Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2149-2185.

In this paper we study the Brinkman model as a unified framework to allow the transition between the Darcy and the Stokes problems. We propose an unconditionally stable low-order finite element approach, which is robust with respect to the whole range of physical parameters, and is based on the combination of stabilized equal-order finite elements with a non-symmetric penalty-free Nitsche method for the weak imposition of essential boundary conditions. In particular, we study the properties of the penalty-free Nitsche formulation for the Brinkman setting, extending a recently reported analysis for the case of incompressible elasticity (Boiveau and Burman, IMA J. Numer. Anal. 36 (2016) 770-795). Focusing on the two-dimensional case, we obtain optimal a priori error estimates in a mesh-dependent norm, which, converging to natural norms in the cases of Stokes or Darcy ows, allows to extend the results also to these limits. Moreover, we show that, in order to obtain robust estimates also in the Darcy limit, the formulation shall be equipped with a Grad-Div stabilization and an additional stabilization to control the discontinuities of the normal velocity along the boundary. The conclusions of the analysis are supported by numerical simulations.

DOI : 10.1051/m2an/2018063
Classification : 65N30, 65N12, 65N15
Mots-clés : Brinkman problem, penalty-free Nitsche method, weak boundary conditions, stabilized finite elements
Blank, Laura 1 ; Caiazzo, Alfonso 1 ; Chouly, Franz 1 ; Lozinski, Alexei 1 ; Mura, Joaquin 1

1
@article{M2AN_2018__52_6_2149_0,
     author = {Blank, Laura and Caiazzo, Alfonso and Chouly, Franz and Lozinski, Alexei and Mura, Joaquin},
     title = {Analysis of a stabilized penalty-free {Nitsche} method for the {Brinkman,} {Stokes,} and {Darcy} problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {2149--2185},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {6},
     year = {2018},
     doi = {10.1051/m2an/2018063},
     zbl = {1417.65196},
     mrnumber = {3905192},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2018063/}
}
TY  - JOUR
AU  - Blank, Laura
AU  - Caiazzo, Alfonso
AU  - Chouly, Franz
AU  - Lozinski, Alexei
AU  - Mura, Joaquin
TI  - Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2018
SP  - 2149
EP  - 2185
VL  - 52
IS  - 6
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/m2an/2018063/
DO  - 10.1051/m2an/2018063
LA  - en
ID  - M2AN_2018__52_6_2149_0
ER  - 
%0 Journal Article
%A Blank, Laura
%A Caiazzo, Alfonso
%A Chouly, Franz
%A Lozinski, Alexei
%A Mura, Joaquin
%T Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2018
%P 2149-2185
%V 52
%N 6
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2018063/
%R 10.1051/m2an/2018063
%G en
%F M2AN_2018__52_6_2149_0
Blank, Laura; Caiazzo, Alfonso; Chouly, Franz; Lozinski, Alexei; Mura, Joaquin. Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2149-2185. doi : 10.1051/m2an/2018063. https://www.numdam.org/articles/10.1051/m2an/2018063/

[1] R.A. Adams, Sobolev Spaces. In Vol. 65 of Pure and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975). | MR | Zbl

[2] I. Babuška, The finite element method with Lagrangian multipliers. Numer. Math. 20 (1972/1973) 179–192. | DOI | MR | Zbl

[3] S. Badia and R. Codina, Unified stabilized finite element formulations for the Stokes and the Darcy problems. SIAM J. Numer. Anal. 47 (2009) 1971–2000. | DOI | MR | Zbl

[4] T. Boiveau, Penalty-free Nitsche method for interface problems in computational mechanics, Ph.D. thesis, University College London (2016).

