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Journal articles on the topic 'Unfitted Finite Elements'

Author: Grafiati

Published: 4 June 2025

Last updated: 15 July 2025

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1

Formaggia, Luca, Christian Vergara, and Stefano Zonca. "Unfitted extended finite elements for composite grids." Computers & Mathematics with Applications 76, no. 4 (2018): 893–904. http://dx.doi.org/10.1016/j.camwa.2018.05.028.

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2

Guzmán, Johnny, and Maxim Olshanskii. "Inf-sup stability of geometrically unfitted Stokes finite elements." Mathematics of Computation 87, no. 313 (2017): 2091–112. http://dx.doi.org/10.1090/mcom/3288.

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3

Badia, Santiago, Pere A. Martorell, and Francesc Verdugo. "Geometrical discretisations for unfitted finite elements on explicit boundary representations." Journal of Computational Physics 460 (July 2022): 111162. http://dx.doi.org/10.1016/j.jcp.2022.111162.

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4

Badia, Santiago, Pere A. Martorell, and Francesc Verdugo. "Space–time unfitted finite elements on moving explicit geometry representations." Computer Methods in Applied Mechanics and Engineering 428 (August 2024): 117091. http://dx.doi.org/10.1016/j.cma.2024.117091.

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5

Dziuk, Gerhard, and Charles M. Elliott. "Finite element methods for surface PDEs." Acta Numerica 22 (April 2, 2013): 289–396. http://dx.doi.org/10.1017/s0962492913000056.

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In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples.

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6

Badia, Santiago, Eric Neiva, and Francesc Verdugo. "Robust high-order unfitted finite elements by interpolation-based discrete extension." Computers & Mathematics with Applications 127 (December 2022): 105–26. http://dx.doi.org/10.1016/j.camwa.2022.09.027.

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7

Burman, Erik, Peter Hansbo, Mats G. Larson, and Sara Zahedi. "Cut finite element methods." Acta Numerica 34 (July 2025): 1–121. https://doi.org/10.1017/s0962492925000017.

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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.

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8

Burman, Erik, and Janosch Preuss. "Unique continuation for an elliptic interface problem using unfitted isoparametric finite elements." SMAI Journal of computational mathematics 11 (March 24, 2025): 165–202. https://doi.org/10.5802/smai-jcm.122.

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9

Wegert, Zachary J., Jordi Manyer, Connor N. Mallon, Santiago Badia, and Vivien J. Challis. "Level-set topology optimisation with unfitted finite elements and automatic shape differentiation." Computer Methods in Applied Mechanics and Engineering 445 (October 2025): 118203. https://doi.org/10.1016/j.cma.2025.118203.

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10

Neiva, Eric, and Santiago Badia. "Robust and scalable h-adaptive aggregated unfitted finite elements for interface elliptic problems." Computer Methods in Applied Mechanics and Engineering 380 (July 2021): 113769. http://dx.doi.org/10.1016/j.cma.2021.113769.

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11

Cattaneo, Laura, Luca Formaggia, Guido Francesco Iori, Anna Scotti, and Paolo Zunino. "Stabilized extended finite elements for the approximation of saddle point problems with unfitted interfaces." Calcolo 52, no. 2 (2014): 123–52. http://dx.doi.org/10.1007/s10092-014-0109-9.

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12

Gahima, Stephan, Pedro Díez, Marco Stefanati, José Félix Rodríguez Matas, and Alberto García-González. "An Unfitted Method with Elastic Bed Boundary Conditions for the Analysis of Heterogeneous Arterial Sections." Mathematics 11, no. 7 (2023): 1748. http://dx.doi.org/10.3390/math11071748.

