Abstract
This paper studies inf-sup stable finite element discretizations of the evolutionary Navier–Stokes equations with a grad-div type stabilization. The analysis covers both the case in which the solution is assumed to be smooth and consequently has to satisfy nonlocal compatibility conditions as well as the practically relevant situation in which the nonlocal compatibility conditions are not satisfied. The constants in the error bounds obtained do not depend on negative powers of the viscosity. Taking into account the loss of regularity suffered by the solution of the Navier–Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order \(\mathcal O(h^{2})\) in space are proved. The analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. Both the continuous-in-time case and the fully discrete scheme with the backward Euler method as time integrator are analyzed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams, R.A.: Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975). Pure and Applied Mathematics, vol. 65
Ahmed, N., Rebollo, T.C., 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. (to appear)
Arndt, D., Dallmann, H., Lube, G: Local projection FEM stabilization for the time-dependent incompressible Navier-Stokes problem. Numer. Methods Partial Diff. Equa. 31(4), 1224–1250 (2015)
Ayuso, B., García-archilla, B., Novo, J.: The postprocessed mixed finite-element method for the Navier-Stokes equations. SIAM J. Numer. Anal. 43(3), 1091–1111 (2005)
Brenner, S.C., Ridgway Scott, L.: The Mathematical Theory of Finite Element Methods, Volume 15 of Texts in Applied Mathematics, 3rd edn. Springer, New York (2008)
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)
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(1), 39–77 (2007)
Chen, H.: Pointwise error estimates for finite element solutions of the Stokes problem. SIAM J. Numer. Anal. 44(1), 1–28 (2006)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Publishing Co., Amsterdam (1978). Studies in Mathematics and its Applications, vol. 4
Constantin, P., Foias, C.: Navier–Stokes Equations. The Unviersity of Chicago Press, Chicago (1988)
Dallmann, H., Arndt, D., Lube, G.: Local projection stabilization for the O,seen problem. IMA J. Numer Anal. 36(2), 796–823 (2016)
de Frutos, J., García-Archilla, B., Novo, J.: The postprocessed mixed finite-element method for the Navier-Stokes equations: Refined error bounds. SIAM J. Numer. Anal. 46(1), 201–230 (2007/08)
de Frutos, J., García-Archilla, B., Novo, J.: Postprocessing finite-element methods for the Navier-Stokes equations: the fully discrete case. SIAM J. Numer. Anal. 47(1), 596–621 (2008/09)
de Frutos, J, García-Archilla, B, John, V, Novo, J: Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements. J. Sci. Comput. 66(3), 991–1024 (2016)
Franca, L.P., Hughes, T.J.R. : Two classes of mixed finite element methods. Comput. Methods Appl. Mech Engrg. 69(1), 89–129 (1988)
Girault, V., Pierre-Arnaud, R: Finite Element Methods for Navier-Stokes Equations, Volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986). Theory and algorithms
Heywood, J.G., Rannacher, R: Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19(2), 275–311 (1982)
Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem. III. Smoothing property and higher order error estimates for spatial discretization. SIAM J. Numer. Anal. 25(3), 489–512 (1988)
Heywood, J.G., Rannacher, R: Finite element approximation of the nonstationary N,avier-Stokes problem. IV. Error analysis for second order time discretization. SIAM J. Numer Anal. 27(2), 353–384 (1990)
Jenkins, E.W., John, V., Linke, A., Rebholz, L.G.: On the parameter choice in grad-div stabilization for the Stokes equations. Adv. Comput. Math. 40, 491–516 (2014)
John, V: Finite Element Methods for Incompressible Flow Problems, Volume 51 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (2016)
John, V., Kindl, A.: Numerical studies of finite element variational multiscale methods for turbulent flow simulations. Comput. Methods Appl. Mech Engrg. 199 (13–16), 841–852 (2010)
John, V., Linke, A., Merdon, C., Neilan, M, Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. (to appear) (2016)
John, V., Gunar, M.: MooNMD—a program package based on mapped finite element methods. Comput. Vis. Sci. 6(2–3), 163–169 (2004)
Lube, G., Arndt, D., Dallmann, H.: Understanding the limits of inf-sup stable Galerkin-FEM for incompressible flows. In: Knobloch, P. (ed.) Boundary and Interior Layers, Computational and Asymptotic Methods - BAIL 2014, pp. 147–169 (2016)
Olshanskii, MA.: A low order Galerkin finite element method for the Navier-Stokes equations of steady incompressible flow: a stabilization issue and iterative methods. Comput. Methods Appl. Mech. Engrg. 191, 5515–5536 (2002)
Olshanskii, M.A., Reusken, A.: Grad-div stabilization for Stokes equations. Math Comp. 73, 1699–1718 (2004)
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. Engrg. 199(37–40), 2331–2342 (2010)
Schoroeder, P.W., Lube, G.: Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier–Stokes flows. J. Num. Anal., in press (2017)
Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis- Studies in Mathematics and its Applications, vol. 2. North-Holland (1979)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Karsten Urban
Javier de Frutos research was supported by Spanish MINECO under grants MTM2013-42538-P (MINECO, ES) and MTM2016-78995-P (AEI/FEDER, UE).
Bosco García-Archilla research was supported by Spanish MINECO under grant MTM2015-65608-P. Julia Novo research was supported by Spanish MINECO under grants MTM2013-42538-P (MINECO, ES) and MTM2016-78995-P (AEI/FEDER, UE).
Rights and permissions
About this article
Cite this article
de Frutos, J., García-Archilla, B., John, V. et al. 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). https://doi.org/10.1007/s10444-017-9540-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-017-9540-1
Keywords
- Incompressible Navier–Stokes equations
- Inf-sup stable finite element methods
- Grad-div stabilization
- Error bounds independent of the viscosity
- Nonlocal compatibility condition
- Backward Euler method