma.2015.05.010CrossRefGoogle Scholar
Hansbo, P., Larson, M. G. and Zahedi, S. (2015b), Stabilized finite element approximation of the mean curvature vector on closed surfaces, SIAM J. Numer. Anal. 53, 1806–1832.10.1137/140982696CrossRefGoogle Scholar
Hansbo, P., Larson, M. G. and Zahedi, S. (2016), A cut finite element method for coupled bulk–surface problems on time-dependent domains, Comput. Methods Appl. Mech. Engrg 307, 96–116.10.1016/j.cma.2016.04.012CrossRefGoogle Scholar
Haslinger, J. and Renard, Y. (2009), A new fictitious domain approach inspired by the extended finite element method, SIAM J. Numer. Anal. 47, 1474–1499.10.1137/070704435CrossRefGoogle Scholar
Heimann, F., Lehrenfeld, C. and Preuß, J. (2023), Geometrically higher order unfitted space–time methods for PDEs on moving domains, SIAM J. Sci. Comput. 45, B139–B165.10.1137/22M1476034CrossRefGoogle Scholar
Huang, P., Wu, H. and Xiao, Y. (2017), An unfitted interface penalty finite element method for elliptic interface problems, Comput. Methods Appl. Mech. Engrg 323, 439–460.10.1016/j.cma.2017.06.004CrossRefGoogle Scholar
Ji, H., Wang, F. and Chen, J. (2017), Unfitted finite element methods for the heat conduction in composite media with contact resistance, Numer. Methods Partial Differential Equations 33, 354–380.10.1002/num.22111CrossRefGoogle Scholar
Johansson, A. and Larson, M. G. (2013), A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary, Numer. Math. 10.1007/s00211-012-0497-1Google Scholar
Jonsson, T., Larson, M. G. and Larsson, K. (2017), Cut finite element methods for elliptic problems on multipatch parametric surfaces, Comput. Methods Appl. Mech. Engrg 324, 366–394.10.1016/j.cma.2017.06.018CrossRefGoogle Scholar
Karatzas, E. N., Nonino, M., Ballarin, F. and Rozza, G. (2022), A reduced order cut finite element method for geometrically parametrized steady and unsteady Navier–Stokes problems, Comput. Math. Appl. 116, 140–160.10.1016/j.camwa.2021.07.016CrossRefGoogle Scholar
Kerfriden, P., Claus, S. and Mihai, I. (2020), A mixed-dimensional CutFEM methodology for the simulation of fibre-reinforced composites, Adv. Model. Simul. Eng. Sci. 7, art. 18.10.1186/s40323-020-00154-5CrossRefGoogle Scholar
Kirchhart, M., Gross, S. and Reusken, A. (2016), Analysis of an XFEM discretization for Stokes interface problems, SIAM J. Sci. Comput. 38, A1019–A1043.10.1137/15M1011779CrossRefGoogle Scholar
Köppel, M., Martin, V. and Roberts, J. E. (2019a), A stabilized Lagrange multiplier finite-element method for flow in porous media with fractures, GEM Int. J. Geomath. 10, art. 7.10.1007/s13137-019-0117-7CrossRefGoogle Scholar
Köppel, M., Martin, V., Jaffré, J. and Roberts, J. E. (2019b), A Lagrange multiplier method for a discrete fracture model for flow in porous media, Comput. Geosci. 23, 239–253.10.1007/s10596-018-9779-8CrossRefGoogle Scholar
Kothari, H. and Krause, R. (2022), A generalized multigrid method for solving contact problems in Lagrange multiplier based unfitted finite element method, Comput. Methods Appl. Mech. Engrg 392, art. 114630.10.1016/j.cma.2022.114630CrossRefGoogle Scholar
Larson, M. G. and Zahedi, S. (2019), Stabilization of high order cut finite element methods on surfaces, IMA J. Numer. Anal. 40, 1702–1745.10.1093/imanum/drz021CrossRefGoogle Scholar
Larson, M. G. and Zahedi, S. (2023), Conservative cut finite element methods using macroelements, Comput. Methods Appl. Mech. Engrg 414, art. 116141.10.1016/j.cma.2023.116141CrossRefGoogle Scholar
Lehrenfeld, C. (2015), The Nitsche XFEM-DG space–time method and its implementation in three space dimensions, SIAM J. Sci. Comput. 37, A245–A270.10.1137/130943534CrossRefGoogle Scholar