[5] T. Boiveau and E. Burman, A penalty-free Nitsche method for the weak imposition of boundary conditions in compressible and incompressible elasticity. IMA J. Numer. Anal. 36 (2016) 770–795. | DOI | MR | Zbl

[6] S.C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd edition. In Vol. 15 of Texts in Applied Mathematics. Springer, New York, NY (2008). | MR | Zbl

[7] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 8 (1974) 129–151. | Numdam | MR | Zbl

[8] F. Brezzi and Pitkäranta J., On the stabilization of finite element approximations of the Stokes equations. In: Efficient Solutions of Elliptic Systems (Kiel, 1984). In Vol. 10 of Notes Numer. Fluid Mech. Friedr. Vieweg, Braunschweig (1984) 11–19. | DOI | MR | Zbl

[9] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. In Vol. 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York, NY (1991). | DOI | MR | Zbl

[10] H.C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Flow Turbul. Combust. 1 (1949) 27. | DOI | Zbl

[11] E. Burman, A penalty-free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions. SIAM J. Numer. Anal. 50 (2012) 1959–1981. | DOI | MR | Zbl

[12] E. Burman and P. Hansbo, A unified stabilized method for Stokes’ and Darcy’s equations. J. Comput. Appl. Math. 198 (2007) 35–51. | DOI | MR | Zbl

[13] P.G. Ciarlet, The finite element method for elliptic problems. Reprint of the 1978 original [North-Holland, Amsterdam]. In Vol. 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). | MR | Zbl

[14] P. Clément, Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975) 77–84. | Numdam | MR | Zbl

[15] C. D’Angelo and P. Zunino, Robust numerical approximation of coupled Stokes’ and Darcy’s flows applied to vascular hemodynamics and biochemical transport. ESAIM: M2AN 45 (2011) 447–476. | DOI | Numdam | MR | Zbl

[16] M. Discacciati, Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows, Ph.D. thesis, EPFL, Lausanne)2004).

[17] J. Douglas Jr and J.P. Wang, An absolutely stabilized finite element method for the Stokes problem. Math. Comp. 52 (1989) 495–508. | DOI | MR | Zbl

[18] A. Ern and J.-L. Guermond, Theory and practice of finite elements. In Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York, NY (2004). | DOI | MR | Zbl

[19] L.P. Franca and T.J.R. Hughes, Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Eng. 69 (1988) 89–129. | DOI | MR | Zbl

[20] Freund J. and Stenberg R., On weakly imposed boundary conditions for second order problems. In: Proceedings of the international Conference on Finite Elements in Fluids – New trends and Applications, Venezia (1995).

[21] V. Girault and P.-A. Raviart, Finite element methods for Navier–Stokes equations, Theory and Algorithms. In Vol. 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986). | DOI | MR | Zbl

[22] A. Hannukainen, M. Juntunen, R. Stenberg, Computations with finite element methods for the Brinkman problem. Comput. Geosci. 15 (2011) 155–166. | DOI | Zbl

[23] P. Hansbo and M. Juntunen, Weakly imposed Dirichlet boundary conditions for the Brinkman model of porous media flow. Appl. Numer. Math. 59 (2009) 1274–1289. | DOI | MR | Zbl

[24] Q. Hu, F. Chouly, P. Hu, G. Cheng, S.P.A. Bordas, Skew-symmetric Nitsches formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact. Comput. Methods Appl. Mech. Eng. 341 (2018) 188–220. | DOI | MR | Zbl

[25] T.J.R. Hughes and L.P. Franca, A new finite sed bouelement formulation for computational fluid dynamics. VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces. Comput. Methods Appl. Mech. Eng. 65 (1987) 85–96. | DOI | MR | Zbl

[26] E.W. Jenkins, V. John, A. Linke, L.G. Rebholz, On the parameter choice in grad-div stabilization for the Stokes equations. Adv. Comput. Math. 40 (2014) 491–516. | DOI | MR | Zbl

[27] M. Juntunen and R. Stenberg, Nitsche’s method for general boundary conditions. Math. Comput. 78 (2009) 1353–1374. | DOI | MR | Zbl

[28] M. Juntunen and R. Stenberg, Analysis of finite element methods for the Brinkman problem. Calcolo 47 (2010) 129–147. | DOI | MR | Zbl

[29] K.A. Mardal, X.-C. Tai, R. Winther, A robust finite element method for Darcy-Stokes ow. SIAM J. Numer. Anal. 40 (2002) 1605–1631. | DOI | MR | Zbl

[30] A. Masud and T.J.R. Hughes, A stabilized mixed finite element method for Darcy flow. Comput. Methods Appl. Mech. Eng. 191 (2002) 4341–4370. | DOI | MR | Zbl

