# Stabilized Lagrange–Galerkin Schemes of First- and Second-Order in Time for the Navier–Stokes Equations

Chapter
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

## Abstract

Two stabilized Lagrange–Galerkin schemes for the Navier–Stokes equations are reviewed. The schemes are based on a combination of the Lagrange–Galerkin method and Brezzi–Pitkäranta’s stabilization method. They maintain the advantages of both methods: (i) They are robust for convection-dominated problems and the systems of linear equations to be solved are symmetric; and (ii) Since the P1 finite element is employed for both velocity and pressure, the numbers of degrees of freedom are much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the schemes are efficient especially for three-dimensional problems. The one of the schemes is of first-order in time by Euler’s method and the other is of second-order by Adams–Bashforth’s method. In the second-order scheme an additional initial velocity is required. A convergence analysis is done for the choice of the velocity obtained by the first-order scheme, whose theoretical result is also recognized numerically.

## References

1. 1.
Achdou, Y., Guermond, J.-L.: Convergence analysis of a finite element projection/Lagrange–Galerkin method for the incompressible Navier–Stokes equations. SIAM J. Numer. Anal. 37, 799–826 (2000)
2. 2.
Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia (1994)
3. 3.
Boukir, K., Maday, Y., Métivet, B., Razafindrakoto, E.: A high-order characteristics/finite element method for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 25, 1421–1454 (1997)
4. 4.
Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations of the Stokes equations. In: Hackbusch, W. (ed.) Efficient Solutions of Elliptic Systems. Vieweg, Wiesbaden (1984)Google Scholar
5. 5.
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
6. 6.
Ewing, R.E., Russell, T.F.: Multistep Galerkin methods along characteristics for convection-diffusion problems. In: Vichnevetsky, R., Stepleman, R.S. (eds.) Advances in Computer Methods for Partial Differential Equations. IMACS, vol. IV, pp. 28–36 (1981)Google Scholar
7. 7.
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)
8. 8.
Morton, K.W., Priestley, A., Süli, E.: Stability of the Lagrange–Galerkin method with non-exact integration. ESAIM: Math. Model. Numer. Anal. 22, 625–653 (1988)
9. 9.
Notsu, H.: Numerical computations of cavity flow problems by a pressure stabilized characteristic-curve finite element scheme. Trans. Jpn. Soc. Comput. Eng. Sci. 2008, 20080032 (2008)Google Scholar
10. 10.
Notsu, H., Tabata, M.: A combined finite element scheme with a pressure stabilization and a characteristic-curve method for the Navier–Stokes equations. Trans. Jpn. Soc. Ind. Appl. Math. 18, 427–445 (2008) (in Japanese)Google Scholar
11. 11.
Notsu, H., Tabata, M.: Error estimates of a pressure-stabilized characteristics finite element scheme for the Oseen equations. J. Sci. Comput. 65, 940–955 (2015)
12. 12.
Notsu, H., Tabata, M.: Error estimates of a stabilized Lagrange–Galerkin scheme for the Navier–Stokes equations. ESAIM: Math. Model. Numer. Anal. 50, 361–380 (2016)
13. 13.
Notsu, H., Tabata, M.: Error estimates of a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. In: Shibata, Y., Suzuki, Y. (eds.) Mathematical Fluid Dynamics, Present and Future. Springer, Berlin (to appear)Google Scholar
14. 14.
Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier–Stokes equations. Numer. Math. 38, 309–332 (1982)
15. 15.
Ravindran, S.S.: Convergence of extrapolated BDF2 finite element schemes for unsteady penetrative convection model. Numer. Funct. Anal. Optim. 33, 48–79 (2012)
16. 16.
Rui, H., Tabata, M.: A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math. 92, 161–177 (2002)
17. 17.
Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs, NJ (1971)
18. 18.
Süli, E.: Convergence and nonlinear stability of the Lagrange–Galerkin method for the Navier–Stokes equations. Numer. Math. 53, 459–483 (1988)
19. 19.
Tabata, M.: Discrepancy between theory and real computation on the stability of some finite element schemes. J. Comput. Appl. Math. 199, 424–431 (2007)
20. 20.
Tabata, M., Uchiumi, S.: A Lagrange–Galerkin scheme with a locally linearized velocity for the Navier–Stokes equations. arXiv:1505.06681 [math.NA] (2015)Google Scholar