Lehrenfeld, C. (2016), High order unfitted finite element methods on level set domains using isoparametric mappings, Comput. Methods Appl. Mech. Engrg 300, 716–733.10.1016/j.cma.2015.12.005CrossRefGoogle Scholar
Lehrenfeld, C. and Olshanskii, M. (2019), An Eulerian finite element method for PDEs in time-dependent domains, ESAIM Math. Model. Numer. Anal. 53, 585–614.10.1051/m2an/2018068CrossRefGoogle Scholar
Lehrenfeld, C. and Reusken, A. (2013), Analysis of a Nitsche XFEM-DG discretization for a class of two-phase mass transport problems, SIAM J. Numer. Anal. 51, 958–983.10.1137/120875260CrossRefGoogle Scholar
Lehrenfeld, C. and Reusken, A. (2017), Optimal preconditioners for Nitsche-XFEM discretizations of interface problems, Numer. Math. 135, 313–332.10.1007/s00211-016-0801-6CrossRefGoogle Scholar
Lehrenfeld, C. and Reusken, A. (2018), Analysis of a high-order unfitted finite element method for elliptic interface problems, IMA J. Numer. Anal. 38, 1351–1387.10.1093/imanum/drx041CrossRefGoogle Scholar
Lehrenfeld, C., van Beeck, T. and Voulis, I. (2023), Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem. Available at arXiv:2306.12722.Google Scholar
Liu, B. (2021), A Nitsche stabilized finite element method: Application for heat and mass transfer and fluid–structure interaction, Comput. Methods Appl. Mech. Engrg 386, art. 114101.10.1016/j.cma.2021.114101CrossRefGoogle Scholar
Liu, H., Neilan, M. and Olshanskii, M. (2023), A CutFEM divergence-free discretization for the Stokes problem, ESAIM Math. Model. Numer. Anal. 57, 143–165.10.1051/m2an/2022072CrossRefGoogle Scholar
Lou, Y. and Lehrenfeld, C. (2022), Isoparametric unfitted BDF–finite element method for PDEs on evolving domains, SIAM J. Numer. Anal. 60, 2069–2098.10.1137/21M142126XCrossRefGoogle Scholar
Ludescher, T., Gross, S. and Reusken, A. (2020), A multigrid method for unfitted finite element discretizations of elliptic interface problems, SIAM J. Sci. Comput. 42, A318–A342.10.1137/18M1203353CrossRefGoogle Scholar
Ma, C., Zhang, Q. and Zheng, W. (2022), A fourth-order unfitted characteristic finite element method for solving the advection–diffusion equation on time-varying domains, SIAM J. Numer. Anal. 60, 2203–2224.10.1137/22M1483475CrossRefGoogle Scholar
Main, A. and Scovazzi, G. (2018), The shifted boundary method for embedded domain computations, Part I: Poisson and Stokes problems, J. Comput. Phys. 372, 972–995.10.1016/j.jcp.2017.10.026CrossRefGoogle Scholar
Massing, A., Larson, M. G., Logg, A. and Rognes, M. E. (2014), A stabilized Nitsche fictitious domain method for the Stokes problem, J. Sci. Comput. 61, 604–628.10.1007/s10915-014-9838-9CrossRefGoogle Scholar
Massing, A., Larson, M. G., Logg, A. and Rognes, M. E. (2015), A Nitsche-based cut finite element method for a fluid–structure interaction problem, Commun. Appl. Math. Comput. Sci. 10, 97–120.10.2140/camcos.2015.10.97CrossRefGoogle Scholar
Massing, A., Schott, B. and Wall, W. A. (2018), A stabilized Nitsche cut finite element method for the Oseen problem, Comput. Methods Appl. Mech. Engrg 328, 262–300.10.1016/j.cma.2017.09.003CrossRefGoogle Scholar
Massjung, R. (2012), An unfitted discontinuous Galerkin method applied to elliptic interface problems, SIAM J. Numer. Anal. 50, 3134–3162.10.1137/090763093CrossRefGoogle Scholar
Maury, B. (2009), Numerical analysis of a finite element/volume penalty method, SIAM J. Numer. Anal. 47, 1126–1148.10.1137/080712799CrossRefGoogle Scholar