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This manuscript presents a novel formulation for a linear elastic model of a heterogeneous arterial section undergoing uniform pressure in a quasi-static regime. The novelties are twofold. First, an elastic bed support on the external boundary (elastic bed boundary condition) replaces the classical Dirichlet boundary condition (i.e., blocking displacements at arbitrarily selected nodes) for elastic solids to ensure a solvable problem. In addition, this modeling approach can be used to effectively account for the effect of the surrounding material on the vessel. Secondly, to study many geometrical configurations corresponding to different patients, we devise an unfitted strategy based on the Immersed Boundary (IB) framework. It allows using the same (background) mesh for all possible configurations both to describe the geometrical features of the cross-section (using level sets) and to compute the solution of the mechanical problem. Results on coronary arterial sections from realistic segmented images demonstrate that the proposed unfitted IB-based approach provides results equivalent to the standard finite elements (FE) for the same number of active degrees of freedom with an average difference in the displacement field of less than 0.5%. However, the proposed methodology does not require the use of a different mesh for every configuration. Thus, it is paving the way for dimensionality reduction.

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13

Jiang, Ying, Minghui Nian, and Qinghui Zhang. "A Stable Generalized Finite Element Method Coupled with Deep Neural Network for Interface Problems with Discontinuities." Axioms 11, no. 8 (2022): 384. http://dx.doi.org/10.3390/axioms11080384.

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The stable generalized finite element method (SGFEM) is an improved version of generalized or extended FEM (GFEM/XFEM), which (i) uses simple and unfitted meshes, (ii) reaches optimal convergence orders, and (iii) is stable and robust in the sense that conditioning is of the same order as that of FEM and does not get bad as interfaces approach boundaries of elements. This paper designs the SGFEM for the discontinuous interface problem (DIP) by coupling a deep neural network (DNN). The main idea is to construct a function using the DNN, which captures the discontinuous interface condition, and transform the DIP to an (approximately) equivalent continuous interface problem (CIP) based on the DNN function such that the SGFEM for CIPs can be applied. The SGFEM for the DIP is a conforming method that maintains the features (i)–(iii) of SGFEM and is free from penalty terms. The approximation error of the proposed SGFEM is analyzed mathematically, which is split into an error of SGFEM of the CIP and a learning error of the DNN. The learning dimension of DNN is one dimension less than that of the domain and can be implemented efficiently. It is known that the DNN enjoys advantages in nonlinear approximations and high-dimensional problems. Therefore, the proposed SGFEM coupled with the DNN has great potential in the high-dimensional interface problem with interfaces of complex geometries. Numerical experiments verify the efficiency and optimal convergence of the proposed method.

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14

Auricchio, F., D. Boffi, L. Gastaldi, A. Lefieux, and A. Reali. "A study on unfitted 1D finite element methods." Computers & Mathematics with Applications 68, no. 12 (2014): 2080–102. http://dx.doi.org/10.1016/j.camwa.2014.08.018.

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15

Bastian, Peter, and Christian Engwer. "An unfitted finite element method using discontinuous Galerkin." International Journal for Numerical Methods in Engineering 79, no. 12 (2009): 1557–76. http://dx.doi.org/10.1002/nme.2631.

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16

Lehrenfeld, Christoph, and Arnold Reusken. "L2-error analysis of an isoparametric unfitted finite element method for elliptic interface problems." Journal of Numerical Mathematics 27, no. 2 (2019): 85–99. http://dx.doi.org/10.1515/jnma-2017-0109.

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AbstractIn the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. In the paper [C. Lehrenfeld and A. Reusken,IMA J. Numer. Anal.38(2018), No. 3, 1351–1387] ana priorierror analysis of the method applied to an interface problem is presented. The analysis reveals optimal order discretization error bounds in theH1-norm. In this paper we extend this analysis and derive optimalL2-error bounds.

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17

Mifune, Takeshi. "Interface Homogenization Technique for Unfitted Finite Element Electromagnetic Analysis." IEEE Transactions on Magnetics 51, no. 3 (2015): 1–4. http://dx.doi.org/10.1109/tmag.2014.2361321.

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18

Badia, Santiago, Francesc Verdugo, and Alberto F. Martín. "The aggregated unfitted finite element method for elliptic problems." Computer Methods in Applied Mechanics and Engineering 336 (July 2018): 533–53. http://dx.doi.org/10.1016/j.cma.2018.03.022.

