A Second Order Local Projection Lagrange-Galerkin Method for Navier-Stokes Equations at High Reynolds Numbers

  • Rodolfo Bermejo
  • Laura SaavedraEmail author
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 8)


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(h2 +Δ t2) whereas the error for the pressure in the l2(L2(Ω)))-norm is O(h2 +Δ t2), 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.


High Reynolds Number Quadrature Rule Local Projection Material Derivative Finite Element Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Bermejo, R., Saavedra, L.: Modified Lagrange-Galerkin methods to integrate time dependent Navier-Stokes equations. SIAM J. Sci. Comput. (2014, submitted)Google Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ganesan, S., Tobiska, L.: Stabilization by local projection for convection–diffusion and incompressible flow problems. J. Sci. Comput. 43, 326–342 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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 1970Google Scholar
  6. 6.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)Google Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rumsey, C.: Langley Research Center. Turbulence Modeling Resource: Last updated: 09/2014

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada, E.T.S.I. IndustrialesUniversidad Politécnica de MadridMadridSpain
  2. 2.Departamento de Matemática Aplicada a la Ingeniería Aeroespacial, E.T.S.I. Aeronáutica y del EspacioUniversidad Poliécnica de MadridMadridSpain

Personalised recommendations