A variant of Nitsche's method

Roland Becker

doi:10.1016/j.crma.2018.11.002

Numerical analysis

A variant of Nitsche's method

Presented by: the Editorial Board

Roland Becker ¹

¹ LMAP UMR 5142, Université de Pau, 64013 Pau cedex, France

Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1236-1242.

Abstract
Résumé

We present a method that aims to reconcile Nitsche's method with the traditional finite element method ('weak' versus 'strong implementation' of essential boundary conditions). We retain the original idea of a variational formulation based on an extended energy, but replace the original boundary terms by domain terms involving weak derivatives. The solution of the proposed method coincides, for the Poisson problem, with the one of the traditional method, which in particular shows monotonicity under the standard angle condition for the Courant element. For more general second-order problems, it allows for the weighting of boundary terms inherent to Nitsche's method. This is of particular interest for singularly perturbed problems.

Nous présentons une méthode qui vise à concilier la méthode de Nitsche avec la méthode traditionnelle (implémentation « faible » versus « forte » de conditions aux limites essentielles). L'idée originelle d'une formulation faible basée sur une énergie étendue est préservée, mais les termes de bord sont remplacés par des termes sur le domaine utilisant des dérivées faibles. La solution de la méthode proposée coïncide, pour le problàme de Poisson, avec celle de la méthode traditionnelle ; cela montre, en particulier, la monotonie sous la condition d'angle maximal pour l'élément de Courant. Pour des problèmes plus généraux, notre modification permet une pondération des termes de bord comme la méthode de Nitsche. Cela est particulièrement intéressant pour les perturbations singulières.

Received: 2018-04-25
Accepted: 2018-11-06
Published online: 2018-11-01

DOI: 10.1016/j.crma.2018.11.002

Author's affiliations:

Roland Becker ¹

¹ LMAP UMR 5142, Université de Pau, 64013 Pau cedex, France

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     title = {A variant of {Nitsche's} method},
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     pages = {1236--1242},
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     doi = {10.1016/j.crma.2018.11.002},
     language = {en},
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Roland Becker. A variant of Nitsche's method. Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1236-1242. doi : 10.1016/j.crma.2018.11.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.11.002/

References
Cited by

[1] D. Arnold An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., Volume 19 (1982), pp. 742-760

[2] I. Babuška The finite element method with Lagrangian multipliers, Numer. Math., Volume 20 (1972–1973), pp. 179-192

[3] I. Babuška The finite element method with penalty, Math. Comput., Volume 27 (1973), pp. 221-228

[4] J.W. Barrett; C.M. Elliott Total flux estimates for a finite-element approximation of elliptic equations, IMA J. Numer. Anal., Volume 7 (1987), pp. 129-148

[5] R. Becker; P. Hansbo; R. Stenberg A finite element method for domain decomposition with non-matching grids, Math. Model. Numer. Anal., Volume 37 (2003), pp. 209-225

[6] F. Ben Belgacem The mortar finite element method with Lagrange multipliers, Numer. Math., Volume 84 (1999), pp. 173-197

[7] E. Burman A penalty-free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions, SIAM J. Numer. Anal., Volume 50 (2012), pp. 1959-1981

[8] E. Burman; M.A. Fernández Stabilized explicit coupling for fluid–structure interaction using Nitsche's method, C. R. Acad. Sci. Paris, Ser. I, Volume 345 (2007), pp. 467-472

[9] F. Chouly; P. Hild A Nitsche-based method for unilateral contact problems: numerical analysis, SIAM J. Numer. Anal., Volume 51 (2013), pp. 1295-1307

[10] M. Giles; M. Larson; M. Levenstam; E. Süli Adaptive Error Control for Finite Element Approximations of the Lift and Drag Coefficients in Viscous Flow, Oxford University Computing Laboratory, 1997 (Technical Report NA-76/06)

[11] A. Hansbo; P. Hansbo An unfitted finite element method, based on Nitsche's method, for elliptic interface problems, Comput. Methods Appl. Mech. Eng., Volume 191 (2002), pp. 5537-5552

[12] A. Hansbo; P. Hansbo; M.G. Larson A finite element method on composite grids based on Nitsche's method, Math. Model. Numer. Anal., Volume 37 (2003), pp. 495-514

[13] J. Nitsche Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Semin. Univ. Hamb., Volume 36 (1971), pp. 9-15

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