Fractional-step methods and finite elements with symmetric stabilization for the transient Oseen problem

Burman, Erik; Ern, Alexandre; Fernández, Miguel A.

doi:10.1051/m2an/2016028

Burman, Erik ¹ ; Ern, Alexandre ² ; Fernández, Miguel A.^3,⁴

¹ Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK.
² Université Paris-Est, CERMICS (ENPC), 77455 Marne la Vallée cedex 2, France.
³ Inria, 75012 Paris, France.
⁴ Sorbonnes Universités, UPMC, Laboratoire Jacques-Louis Lions, 75005 Paris, France.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 487-507.

Résumé

This paper deals with the spatial and time discretization of the transient Oseen equations. Finite elements with symmetric stabilization in space are combined with several time-stepping schemes (monolithic and fractional-step). Quasi-optimal (in space) and optimal (in time) error estimates are established for smooth solutions in all flow regimes. We first analyze monolithic time discretizations using the Backward Differentation Formulas of order 1 and 2 (BDF1 and BDF2). We derive a new estimate on the time-average of the pressure error featuring the same robustness with respect to the Reynolds number as the velocity estimate. Then, we analyze fractional-step pressure-projection methods using BDF1. The stabilization of velocities and pressures can be treated either implicitly or explicitly. Numerical results illustrate the main theoretical findings.

Reçu le : 2015-10-21
Accepté le : 2016-04-21

MR Zbl

DOI : 10.1051/m2an/2016028

Classification : 65M12, 65M60, 76D07, 76M10
Mots-clés : Oseen equations, stabilized finite elements, fractional-step methods, pressure-correction methods, error estimates, high Reynolds number

Affiliations des auteurs :

Burman, Erik ¹ ; Ern, Alexandre ² ; Fernández, Miguel A. ^{3, 4}

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     author = {Burman, Erik and Ern, Alexandre and Fern\'andez, Miguel A.},
     title = {Fractional-step methods and finite elements with symmetric stabilization for the transient {Oseen} problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {487--507},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {2},
     year = {2017},
     doi = {10.1051/m2an/2016028},
     mrnumber = {3626408},
     zbl = {1398.76097},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2016028/}
}

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Burman, Erik; Ern, Alexandre; Fernández, Miguel A. Fractional-step methods and finite elements with symmetric stabilization for the transient Oseen problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 487-507. doi : 10.1051/m2an/2016028. http://www.numdam.org/articles/10.1051/m2an/2016028/

Bibliographie
Cité par

R. Becker and M. Braack, A two-level stabilization scheme for the Navier–Stokes equations. In Numerical Mathematics and Advanced Applications. Springer, Berlin (2004) 123–130. | MR | Zbl

M. Braack, E. Burman, V. John and G. Lube, Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853–866. | DOI | MR | Zbl

E. Burman, Interior penalty variational multiscale method for the incompressible Navier–Stokes equation: monitoring artificial dissipation. Comput. Methods Appl. Mech. Engrg. 196 (2007) 4045–4058. | DOI | MR | Zbl

E. Burman and M.A. Fernández, Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence. Numer. Math. 107 (2007) 39–77. | DOI | MR | Zbl

E. Burman and M.A. Fernández, Galerkin finite element methods with symmetric pressure stabilization for the transient Stokes equations: stability and convergence analysis. SIAM J. Numer. Anal. 47 (2008/09) 409–439. | DOI | MR | Zbl

E. Burman and P. Hansbo, Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1437–1453. | DOI | MR | Zbl

E. Burman, M.A. Fernández and P. Hansbo, Continuous interior penalty finite element method for Oseen’s equations. SIAM J. Numer. Anal. 44 (2006) 1248–1274. | DOI | MR | Zbl

T. Chacón Rebollo, M. Gómez Mármol and M. Restelli, Numerical analysis of penalty stabilized finite element discretizations of evolution Navier–Stokes equations. J. Sci. Comput. 63 (2015) 885–912. | DOI | MR | Zbl

R. Codina, Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods. Comput. Methods Appl. Mech. Engrg. 190 (2000) 1579–1599. | DOI | MR | Zbl

R. Codina and S. Badia, On some pressure segregation methods of fractional-step type for the finite element approximation of incompressible flow problems. Comput. Methods Appl. Mech. Engrg. 195 (2006) 2900–2918. | DOI | MR | Zbl

D.A. Di Pietro and A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations. Math. Comp. 79 (2010) 1303–1330. | DOI | MR | Zbl

D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Vol. 69 of Math. Appl. Springer-Verlag, Berlin (2012). | MR | Zbl

A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory. SIAM J. Numer. Anal. 44 (2006) 753–778. | DOI | MR | Zbl

A. Ern and J.-L. Guermond, Weighting the edge stabilization. SIAM J. Numer. Anal. 51 (2013) 1655–1677. | DOI | MR | Zbl

V. Girault, B. Rivière and M.F. Wheeler, A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier–Stokes problems. Math. Comput. 74 (2005) 53–84. | DOI | MR | Zbl

V. Gravemeier, W.A. Wall and E. Ramm, Large eddy simulation of turbulent incompressible flows by a three-level finite element method. Int. J. Numer. Methods Fluids 48 (2005) 1067–1099. | DOI | MR | Zbl

J.-L. Guermond, Stabilization of Galerkin approximations of transport equations by subgrid modeling. ESAIM: M2AN 33 (1999) 1293–1316. | DOI | Numdam | MR | Zbl

J.-L. Guermond and L. Quartapelle, On the approximation of the unsteady Navier–Stokes equations by finite element projection methods. Numer. Math. 80 (1998) 207–238. | DOI | MR | Zbl

J.-L. Guermond, A. Marra and L. Quartapelle, Subgrid stabilized projection method for 2D unsteady flows at high Reynolds numbers. Comput. Methods Appl. Mech. Engrg. 195 (2006) 5857–5876. | DOI | MR | Zbl

J.L. Guermond, P. Minev and Jie Shen. An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg. 195 (2006) 6011–6045. | DOI | MR | Zbl

V. John, An assessment of two models for the subgrid scale tensor in the rational LES model. J. Comput. Appl. Math. 173 (2005) 57–80. | DOI | MR | Zbl

C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46 (1986) 1–26. | DOI | MR | Zbl

C. Johnson, U. Nävert and J. Pitkäranta, Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 45 (1984) 285–312. | DOI | MR | Zbl

M. Lesieur, C. Staquet, P. Le Roy and P. Comte, The mixing layer and its coherence examined from the point of view of two-dimensional turbulence. J. Fluid Mech. 192 (1988) 511–534. | DOI | MR

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

A. Prohl, Projection and quasi-compressibility methods for solving the incompressible Navier–Stokes equations. Advances in Numerical Mathematics. B. G. Teubner, Stuttgart (1997). | MR | Zbl

S. Turek and M. Schäfer, Benchmark computations of laminar flow around cylinder. Flow Simulation with High-Performance Computers II. Vieweg (1996). | Zbl

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