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19

Bergbauer, Maximilian, Peter Munch, Wolfgang A. Wall, and Martin Kronbichler. "High-Performance Matrix-Free Unfitted Finite Element Operator Evaluation." SIAM Journal on Scientific Computing 47, no. 3 (2025): B665—B689. https://doi.org/10.1137/24m1653689.

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20

He, Cuiyu, and Xu Zhang. "Residual-Based a Posteriori Error Estimation for Immersed Finite Element Methods." Journal of Scientific Computing 81, no. 3 (2019): 2051–79. http://dx.doi.org/10.1007/s10915-019-01071-5.

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Abstract In this paper we introduce and analyze the residual-based a posteriori error estimation of the partially penalized immersed finite element method for solving elliptic interface problems. The immersed finite element method can be naturally utilized on interface-unfitted meshes. Our a posteriori error estimate is proved to be both reliable and efficient with both reliability and efficiency constants independent of the location of the interface. Numerical results indicate that the error estimation is robust with respect to the coefficient contrast.

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21

Liu, Zhijun, Yimin Zhang, Yao Jiang, Han Yang, and Yongtao Yang. "Unfitted finite element method for fully coupled poroelasticity with stabilization." Computer Methods in Applied Mechanics and Engineering 397 (July 2022): 115132. http://dx.doi.org/10.1016/j.cma.2022.115132.

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22

Verdugo, Francesc, Alberto F. Martín, and Santiago Badia. "Distributed-memory parallelization of the aggregated unfitted finite element method." Computer Methods in Applied Mechanics and Engineering 357 (December 2019): 112583. http://dx.doi.org/10.1016/j.cma.2019.112583.

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23

Martorell, Pere A., and Santiago Badia. "High order unfitted finite element discretizations for explicit boundary representations." Journal of Computational Physics 511 (August 2024): 113127. http://dx.doi.org/10.1016/j.jcp.2024.113127.

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24

Lou, Yimin, and Christoph Lehrenfeld. "Isoparametric Unfitted BDF--Finite Element Method for PDEs on Evolving Domains." SIAM Journal on Numerical Analysis 60, no. 4 (2022): 2069–98. http://dx.doi.org/10.1137/21m142126x.

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25

Lehrenfeld, Christoph, Fabian Heimann, Janosch Preuß, and Henry von Wahl. "ngsxfem: Add-on to NGSolve for geometrically unfitted finite element discretizations." Journal of Open Source Software 6, no. 64 (2021): 3237. http://dx.doi.org/10.21105/joss.03237.

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26

Huang, Peiqi, Haijun Wu, and Yuanming Xiao. "An unfitted interface penalty finite element method for elliptic interface problems." Computer Methods in Applied Mechanics and Engineering 323 (August 2017): 439–60. http://dx.doi.org/10.1016/j.cma.2017.06.004.

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27

Sinha, R. K., and B. Deka. "An unfitted finite-element method for elliptic and parabolic interface problems." IMA Journal of Numerical Analysis 27, no. 3 (2006): 529–49. http://dx.doi.org/10.1093/imanum/drl029.

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28

Legrain, Grégory. "A NURBS enhanced extended finite element approach for unfitted CAD analysis." Computational Mechanics 52, no. 4 (2013): 913–29. http://dx.doi.org/10.1007/s00466-013-0854-7.

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29

Badia, Santiago, and Francesc Verdugo. "Robust and scalable domain decomposition solvers for unfitted finite element methods." Journal of Computational and Applied Mathematics 344 (December 2018): 740–59. http://dx.doi.org/10.1016/j.cam.2017.09.034.

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30

Lehrenfeld, Christoph, and Maxim Olshanskii. "An Eulerian finite element method for PDEs in time-dependent domains." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 2 (2019): 585–614. http://dx.doi.org/10.1051/m2an/2018068.