May, S. and Berger, M. (2017), An explicit implicit scheme for cut cells in embedded boundary meshes, J. Sci. Comput. 71, 919–943.10.1007/s10915-016-0326-2CrossRefGoogle Scholar
May, S. and Laakmann, F. (2024), Accuracy analysis for explicit–implicit finite volume schemes on cut cell meshes, Commun. Appl. Math. Comput. 6, 2239–2264.10.1007/s42967-023-00345-yCrossRefGoogle Scholar
McCracken, M. and Peskin, C. (1980), A vortex method for blood flow through heart valves, J. Comput. Phys. 35, 183–205.10.1016/0021-9991(80)90085-6CrossRefGoogle Scholar
Mikaeili, E., Claus, S. and Kerfriden, P. (2022), Concurrent multiscale analysis without meshing: Microscale representation with CutFEM and micro/macro model blending, Comput. Methods Appl. Mech. Engrg 393, art. 114807.10.1016/j.cma.2022.114807CrossRefGoogle Scholar
Moës, N., Dolbow, J. and Belytschko, T. (1999), A finite element method for crack growth without remeshing, Int. J. Numer. Methods Engrg 46, 131–150.10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J3.0.CO;2-J>CrossRefGoogle Scholar
Myrbäck, S. (2022), Conservative discontinuous cut finite element methods: Convection–diffusion problems in evolving bulk-interface domains. Master’s thesis, KTH Royal Institute of Technology.Google Scholar
Myrbäck, S. and Zahedi, S. (2024), A high-order conservative cut finite element method for problems in time-dependent domains, Comput. Methods Appl. Mech. Engrg 431, art. 117245.10.1016/j.cma.2024.117245CrossRefGoogle Scholar
Nédélec, J.-C. (1976), Curved finite element methods for the solution of singular integral equations on surfaces in R3, Comput. Methods Appl. Mech. Engrg 8, 61–80.10.1016/0045-7825(76)90053-0CrossRefGoogle Scholar
Nitsche, J. A. (1971), Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Univ. Hamburg 36, 9–15.10.1007/BF02995904CrossRefGoogle Scholar
Odsæter, L. H., Kvamsdal, T. and Larson, M. G. (2019), A simple embedded discrete fracture-matrix model for a coupled flow and transport problem in porous media, Comput. Methods Appl. Mech. Engrg 343, 572–601.10.1016/j.cma.2018.09.003CrossRefGoogle Scholar
Olshanskii, M. A. and Reusken, A. (2014), Error analysis of a space–time finite element method for solving PDEs on evolving surfaces, SIAM J. Numer. Anal. 52, 2092–2120.10.1137/130936877CrossRefGoogle Scholar
Olshanskii, M. A. and Reusken, A. (2017), Trace finite element methods for PDEs on surfaces, in Geometrically Unfitted Finite Element Methods and Applications, Vol. 121 of Lecture Notes in Computational Science and Engineering, Springer, pp. 211–258.10.1007/978-3-319-71431-8_7CrossRefGoogle Scholar
Olshanskii, M. A. and Safin, D. (2016), Numerical integration over implicitly defined domains for higher order unfitted finite element methods, Lobachevskii J. Math. 37, 582–596.10.1134/S1995080216050103CrossRefGoogle Scholar
Olshanskii, M. A., Quaini, A., Reusken, A. and Yushutin, V. (2018), A finite element method for the surface Stokes problem, SIAM J. Sci. Comput. 40, A2492–A2518.10.1137/18M1166183CrossRefGoogle Scholar
Olshanskii, M. A., Reusken, A. and Grande, J. (2009), A finite element method for elliptic equations on surfaces, SIAM J. Numer. Anal. 47, 3339–3358.10.1137/080717602CrossRefGoogle Scholar
Olshanskii, M. A., Reusken, A. and Xu, X. (2014), An Eulerian space–time finite element method for diffusion problems on evolving surfaces, SIAM J. Numer. Anal. 52, 1354–1377.10.1137/130918149CrossRefGoogle Scholar
Olshanskii, M. and von Wahl, H. (2024), A conservative Eulerian finite element method for transport and diffusion in moving domains. Available at arXiv:2404.07130.Google Scholar
