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.
Mots-clés : 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 },
pages = {859--874},
publisher = {EDP-Sciences},
volume = {48},
number = {3},
year = {2014},
doi = {10.1051/m2an/2013123},
mrnumber = {3264337},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2013123/}
}
TY - JOUR AU - Burman, Erik AU - Hansbo, Peter TI - Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes' problem JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 859 EP - 874 VL - 48 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013123/ DO - 10.1051/m2an/2013123 LA - en ID - M2AN_2014__48_3_859_0 ER -
%0 Journal Article %A Burman, Erik %A Hansbo, Peter %T Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes' problem %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 859-874 %V 48 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013123/ %R 10.1051/m2an/2013123 %G en %F 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 , Tome 48 (2014) no. 3, pp. 859-874. doi : 10.1051/m2an/2013123. http://www.numdam.org/articles/10.1051/m2an/2013123/
[1] , , , and , Numerical convergence and stability of mixed formulation with X-FEM cut-off. Eur. J. Comput. Mech. 21 (2012) 160-73.
[2] , and , A local projection stabilization of fictitious domain method for elliptic boundary value problems. Preprint, hal.archives-ouvertes.fr: hal-00713115 (2012) | MR | Zbl
[3] , 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 | Zbl
[4] , and , A stable finite element for the Stokes equation. Calcolo 21 (1984) 337-344. | MR | Zbl
[5] and , A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38 (2001) 173-199. | MR | Zbl
[6] , and , A finite element time relaxation method. C.R. Math. Acad. Sci. Paris 349 (2011) 353-356. | MR
[7] , and , A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3352-3360. | MR | Zbl
[8] and , A simple pressure stabilization method for the Stokes equation. Commun. Numer. Methods Eng. 24 (2008) 1421-1430. | MR | Zbl
[9] and , Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math. 33 (1979) 211-224. | MR | Zbl
[10] , and , Analysis of the fully discrete fat boundary method. Numer. Math. 118 (2011) 49-77. | MR | Zbl
[11] , , , , and , 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 | Zbl
[12] and , 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 | Zbl
[13] and , Stability of higher-order Hood-Taylor methods. SIAM J. Numer. Anal. 28 (1991) 581-590. | MR | Zbl
[14] and , Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62 (2012) 328-341. | MR
[15] and , Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Engrg. 195 (2006) 2393-2410. | MR | Zbl
[16] , Pressure projection stabilizations for Galerkin approximations of Stokes' and Darcy's problem. Numer. Methods Part. Differ. Eqs. 24 (2008) 127-143. | MR | Zbl
[17] , Ghost penalty. C.R. Math. Acad. Sci. Paris 348 (2010) 1217-1220. | MR | Zbl
[18] and , Mathematical Analysis and Numerical Methods for Science and Technology, vol. 2. Functional and Variational Methods. Springer-Verlag, Berlin (1988) | MR | Zbl
[19] and , A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Methods Fluids 46 (2004) 183-201. | MR | Zbl
[20] , and , A fictitious-domain method with distributed multiplier for the Stokes problem, in Appl. Nonlinear Anal. Kluwer/Plenum, New York (1999) 159-174. | MR | Zbl
[21] and , An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput. Methods Appl. Mech. Engrg. 47 (2009) 5537-5552. | MR | Zbl
[22] and , A new fictitious domain approach inspired by the extended finite element method. SIAM J. Numer. Anal. 191 (2002) 1474-1499. | MR | Zbl
[23] , and , Stability of incompressible formulations enriched with X-FEM. Comput. Methods Appl. Mech. Engrg. 197 (2008) 1835-1849. | MR | Zbl
[24] , 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 | Zbl
Cité par Sources :