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The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in a triangulated computational domain and can overlap the time-independent background mesh in an arbitrary way. The numerical method is based on finite difference discretizations of time derivatives and a standard geometrically unfitted finite element method with an additional stabilization term in the spatial domain. The performance and analysis of the method rely on the fundamental extension result in Sobolev spaces for functions defined on bounded domains. This paper includes a complete stability and error analysis, which accounts for discretization errors resulting from finite difference and finite element approximations as well as for geometric errors coming from a possible approximate recovery of the physical domain. Several numerical examples illustrate the theory and demonstrate the practical efficiency of the method.

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31

Olshanskii, Maxim, Annalisa Quaini, and Qi Sun. "A Finite Element Method for Two-Phase Flow with Material Viscous Interface." Computational Methods in Applied Mathematics 22, no. 2 (2021): 443–64. http://dx.doi.org/10.1515/cmam-2021-0185.

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Abstract This paper studies a model of two-phase flow with an immersed material viscous interface and a finite element method for the numerical solution of the resulting system of PDEs. The interaction between the bulk and surface media is characterized by no-penetration and slip with friction interface conditions. The system is shown to be dissipative, and a model stationary problem is proved to be well-posed. The finite element method applied in this paper belongs to a family of unfitted discretizations. The performance of the method when model and discretization parameters vary is assessed. Moreover, an iterative procedure based on the splitting of the system into bulk and surface problems is introduced and studied numerically.

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32

Xiao, Haijun Wu and Yuanming. "An Unfitted hp-Interface Penalty Finite Element Method for Elliptic Interface Problems." Journal of Computational Mathematics 37, no. 3 (2019): 316–39. http://dx.doi.org/10.4208/jcm.1802-m2017-0219.

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33

Yang, Chaochao. "An Interface-Unfitted Finite Element Method for Elliptic Interface Optimal Control Problems." Numerical Mathematics: Theory, Methods and Applications 12, no. 3 (2019): 727–49. http://dx.doi.org/10.4208/nmtma.oa-2018-0031.

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34

Chen, Zhiming, Ke Li, and Xueshuang Xiang. "An adaptive high-order unfitted finite element method for elliptic interface problems." Numerische Mathematik 149, no. 3 (2021): 507–48. http://dx.doi.org/10.1007/s00211-021-01243-2.

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35

Badia, Santiago, Alberto F. Martín, Eric Neiva, and Francesc Verdugo. "The Aggregated Unfitted Finite Element Method on Parallel Tree-Based Adaptive Meshes." SIAM Journal on Scientific Computing 43, no. 3 (2021): C203—C234. http://dx.doi.org/10.1137/20m1344512.

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36

Laymuns, Genaro, and Manuel A. Sánchez. "Corrected finite element methods on unfitted meshes for Stokes moving interface problem." Computers & Mathematics with Applications 108 (February 2022): 159–74. http://dx.doi.org/10.1016/j.camwa.2021.12.018.

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37

Deckelnick, Klaus, Charles M. Elliott, and Thomas Ranner. "Unfitted Finite Element Methods Using Bulk Meshes for Surface Partial Differential Equations." SIAM Journal on Numerical Analysis 52, no. 4 (2014): 2137–62. http://dx.doi.org/10.1137/130948641.

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38

BARRETT, JOHN W., and CHARLES M. ELLIOTT. "Fitted and Unfitted Finite-Element Methods for Elliptic Equations with Smooth Interfaces." IMA Journal of Numerical Analysis 7, no. 3 (1987): 283–300. http://dx.doi.org/10.1093/imanum/7.3.283.

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39

Wang, Qiuliang, and Jinru Chen. "A new unfitted stabilized Nitsche’s finite element method for Stokes interface problems." Computers & Mathematics with Applications 70, no. 5 (2015): 820–34. http://dx.doi.org/10.1016/j.camwa.2015.05.024.

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40

Nürnberg, Robert, and Andrea Sacconi. "An unfitted finite element method for the numerical approximation of void electromigration." Journal of Computational and Applied Mathematics 270 (November 2014): 531–44. http://dx.doi.org/10.1016/j.cam.2013.11.023.

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41

Ludescher, Thomas, Sven Gross, and Arnold Reusken. "A Multigrid Method for Unfitted Finite Element Discretizations of Elliptic Interface Problems." SIAM Journal on Scientific Computing 42, no. 1 (2020): A318—A342. http://dx.doi.org/10.1137/18m1203353.

