F Nous contacter

Multi angle

H 1 Combining cut element methods and hybridization

Auteurs : Burman, Erik (Auteur de la Conférence)
CIRM (Editeur )

    Loading the player...

    Résumé : Recently there has been a surge in interest in cut, or unfitted, finite element methods. In this class of methods typically the computational mesh is independent of the geometry. Interfaces and boundaries are allowed to cut through the mesh in a very general fashion. Constraints on the boundaries such as boundary or transmission conditions are typically imposed weakly using Nitsche’s method. In this talk we will discuss how these ideas can be combined in a fruitful way with the idea of hybridization, where additional degrees of freedom are added on the interfaces to further improve the decoupling of the systems, allowing for static condensation of interior unknowns. In the first part of the talk we will discuss how hybridization can be combined with the classical cut finite element method, using standard H1 -conforming finite elements in each subdomain, leading to a robust method allowing for the integration of polytopal geometries, where the subdomains are independent of the underlying mesh. This leads to a framework where it is easy to integrate multiscale features such as strongly varying coefficients, or multidimensional coupling, as in flow in fractured domains. Some examples of such applications will be given. In the second part of the talk we will focus on the Hybridized High Order Method (HHO) and show how cut techniques can be introduced in this context. The HHO is a recently introduced nonconforming method that allows for arbitrary order discretization of diffusive problems on polytopal meshes. HHO methods have hybrid unknowns, made of polynomials in the mesh elements and on the faces, without any continuity requirement. They rely on high-order local reconstructions, which are used to build consistent Galerkin contributions and appropriate stabilization terms designed to preserve the high-order approximation properties of the local reconstructions. Here we will show how cut element techniques can be introduced as a tool for the handling of (possibly curved) interfaces or boundaries that are allowed to cut through the polytopal mesh. In this context the cut element method plays the role of a local interface model, where the associated degrees of freedom are eliminated in the static condensation step. Issues of robustness and accuracy will be discussed and illustrated by some numerical examples.

    Codes MSC :
    65N30 - Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
    34A38 - Hybrid systems

    Ressources complémentaires :

      Informations sur la Vidéo

      Réalisateur : Hennenfent, Guillaume
      Langue : Anglais
      Date de publication : 28/05/2019
      Date de captation : 01/05/2019
      Collection : Research talks ; Numerical Analysis and Scientific Computing ; Partial Differential Equations
      Format : MP4
      Durée : 00:42:37
      Domaine : PDE ; Numerical Analysis & Scientific Computing
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2019-05-01_Burman.mp4

    Informations sur la rencontre

    Nom de la rencontre : POEMs - POlytopal Element Methods in Mathematics and Engineering
    Organisateurs de la rencontre : Antonietti, Paola ; Beirão da Veiga, Lourenço ; Di Pietro, Daniele ; Droniou, Jérôme ; Krell, Stella
    Dates : 29/04/2019 - 03/05/2019
    Année de la rencontre : 2019
    URL Congrès : https://conferences.cirm-math.fr/1954.html

    Citation Data

    DOI : 10.24350/CIRM.V.19528803
    Cite this video as: Burman, Erik (2019). Combining cut element methods and hybridization. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19528803
    URI : http://dx.doi.org/10.24350/CIRM.V.19528803

    Voir aussi


    1. Girault, V. & Glowinski, R. Japan J. Indust. Appl. Math. (1995) - https://doi.org/10.1007/BF03167240

    2. Silvia Bertoluzza, Mourad Ismail, Bertrand Maury. The Fat Boundary Method: Semi-Discrete Scheme and Some Numerical Experiments. DOMAIN DECOMPOSITION METHODS IN SCIENCE AND ENGINEERING, springer, pp.513, 2005, Lecture Notes in Computational Science and Engineering. - https://hal.archives-ouvertes.fr/hal-00666064

    3. HASLINGER, Jaroslav et RENARD, Yves. A new fictitious domain approach inspired by the extended finite element method. SIAM Journal on Numerical Analysis, 2009, vol. 47, no 2, p. 1474-1499. - https://epubs.siam.org/doi/pdf/10.1137/070704435

    4. BARRETT, John W. et ELLIOTT, Charles M. Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces. IMA journal of numerical analysis, 1987, vol. 7, no 3, p. 283-300. - https://doi.org/10.1093/imanum/7.3.283