Olshanskii, M., Palzhanov, Y. and Quaini, A. (2023), A scalar auxiliary variable unfitted FEM for the surface Cahn–Hilliard equation, J. Sci. Comput. 97, art. 57.10.1007/s10915-023-02370-8CrossRefGoogle Scholar
Olshanskii, M., Quaini, A. and Sun, Q. (2021), An unfitted finite element method for two-phase Stokes problems with slip between phases, J. Sci. Comput. 89, art. 41.10.1007/s10915-021-01658-xCrossRefGoogle Scholar
Osher, S. and Fedkiw, R. P. (2001), Level set methods: An overview and some recent results, J. Comput. Phys. 169, 463–502.10.1006/jcph.2000.6636CrossRefGoogle Scholar
Parvizian, J., Düster, A. and Rank, E. (2007), Finite cell method: h- and p-extension for embedded domain problems in solid mechanics, Comput. Mech. 41, 121–133.10.1007/s00466-007-0173-yCrossRefGoogle Scholar
Peskin, C. S. (2002), The immersed boundary method, Acta Numer. 11, 479–517.10.1017/S0962492902000077CrossRefGoogle Scholar
Pitkäranta, J. (1980), Local stability conditions for the Babuška method of Lagrange multipliers, Math. Comp. 35, 1113–1129.Google Scholar
Poluektov, M. and Figiel, Ł. (2022), A cut finite-element method for fracture and contact problems in large-deformation solid mechanics, Comput. Methods Appl. Mech. Engrg 388, art. 114234.10.1016/j.cma.2021.114234CrossRefGoogle Scholar
Preuß, J. (2018), Higher order unfitted isoparametric space–time FEM on moving domains. Master’s thesis, University of Göttingen.Google Scholar
Reusken, A. (2008), Analysis of an extended pressure finite element space for two-phase incompressible flows, Comput. Vis. Sci. 11, 293–305.10.1007/s00791-008-0099-8CrossRefGoogle Scholar
Reusken, A. (2015), Analysis of trace finite element methods for surface partial differential equations, IMA J. Numer. Anal. 35, 1568–1590.10.1093/imanum/dru047CrossRefGoogle Scholar
Ruess, M., Schillinger, D., Bazilevs, Y., Varduhn, V. and Rank, E. (2013), Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method, Int. J. Numer. Methods Engrg 95, 811–846.10.1002/nme.4522CrossRefGoogle Scholar
Saye, R. I. (2015), High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles, SIAM J. Sci. Comput. 37, A993–A1019.10.1137/140966290CrossRefGoogle Scholar
Saye, R. I. (2022), High-order quadrature on multi-component domains implicitly defined by multivariate polynomials, J. Comput. Phys. 448, art. 110720.10.1016/j.jcp.2021.110720CrossRefGoogle Scholar
Schott, B., Ager, C. and Wall, W. A. (2019), A monolithic approach to fluid–structure interaction based on a hybrid Eulerian-ALE fluid domain decomposition involving cut elements, Int. J. Numer. Methods Engrg 119, 208–237.10.1002/nme.6047CrossRefGoogle Scholar
Sethian, J. A. (2001), Evolution, implementation, and application of level set and fast marching methods for advancing fronts, J. Comput. Phys. 169, 503–555.10.1006/jcph.2000.6657CrossRefGoogle Scholar
Stein, E. M. (1970), Singular Integrals and Differentiability Properties of Functions, Vol. 30 of Princeton Mathematical Series, Princeton University Press.Google Scholar
Stenberg, R. (1995), On some techniques for approximating boundary conditions in the finite element method, J. Comput. Appl. Math. 63, 139–148.10.1016/0377-0427(95)00057-7CrossRefGoogle Scholar
Sticko, S. and Kreiss, G. (2016), A stabilized Nitsche cut element method for the wave equation, Comput. Methods Appl. Mech. Engrg 309, 364–387.10.1016/j.cma.2016.06.001CrossRefGoogle Scholar
Sticko, S. and Kreiss, G. (2019), Higher order cut finite elements for the wave equation, J. Sci. Comput. 80, 1867–1887.10.1007/s10915-019-01004-2CrossRefGoogle Scholar