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42

He, Xiaoxiao. "Superconvergence of a Nonconforming Interface Penalty Finite Element Method for Elliptic Interface Problems." Axioms 14, no. 5 (2025): 364. https://doi.org/10.3390/axioms14050364.

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In our previous works, we developed the superconvergence of a nonconforming finite element method based on unfitted meshes for an elliptic interface problem and elliptic problem, respectively. In this paper, a nonconforming interface penalty finite element method (NIPFEM) based on body-fitted meshes is explored for elliptic interface problems, which allows us to use different meshes in different sub-domains separated by the interface. A nonconforming finite element method based on rectangular meshes is studied and the supercloseness property between the gradient of the numerical solution and the gradient of the interpolation of the exact solution is proven for both symmetric NIPFEM and nonsymmetric NIPFEM. Then, the global superconvergence rate O(hi32) between the postprocessed numerical solution of NIPFEM and the exact solution is derived by using an interpolation postprocessing technique. Numerical examples are carried out to demonstrate the theoretical results.

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43

Wang, Tao, Chaochao Yang, and Xiaoping Xie. "A Nitsche-eXtended Finite Element Method for Distributed Optimal Control Problems of Elliptic Interface Equations." Computational Methods in Applied Mathematics 20, no. 2 (2020): 379–93. http://dx.doi.org/10.1515/cmam-2018-0256.

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AbstractThis paper analyzes an interface-unfitted numerical method for distributed optimal control problems governed by elliptic interface equations. We follow the variational discretization concept to discretize the optimal control problems and apply a Nitsche-eXtended finite element method to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around the interface are enriched into the standard linear element space. Optimal error estimates of the state, co-state and control in a mesh-dependent norm and the {L^{2}} norm are derived. Numerical results are provided to verify the theoretical results.

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44

Badia, Santiago, Manuel A. Caicedo, Alberto F. Martín, and Javier Principe. "A robust and scalable unfitted adaptive finite element framework for nonlinear solid mechanics." Computer Methods in Applied Mechanics and Engineering 386 (December 2021): 114093. http://dx.doi.org/10.1016/j.cma.2021.114093.

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45

Olshanskii, M. A., and D. Safin. "Numerical integration over implicitly defined domains for higher order unfitted finite element methods." Lobachevskii Journal of Mathematics 37, no. 5 (2016): 582–96. http://dx.doi.org/10.1134/s1995080216050103.

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46

Chen, Zhiming, Ke Li, Maohui Lyu null, and Xueshaung Xiang. "A High Order Unfitted Finite Element Method for Time-Harmonic Maxwell Interface Problems." International Journal of Numerical Analysis and Modeling 21, no. 6 (2024): 822–49. http://dx.doi.org/10.4208/ijnam2024-1033.

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47

Suvin, V. S., M. Arrutselvi, Ean Tat Ooi, Chongmin Song, and Sundararajan Natarajan. "Fitted meshes on an unfitted grid based on scaled boundary finite element analysis." Engineering Analysis with Boundary Elements 166 (September 2024): 105844. http://dx.doi.org/10.1016/j.enganabound.2024.105844.

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48

Lehrenfeld, Christoph. "High order unfitted finite element methods on level set domains using isoparametric mappings." Computer Methods in Applied Mechanics and Engineering 300 (March 2016): 716–33. http://dx.doi.org/10.1016/j.cma.2015.12.005.

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49

Lehrenfeld, Christoph, and Arnold Reusken. "Analysis of a high-order unfitted finite element method for elliptic interface problems." IMA Journal of Numerical Analysis 38, no. 3 (2017): 1351–87. http://dx.doi.org/10.1093/imanum/drx041.

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50

Hansbo, Anita, and Peter Hansbo. "An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems." Computer Methods in Applied Mechanics and Engineering 191, no. 47-48 (2002): 5537–52. http://dx.doi.org/10.1016/s0045-7825(02)00524-8.

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