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Journal of Scientific Computing

Erschienen in:

28.05.2019

Fully Discrete Approximations to the Time-Dependent Navier–Stokes Equations with a Projection Method in Time and Grad-Div Stabilization

verfasst von: Javier de Frutos, Bosco García-Archilla, Julia Novo

Erschienen in: Journal of Scientific Computing | Ausgabe 2/2019

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Abstract

This paper studies fully discrete approximations to the evolutionary Navier–Stokes equations by means of inf-sup stable \(H^1\)-conforming mixed finite elements with a grad-div type stabilization and the Euler incremental projection method in time. We get error bounds where the constants do not depend on negative powers of the viscosity. We get the optimal rate of convergence in time of the projection method. For the spatial error we get a bound \(O(h^k)\) for the \(L^2\) error of the velocity, k being the degree of the polynomials in the velocity approximation. We prove numerically that this bound is sharp for this method.

Vorheriger Artikel Mortar Element Method for the Time Dependent Coupling of Stokes and Darcy Flows

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Adams, R.A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65. Academic Press, New York-London (1975)MATH

Ahmed, N., Chacón Rebollo, T., John, V., Rubino, S.: Analysis of full space-time discretization of the Navier–Stokes equations by a local projection stabilization method. IMA J. Numer. Anal. 37, 1437–1467 (2017)MathSciNetMATH

Arndt, D., Dallmann, H., Lube, G.: Local projection FEM stabilization for the time-dependent incompressible Navier–Stokes problem. Numer. Methods Partial Differ. Equ. 31, 1224–1250 (2015)MathSciNetCrossRefMATH

Arndt, D., Dallmann, H., Lube, G.: Quasi-optimal error estimates for the incompressible Navier–Stokes problem discretized by finite element methods and pressure-correction projection with velocity stabilization. arXiv:1609.00807v1

Ayuso, B., García-Archilla, B., Novo, J.: The postprocessed mixed finite-element method for the Navier–Stokes equations. SIAM J. Numer. Anal. 43, 1091–1111 (2005)MathSciNetCrossRefMATH

Badia, S., Codina, R.: Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition. Numer. Math. 107, 533–557 (2007)MathSciNetCrossRefMATH

Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Volume 15 of Texts in Applied Mathematics, 3rd edn. Springer, New York (2008)CrossRef

Botti, L., Di Pietro, D.A.: A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure. J. Comput. Phys. 230, 572–585 (2011)MathSciNetCrossRefMATH

Bowers, A.L., Le Borne, S., Rebholz, L.G.: Error analysis and iterative solvers for Navier–Stokes projection methods with standard and sparse grad-div stabilization. Comput. Methods Appl. Mech. Eng. 275, 1–19 (2014)MathSciNetCrossRefMATH

10.

Burman, E.: Robust error estimates for stabilized finite element approximations of the two dimensional Navier–Stokes’ equations at high Reynolds number. Comput. Methods Appl. Mech. Eng. 288, 2–23 (2015)MathSciNetCrossRefMATH

11.

Burman, E., Ern, A., Fernández, M.A.: Fractional-step methods and finite elements with symmetric stabilization for the transient Oseen problem. ESAIM M2AN 51, 487–507 (2017)MathSciNetCrossRefMATH

12.

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

13.

Burman, E., Fernández, M.A., Hansbo, P.: Continuous interior penalty finite element method for the Oseen’s equations. SIAM J. Numer. Anal. 44, 1437–1453 (2006)MathSciNetMATH

14.

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

15.

Charnyi, S., Heister, T., Olshanskii, M.A., Rebholz, L.G.: On conservation laws of Navier–Stokes Galerkin discretizations. J. Comput. Phys. 337, 289–308 (2017)MathSciNetCrossRefMATH

16.

Chen, H.: Pointwise error estimates for finite element solutions of the Stokes problem. SIAM J. Numer. Anal. 44, 1–28 (2006)MathSciNetCrossRefMATH

17.

Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, vol. 4. North-Holland Publishing Co., Amsterdam (1978)

18.

Codina, R.: Pressure stability in fractional step finite element methods for incompressible flows. J. Comput. Phys. 170, 112–140 (2001)MathSciNetCrossRefMATH

19.

Constantin, P., Foias, C.: Navier–Stokes Equations. The University of Chicago Press, Chicago (1988)MATH

20.

D’Agnillo, E., Rebholz, L.: On the enforcement of discrete mass conservation in incompressible flow simulations with continuous velocity approximations. In: Li, J., Macharro, E., Yang, H. (eds.) Recent Advances in Scientific Computing and Applications. Proceedings of the 8th International Conference on Scientific Computing and Applications, AMS Contemporary Mathematics, vol. 586, pp. 143–152 (2013)

21.

