Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes' problem
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 3, p. 859-874
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

We extend our results on fictitious domain methods for Poisson's problem to the case of incompressible elasticity, or Stokes' problem. The mesh is not fitted to the domain boundary. Instead boundary conditions are imposed using a stabilized Nitsche type approach. Control of the non-physical degrees of freedom, i.e., those outside the physical domain, is obtained thanks to a ghost penalty term for both velocities and pressures. Both inf-sup stable and stabilized velocity pressure pairs are considered.

DOI : https://doi.org/10.1051/m2an/2013123
Classification:  65N12,  65N30
Keywords: finite element methods, stabilized methods, penalty methods, Stokes' problem, fictitious domain
@article{M2AN_2014__48_3_859_0,
     author = {Burman, Erik and Hansbo, Peter},
     title = {Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes' problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {3},
     year = {2014},
     pages = {859-874},
     doi = {10.1051/m2an/2013123},
     mrnumber = {3264337},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_3_859_0}
}
Burman, Erik; Hansbo, Peter. Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes' problem. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 3, pp. 859-874. doi : 10.1051/m2an/2013123. http://www.numdam.org/item/M2AN_2014__48_3_859_0/

[1] S. Amdouni, K. Mansouri, Y. Renard, M. Arfaoui and M. Moakher, Numerical convergence and stability of mixed formulation with X-FEM cut-off. Eur. J. Comput. Mech. 21 (2012) 160-73.

[2] S. Amdouni, M. Moakher and Y. Renard, A local projection stabilization of fictitious domain method for elliptic boundary value problems. Preprint, hal.archives-ouvertes.fr: hal-00713115 (2012) | MR 3131863 | Zbl 1288.65152

[3] Ph. Angot, A fictitious domain model for the Stokes/Brinkman problem with jump embedded boundary conditions. C.R. Math. Acad. Sci. Paris 348 (2010) 697-702. | MR 2652501 | Zbl 1194.35317

[4] D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equation. Calcolo 21 (1984) 337-344. | MR 799997 | Zbl 0593.76039

[5] R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38 (2001) 173-199. | MR 1890352 | Zbl 1008.76036

[6] R. Becker, E. Burman and P. Hansbo, A finite element time relaxation method. C.R. Math. Acad. Sci. Paris 349 (2011) 353-356. | MR 2783334 | Zbl pre05877051

[7] R. Becker, E. Burman and P. Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3352-3360. | MR 2571349 | Zbl 1230.74169

[8] R. Becker and P. Hansbo, A simple pressure stabilization method for the Stokes equation. Commun. Numer. Methods Eng. 24 (2008) 1421-1430. | MR 2474694 | Zbl 1153.76036

[9] M. Bercovier and O. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math. 33 (1979) 211-224. | MR 549450 | Zbl 0423.65058

[10] S. Bertoluzza, M. Ismail and B. Maury, Analysis of the fully discrete fat boundary method. Numer. Math. 118 (2011) 49-77. | MR 2793902 | Zbl 1217.65201

[11] D. Boffi, F. Brezzi, L. Demkowicz, R. Durán, R. Falk and M. Fortin, Mixed finite elements, compatibility conditions, and applications. Lectures given at the C.I.M.E. Summer School held in Cetraro 2006, edited by Boffi and Lucia Gastaldi. In vol. 1939 Lect. Notes Math. Springer-Verlag, Berlin (2008). | MR 2459075 | Zbl 1182.76895

[12] F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations, in Efficient solutions of elliptic systems (Kiel, 1984), vol. 10 of Notes Numer. Fluid Mech. Vieweg, Braunschweig (1984) 11-19. | MR 804083 | Zbl 0552.76002

[13] F. Brezzi and R. Falk, Stability of higher-order Hood-Taylor methods. SIAM J. Numer. Anal. 28 (1991) 581-590. | MR 1098408 | Zbl 0731.76042

[14] E. Burman and P. Hansbo, Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62 (2012) 328-341. | MR 2899249 | Zbl pre06030364

[15] E. Burman and P. Hansbo, Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Engrg. 195 (2006) 2393-2410. | MR 2207476 | Zbl 1125.76038

[16] E. Burman, Pressure projection stabilizations for Galerkin approximations of Stokes' and Darcy's problem. Numer. Methods Part. Differ. Eqs. 24 (2008) 127-143. | MR 2371351 | Zbl 1139.76029

[17] E. Burman, Ghost penalty. C.R. Math. Acad. Sci. Paris 348 (2010) 1217-1220. | MR 2738930 | Zbl 1204.65142

[18] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 2. Functional and Variational Methods. Springer-Verlag, Berlin (1988) | MR 969367 | Zbl 0683.35001

[19] C. Dohrmann and P. Bochev, A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Methods Fluids 46 (2004) 183-201. | MR 2079895 | Zbl 1060.76569

[20] V. Girault, R. Glowinski and T. Pan, A fictitious-domain method with distributed multiplier for the Stokes problem, in Appl. Nonlinear Anal. Kluwer/Plenum, New York (1999) 159-174. | MR 1727447 | Zbl 0954.35127

[21] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput. Methods Appl. Mech. Engrg. 47 (2009) 5537-5552. | MR 1941489 | Zbl 1035.65125

[22] J. Haslinger and Y. Renard, A new fictitious domain approach inspired by the extended finite element method. SIAM J. Numer. Anal. 191 (2002) 1474-1499. | MR 2497337 | Zbl 1205.65322

[23] G. Legrain, N. Moës and A. Huerta, Stability of incompressible formulations enriched with X-FEM. Comput. Methods Appl. Mech. Engrg. 197 (2008) 1835-1849. | MR 2417162 | Zbl 1194.74426

[24] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9-15. | MR 341903 | Zbl 0229.65079