[31] J. Nitsche, über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Collection of articles dedicated to Lothar Collatz on his sixtieth birthday. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15. | DOI | MR | Zbl

[32] L. Ridgway Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. | DOI | MR | Zbl

[33] R. Stenberg, On some techniques for approximating boundary conditions in the finite element method. International Symposium on Mathematical Modelling and Computational Methods Modelling 94 (Prague, 1994). J. Comput. Appl. Math. 63 (1995) 139–148. | DOI | MR | Zbl

[34] V. Thomée, Galerkin finite element methods for parabolic problems, 2nd edition. In Vol. 25 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (2006). | MR | Zbl

[35] U. Wilbrandt, C. Bartsch, N. Ahmed, N. Alia, F. Anker, L. Blank, A. Caiazzo, S. Ganesan, S. Giere, G. Matthies, R. Meesala, A. Shamim, J. Venkatesan, V. John, ParMooN – A modernized program package based on mapped finite elements. Comput. Math. Appl. 74 (2017) 74–88. | DOI | MR | Zbl

  • Dupont, Todd; Guzmán, Johnny; Scott, L. Ridgway Obtaining higher-order Galerkin accuracy when the boundary is polygonally approximated, Journal of Numerical Mathematics, Volume 33 (2025) no. 1, p. 55 | DOI:10.1515/jnma-2023-0135
  • Gerosa, Fannie M.; Marsden, Alison L. A mechanically consistent unified formulation for fluid-porous-structure-contact interaction, Computer Methods in Applied Mechanics and Engineering, Volume 425 (2024), p. 116942 | DOI:10.1016/j.cma.2024.116942
  • Araya, Rodolfo; Caiazzo, Alfonso; Chouly, Franz Stokes problem with slip boundary conditions using stabilized finite elements combined with Nitsche, Computer Methods in Applied Mechanics and Engineering, Volume 427 (2024), p. 117037 | DOI:10.1016/j.cma.2024.117037
  • Araya, Rodolfo; Cárcamo, Cristian; Poza, Abner H.; Vino, Eduardo An adaptive stabilized finite element method for the Stokes–Darcy coupled problem, Journal of Computational and Applied Mathematics, Volume 443 (2024), p. 115753 | DOI:10.1016/j.cam.2024.115753
  • Chouly, Franz A Review on Some Discrete Variational Techniques for the Approximation of Essential Boundary Conditions, Vietnam Journal of Mathematics (2024) | DOI:10.1007/s10013-024-00702-1
  • Pártl, Ondřej; Wilbrandt, Ulrich; Mura, Joaquín; Caiazzo, Alfonso Reconstruction of flow domain boundaries from velocity data via multi-step optimization of distributed resistance, Computers Mathematics with Applications, Volume 129 (2023), p. 11 | DOI:10.1016/j.camwa.2022.11.006
  • Mahbub, Md. Abdullah Al; Shan, Li; Zheng, Haibiao Uncoupling evolutionary groundwater-surface water flows: stabilized mixed methods in both porous media and fluid regions, Numerical Algorithms, Volume 92 (2023) no. 3, p. 1837 | DOI:10.1007/s11075-022-01370-3
  • He, Xiaoxiao; Deng, Weibing An interface penalty parameter free nonconforming cut finite element method for elliptic interface problems, Applied Numerical Mathematics, Volume 173 (2022), p. 434 | DOI:10.1016/j.apnum.2021.12.011
  • Blank, Laura; Meneses Rioseco, Ernesto; Caiazzo, Alfonso; Wilbrandt, Ulrich Modeling, simulation, and optimization of geothermal energy production from hot sedimentary aquifers, Computational Geosciences, Volume 25 (2021) no. 1, p. 67 | DOI:10.1007/s10596-020-09989-8
  • John, Volker; Knobloch, Petr; Wilbrandt, Ulrich Finite Element Pressure Stabilizations for Incompressible Flow Problems, Fluids Under Pressure (2020), p. 483 | DOI:10.1007/978-3-030-39639-8_6

Cité par 10 documents. Sources : Crossref