Dallmann, H., Arndt, D., Lube, G.: Local projection stabilization for the Oseen problem. IMA J. Numer. Anal. 36, 796–823 (2016)MathSciNetCrossRefMATH

22.

de Frutos, J., John, V., Novo, J.: Projection methods for incompressible flow problems with WENO finite difference schemes. J. Comput. Phys. 309, 1–19 (2016)MathSciNetCrossRefMATH

23.

de Frutos, J., Garcí-Archilla, B., John, V., Novo, J.: Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements. J. Sci. Comput. 66, 991–1024 (2016)MathSciNetCrossRefMATH

24.

de Frutos, J., Garcí-Archilla, B., Novo, J.: Error analysis of projection methods for non inf-sup stable mixed finite elements. The transient Stokes problem. Appl. Math. Comput. 322, 154–173 (2018)MathSciNetMATH

25.

de Frutos, J., Garcí-Archilla, B., Novo, J.: Error analysis of projection methods for non inf-sup stable mixed finite elements. The Navier–Stokes equations. J. Sci. Comput. 74, 426–455 (2018)MathSciNetCrossRefMATH

26.

de Frutos, J., Garcí-Archilla, B., John, V., Novo, J.: Analysis of the grad-div stabilization for the time-dependent Navier–Stokes equations with inf-sup stable finite elements. Adv. Comput. Math. 44, 195–225 (2018)MathSciNetCrossRefMATH

27.

de Frutos, J., Garcí-Archilla, B., John, V., Novo, J.: Error analysis of non inf-sup stable discretizations of the time-dependent Navier–Stokes equations with local projection stabilization. IMA J. Numer. Anal. https://doi.org/10.1093/imanum/dry044. (to appear)

28.

Fehn, N., Wall, W.A., Kronbichler, M.: On the stability of projection methods for the incompressible Navier–Stokes equations based on high-order discontinuous Galerkin discretizations. J. Comput. Phys. 351, 392–421 (2017)MathSciNetCrossRefMATH

29.

Franca, L.P., Hughes, T.J.R.: Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Eng. 69(1), 89–129 (1988)MathSciNetCrossRefMATH

30.

Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations Theory and Algorithms. Volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986)MATH

31.

Guermond, J.L., Marra, A., Quartapelle, L.: Subgrid stabilized projection method for 2D unsteady flows at high Reynolds numbers. Comput. Methods Appl. Mech. Eng. 195, 5857–5876 (2006)MathSciNetCrossRefMATH

32.

Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195, 6011–6045 (2006)MathSciNetCrossRefMATH

33.

Guermond, J.L., Quartapelle, L.: On the approximation of the unsteady Navier–Stokes equations by finite element projection methods. Numer. Math. 80, 207–238 (1998)MathSciNetCrossRefMATH

34.

Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem. IV. Error analysis for second order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)MathSciNetCrossRefMATH

35.

John, V.: Reference values for drag and lift of a two-dimensional time-dependent flow around a cylinder. Int. J. Numer. Meth. Fluids 44, 777–788 (2004)CrossRefMATH

36.

John, V., Kindl, A.: Numerical studies of finite element variational multiscale methods for turbulent flow simulations. Comput. Methods Appl. Mech. Engrg. 199, 841–852 (2010)MathSciNetCrossRefMATH

37.

John, V., Knobloch, P., Novo, J.: Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story? J. Comput. Vis. Sci. (2018). https://doi.org/10.1007/s00791-018-0290-5 MathSciNet

38.

Karniadakis, G., Sherwin, S.: Spectral/hp Element Methods for Computational Fluid Dynamics, Scd edn. Oxford University Press, Oxford (2005)CrossRefMATH

39.

Layton, W., Manica, C., Neda, M., Rebholz, L.: On the accuracy of the rotation form in simulations of the Navier–Stokes equations. J. Comput. Phys. 228, 3433–3447 (2009)MathSciNetCrossRefMATH

40.

Linke, A., Rebholz, L.: On a reduced sparsity stabilization of grad-div type for incompressible flow problems. Comput. Methods Appl. Mech. Eng. 261–262, 142–153 (2013)MathSciNetCrossRefMATH

41.

Minev, P.D.: A stabilized incremental projection scheme for the Incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 36, 441–464 (2001)MathSciNetCrossRefMATH

42.

Prohl, A.: Projection and Quasi-compressibility Methods for Solving the Incompressible Navier–Stokes Equations. B. G. Teubner, Stuttgart (1997)CrossRefMATH

43.

Prohl, A.: On pressure approximations via projection methods for nonstationary incompressible Navier–Stokes equations. SIAM J. Numer. Anal. 29, 158–180 (2008)MathSciNetMATH

44.

Rannacher, R.: On Chorin’s Projection Method for the Incompressible Navier–Stokes Equations. Lecture Notes in Mathematics, vol. 1530. Springer, Berlin (1992)MATH

45.

Röhe, L., Lube, G.: Analysis of a variational multiscale method for large-eddy simulation and its application to homogeneous isotropic turbulence. Comput. Methods Appl. Mech. Eng. 199, 2331–2342 (2010)MathSciNetCrossRefMATH

46.

Schäfer, M., Turek, S.: Benchmark computations of laminar flow around a cylinder, With support by Durst, F., Krause, E., Rannacher, R. In: Flow Simulation with High-Performance Computers II. DFG Priority Research Programme Results 1993–1995, Vieweg, Wiesbaden, pp. 547–566 (1996)

47.

Schroeder, P.W., Lube, G.: Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier–Stokes flows. J. Numer. Anal. 25, 249–276 (2017)MathSciNetMATH

48.

Shen, J.: On error estimates of projection methods for Navier–Stokes equations: first-order schemes. SIAM J. Numer. Anal. 29, 57–77 (1992)MathSciNetCrossRefMATH

49.

Shen, J.: Remarks on the pressure error estimates for the projection methods. Numer. Math. 67, 513–520 (1994)MathSciNetCrossRefMATH

Titel: Fully Discrete Approximations to the Time-Dependent Navier–Stokes Equations with a Projection Method in Time and Grad-Div Stabilization
verfasst von: Javier de Frutos
Bosco García-Archilla
Julia Novo
Publikationsdatum: 28.05.2019
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 2/2019
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-019-00980-9

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