Sticko, S., Ludvigsson, G. and Kreiss, G. (2020), High-order cut finite elements for the elastic wave equation, Adv. Comput. Math. 46, art. 45.10.1007/s10444-020-09785-zCrossRefGoogle Scholar
Taylor, M. E. (2023), Partial Differential Equations I: Basic Theory, Vol. 115 of Applied Mathematical Sciences, third edition, Springer.10.1007/978-3-031-33859-5CrossRefGoogle Scholar
Ngueyong, I. Tchinda and Urquiza, J. M. (2024), Fictitious domain method: A stabilized post-processing technique for boundary-flux calculation using cut elements, Comput. Methods Appl. Mech. Engrg 418, art. 116509.10.1016/j.cma.2023.116509CrossRefGoogle Scholar
Ngueyong, I. Tchinda, Urquiza, J. M. and Martin, D. (2024a), A CutFEM method for phase change problems with natural convection, Comput. Methods Appl. Mech. Engrg 420, art. 116713.10.1016/j.cma.2023.116713CrossRefGoogle Scholar
Ngueyong, I. Tchinda, Urquiza, J. M. and Martin, D. (2024b), A Nitsche-based cut finite element solver for two-phase Stefan problems, Int. J. Numer. Methods Engrg 125, art. e7408.10.1002/nme.7408CrossRefGoogle Scholar
Villanueva, C. H. and Maute, K. (2017), CutFEM topology optimization of 3D laminar incompressible flow problems, Comput. Methods Appl. Mech. Engrg 320, 444–473.10.1016/j.cma.2017.03.007CrossRefGoogle Scholar
von Wahl, H. and Richter, T. (2023), Error analysis for a parabolic PDE model problem on a coupled moving domain in a fully Eulerian framework, SIAM J. Numer. Anal. 61, 286–314.10.1137/21M1458417CrossRefGoogle Scholar
von Wahl, H., Richter, T. and Lehrenfeld, C. (2022), An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domains, IMA J. Numer. Anal. 42, 2505–2544.10.1093/imanum/drab044CrossRefGoogle Scholar
Wadbro, E., Zahedi, S., Kreiss, G. and Berggren, M. (2013), A uniformly well-conditioned, unfitted Nitsche method for interface problems, BIT Numer. Math. 53, 791–820.10.1007/s10543-012-0417-xCrossRefGoogle Scholar
Winter, M., Schott, B., Massing, A. and Wall, W. A. (2018), A Nitsche cut finite element method for the Oseen problem with general Navier boundary conditions, Comput. Methods Appl. Mech. Engrg 330, 220–252.10.1016/j.cma.2017.10.023CrossRefGoogle Scholar
Wu, H. and Xiao, Y. (2019), An unfitted hp-interface penalty finite element method for elliptic interface problems, J. Comput. Math. 37, 316–339.Google Scholar
Yang, F. (2024), The least squares finite element method for elasticity interface problem on unfitted mesh, ESAIM Math. Model. Numer. Anal. 58, 695–721.10.1051/m2an/2024015CrossRefGoogle Scholar
Yang, F. and Xie, X. (2024), An unfitted finite element method with direct extension stabilization for time-harmonic Maxwell problems on smooth domains, Adv. Comput. Math. 50, art. 51.10.1007/s10444-024-10148-1CrossRefGoogle Scholar
Zahedi, S. (2017), A space–time cut finite element method with quadrature in time, in Geometrically Unfitted Finite Element Methods and Applications, Vol. 121 of Lecture Notes in Computational Science and Engineering, Springer, pp. 281–306.10.1007/978-3-319-71431-8_9CrossRefGoogle Scholar
Zhang, K., Deng, W. and Wu, H. (2024), A CutFE-LOD method for the multiscale elliptic problems on complex domains, J. Comput. Appl. Math. 445, art. 115820.10.1016/j.cam.2024.115820CrossRefGoogle Scholar
Zhang, L., Gerstenberger, A., Wang, X. and Liu, W. K. (2004), Immersed finite element method, Comput. Methods Appl. Mech. Engrg 193, 2051–2067.10.1016/j.cma.2003.12.044CrossRefGoogle Scholar
Zonca, S., Vergara, C. and Formaggia, L. (2018), An unfitted formulation for the interaction of an incompressible fluid with a thick structure via an XFEM/DG approach, SIAM J. Sci. Comput. 40, B59–B84.10.1137/16M1097602CrossRefGoogle Scholar