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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

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