Abstract
We present a stabilized Backward Difference Formula of order 2- Lagrange Galerkin method for the incompressible Navier-Stokes equations at high Reynolds numbers. The stabilization of the conventional Lagrange-Galerkin method is done via a local projection technique for inf-sup stable finite elements. We have proven that for the Taylor-Hood finite element the a priori error estimate for velocity in the \(l^{\infty }(L^{2}(\varOmega )))\)-norm is O(h 2 +Δ t 2) whereas the error for the pressure in the l 2(L 2(Ω)))-norm is O(h 2 +Δ t 2), with error constants that are independent of the inverse of the Reynolds number. Numerical examples at high Reynolds numbers show the robustness of our method.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bermejo, R., Saavedra, L.: Modified Lagrange-Galerkin methods to integrate time dependent Navier-Stokes equations. SIAM J. Sci. Comput. (2014, submitted)
Bermejo, R., Galán del Sastre, P., Saavedra, L.: A second order in time modified Lagrange-Galerkin finite element method for the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 50, 3084–3109 (2012)
Braack, M., Burman, E.: Local projection stabilization of the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 43, 2544–2566 (2006)
Ganesan, S., Tobiska, L.: Stabilization by local projection for convection–diffusion and incompressible flow problems. J. Sci. Comput. 43, 326–342 (2010)
Gregory, N., O’Reilly, C.L.: Low-speed aerodynamic characteristics of NACA 0012 aerofoil sections including the effects of upper-surface roughness simulation hoar frost. NASA R&M 3726, Jan 1970
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)
John, V., Roland, M.: Simulations of the turbulent channel flow at Re τ = 180 with projection-based finite element variational multiscale methods. Int. J. Numer. Meth. Fluids 55, 407–429 (2007)
John, V., Kaya, S., Kindl, A.: Finite element error analysis for projection-based variational multiscale method with nonlinear eddy viscosity. J. Math. Anal. Appl. 344, 627–641 (2008)
Morton, K.W., Priestley, A., Suli, E.: Stability of the Lagrange-Galerkin method with non-exact integration. M2AN Math. Model. Numer. Anal. 22, 625–653 (1988)
Notsu, H., Tabata, M.: A single-step characteristic-curve finite element scheme of second order in time for the incompressible Navier-Stokes equations. J. Sci. Comput. 38 (1), 1–14 (2009)
Rumsey, C.: Langley Research Center. Turbulence Modeling Resource: http://turbmodels.larc.nasa.gov/index.html. Last updated: 09/2014
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bermejo, R., Saavedra, L. (2016). A Second Order Local Projection Lagrange-Galerkin Method for Navier-Stokes Equations at High Reynolds Numbers. In: Ortegón Gallego, F., Redondo Neble, M., Rodríguez Galván, J. (eds) Trends in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-32013-7_24
Download citation
DOI: https://doi.org/10.1007/978-3-319-32013-7_24
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-32012-0
Online ISBN: 978-3-319-32013-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)