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Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem

Published online by Cambridge University Press:  22 April 2014

Erik Burman  and
Peter Hansbo
Erik Burman
Affiliation:
Department of Mathematics, University College London, UK-WC1E 6BT United Kingdom
Peter Hansbo
Affiliation:
Department of Mechanical Engineering, Jönköping University, 551 11 Jönköping, Sweden. [email protected] .
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Abstract

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.

Keywords

Finite element methodsstabilized methodspenalty methodsStokes’ problemfictitious domain
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 859 - 874
Copyright
© EDP Sciences, SMAI, 2014

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References

Amdouni, S., Mansouri, K., Renard, Y., Arfaoui, M. and Moakher, M., Numerical convergence and stability of mixed formulation with X-FEM cut-off. Eur. J. Comput. Mech. 21 (2012) 160–73. Google Scholar
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)
Angot, Ph., A fictitious domain model for the Stokes/Brinkman problem with jump embedded boundary conditions. C.R. Math. Acad. Sci. Paris 348 (2010) 697–702. Google Scholar
Arnold, D.N., Brezzi, F. and Fortin, M., A stable finite element for the Stokes equation. Calcolo 21 (1984) 337–344. Google Scholar
Becker, R. and Braack, M., A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38 (2001) 173–199. Google Scholar
Becker, R., Burman, E. and Hansbo, P., A finite element time relaxation method. C.R. Math. Acad. Sci. Paris 349 (2011) 353–356. Google Scholar
Becker, R., Burman, E. and Hansbo, P., A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3352–3360. Google Scholar
Becker, R. and Hansbo, P., A simple pressure stabilization method for the Stokes equation. Commun. Numer. Methods Eng. 24 (2008) 1421–1430. Google Scholar
Bercovier, M. and Pironneau, O., Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math. 33 (1979) 211–224. Google Scholar
Bertoluzza, S., Ismail, M. and Maury, B., Analysis of the fully discrete fat boundary method. Numer. Math. 118 (2011) 49–77. Google Scholar
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).
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.
Brezzi, F. and Falk, R., Stability of higher-order Hood-Taylor methods. SIAM J. Numer. Anal. 28 (1991) 581–590. Google Scholar
Burman, E. and Hansbo, P., Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62 (2012) 328–341. Google Scholar
Burman, E. and Hansbo, P., Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Engrg. 195 (2006) 2393–2410. Google Scholar
Burman, E., Pressure projection stabilizations for Galerkin approximations of Stokes’ and Darcy’s problem. Numer. Methods Part. Differ. Eqs. 24 (2008) 127–143. Google Scholar
Burman, E., Ghost penalty. C.R. Math. Acad. Sci. Paris 348 (2010) 1217–1220. Google Scholar
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)
Dohrmann, C. and Bochev, P., A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Methods Fluids 46 (2004) 183–201. Google Scholar
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.
Hansbo, A. and Hansbo, P., An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Engrg. 47 (2009) 5537–5552. Google Scholar
Haslinger, J. and Renard, Y., A new fictitious domain approach inspired by the extended finite element method. SIAM J. Numer. Anal. 191 (2002) 1474–1499. Google Scholar
Legrain, G., Moës, N. and Huerta, A., Stability of incompressible formulations enriched with X-FEM. Comput. Methods Appl. Mech. Engrg. 197 (2008) 1835–1849. Google Scholar
Nitsche, J., 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. Google Scholar