    5. Anita Hansbo, Peter Hansbo. An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2002, 191 (47-48), pp. 5537-5552. - https://doi.org/10.1016/S0045-7825(02)00524-8

    6. OLSHANSKII, Maxim A., REUSKEN, Arnold, et GRANDE, Jörg. A finite element method for elliptic equations on surfaces. SIAM journal on numerical analysis, 2009, vol. 47, no 5, p. 3339-3358. - https://doi.org/10.1137/080717602

    7. BURMAN, Erik. Ghost penalty. Comptes Rendus Mathematique, 2010, vol. 348, no 21-22, p. 1217-1220. - https://doi.org/10.1016/j.crma.2010.10.006doi.org/10.1016/j.crma.2010.10.006

    8. LARSON, Mats G. et ZAHEDI, Sara. Stabilization of high order cut finite element methods on surfaces. arXiv preprint arXiv:1710.03343, 2017. - https://arxiv.org/abs/1710.03343

    9. Daniele Di Pietro, Alexandre Ern. Equilibrated tractions for the Hybrid High-Order method. Comptes Rendus Mathématique, Elsevier Masson, 2015, 353 (3), pp.279--282. - https://doi.org/10.1016/j.crma.2014.12.009

    10. JOHANSSON, August et LARSON, Mats G. A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary. Numerische Mathematik, 2013, vol. 123, no 4, p. 607-628. - https://doi.org/10.1007/s00211-012-0497-1IS

    11. BASTIAN, Peter et ENGWER, Christian. An unfitted finite element method using discontinuous Galerkin. International journal for numerical methods in engineering, 2009, vol. 79, no 12, p. 1557-1576. - https://doi.org/10.1002/nme.2631ISTE

    12. MASSJUNG, Ralf. An unfitted discontinuous Galerkin method applied to elliptic interface problems. SIAM Journal on Numerical Analysis, 2012, vol. 50, no 6, p. 3134-3162. - https://doi.org/10.1137/090763093

    13. GÜRKAN, Ceren et MASSING, André. A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems. Computer Methods in Applied Mechanics and Engineering, 2019, vol. 348, p. 466-499. - https://arxiv.org/abs/1803.06635

    14. CANGIANI, Andrea, GEORGOULIS, Emmanuil, et SABAWI, Younis. Adaptive discontinuous Galerkin methods for elliptic interface problems. Mathematics of Computation, 2018, vol. 87, no 314, p. 2675-2707. - https://doi.org/10.1090/mcom/3322

    15. COCKBURN, Bernardo, QIU, Weifeng, et SOLANO, Manuel. A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity. Mathematics of Computation, 2014, vol. 83, no 286, p. 665-699. - https://doi.org/10.1090/S0025-5718-2013-02747-0

    16. GÜRKAN, Ceren, SALA-LARDIES, Esther, KRONBICHLER, Martin, et al. eXtended hybridizable discontinuous Galerkin (X-HDG) for void and bimaterial problems. In : Advances in Discretization Methods. Springer, Cham, 2016. p. 103-122. - https://doi.org/10.1007/978-3-319-41246-7_5

    17. Anita Hansbo, Peter Hansbo. An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2002, 191 (47-48), pp. 5537-5552. - https://doi.org/10.1016/S0045-7825(02)00524-8

    18. BURMAN, Erik et ERN, Alexandre. An unfitted Hybrid High-Order method for elliptic interface problems. SIAM Journal on Numerical Analysis, 2018, vol. 56, no 3, p. 1525-1546. - https://doi.org/10.1137/17M1154266

    19. BURMAN, Erik, GUZMÁN, Johnny, SÁNCHEZ, Manuel A., et al. Robust flux error estimation of Nitsche’s method for high contrast interface problems. arXiv preprint arXiv:1602.00603, 2016, vol. 48. - https://arxiv.org/abs/1602.00603

    20. HUYNH, L. N. T., NGUYEN, N. C., PERAIRE, J., et al. A high‐order hybridizable discontinuous Galerkin method for elliptic interface problems. International Journal for Numerical Methods in Engineering, 2013, vol. 93, no 2, p. 183-200. - https://doi.org/10.1002/nme.4382