Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. (English) Zbl 0637.76024

Süli, Endre

Numer. Math. 53, No. 4, 459-483 (1988).

The Lagrange-Galerkin method is a numerical technique for solving convection-dominated diffusion problems, based on combining a special discretization of the Lagrangian material derivative along particle trajectories with the Galerkin finite element method. In this paper optimal error estimates are presented for the Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations in a velocity/pressure formulation. The method is shown to be nonlinearly stable.

Reviewer: E.Süli

MSC:

76D05	Navier-Stokes equations for incompressible viscous fluids
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Keywords:

Lagrange-Galerkin method; convection-dominated diffusion problems; special discretisation; Lagrangian material derivative; particle trajectories; Galerkin finite element method; optimal error estimates; Lagrange-Galerkin mixed finite element approximation; Navier-Stokes equations; velocity/pressure formulation

References:

[1]	Adams, R.A.: Sobolev spaces, New York: Academic Press 1975 · Zbl 0314.46030
[2]	Benqué, J.P., Labadie, G., Ronat, J.: A new finite element method for Navier-Stokes equations coupled with a temperature equation. In: T. Kawai (ed.) Proc. 4th Int. Symp. on finite element methods in flow problems, pp. 295-302. Amsterdam: North-Holland 1982 · Zbl 0508.76049
[3]	Bercovier, M., Pironneau, O.: Characteristics and the finite element method. In: T. Kawai (ed.) Proc. 4th Int. Symp. On finite element methods in flow problems, pp. 67-73. Amsterdam. North-Holland 1982 · Zbl 0508.76007
[4]	Bernardi, C., Raugel, G.: Analysis of some finite elements for the Stokes problem. Math. Comput.44, 71-80 (1985) · Zbl 0563.65075
[5]	Chorin, A.J., Marsden, J.E.: A Mathematical introduction to fluid mechanics. Berlin Heidelberg New York: Springer 1984 · Zbl 0417.76002
[6]	Ciarlet, P.: The finite element method for elliptic problems. Amsterdam: North-Holland 1978 · Zbl 0383.65058
[7]	Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér7, 33-76 (1973) · Zbl 0302.65087
[8]	Douglas, J. Jr., Russell, T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal.19, 871-885 (1982) · Zbl 0492.65051
[9]	Federer, H.: Geometric measure theory. Berlin Heidelberg New York: Springer 1969 · Zbl 0176.00801
[10]	Fortin, M.: Old and new elements for incompressible flows. Int. J. Numer. Methods Fluids1, 347-364 (1981) · Zbl 0467.76030
[11]	Girault, V., Raviart, P.-A.: Finite element approximation of the Navier-Stokes equations. Lect. Notes Math. 749. Berlin Heidelberg New York: Springer 1979 · Zbl 0413.65081
[12]	Grisvard, P.: Singularitè des solutions du problème de Stokes dans un polygone. Publications de l’Université de Nice 1978
[13]	Hasbani, Y., Livne, E., Bercovier, M.: Finite elements and characteristics applied to advectiondiffusion equations. Comput. Fluids11, 71-83 (1983) · Zbl 0511.76089
[14]	Heywood, J.G.: The Navier-Stokes equations: on existence, regularity and decay of solutions. Indiana Univ Math. J.29, 639-681 (1980) · Zbl 0494.35077
[15]	Lefschetz, S.: Differential equations: Geometric theory: New York: Dover Publications 1977 · Zbl 0080.06401
[16]	López-Marcos, J.C., Sanz-Serna, J.M.: Stability and convergence in numerical analysis III: Linear investigation of nonlinear stability. IMA J. Numer. Anal.8, 71-84 (1987) · Zbl 0695.65042
[17]	Mansfield, L.: Finite element subspaces with optimal rates of convergence for the stationary Stokes problem. RAIRO Anal. Numér.16, 49-66 (1982) · Zbl 0477.65084
[18]	Maz’ya, V.G.: Sobolev spaces. Berlin Heidelberg New York: Springer 1985
[19]	Morton, K. W., Priestley, A., Süli, E.: Convergence analysis of the Lagrange-Galerkin method with non-exact integration. Oxford University Computing Laboratory Report. No 86/14 (1986) · Zbl 0661.65114
[20]	Pironneau, O.: On the transport-diffusion algorithm and its application to the Navier-Stokes equations. Numer. Math.38, 309-332 (1982) · Zbl 0505.76100
[21]	Russell, T.F.: Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media. Ph.D. Thesis. University of Chicago 1980
[22]	Stenberg, R.: Analysis of some mixed finite elements methods for the Stokes problem: A unified approach. Math. Comput.42, 9-23 (1984) · Zbl 0535.76037
[23]	Stenberg, R.: On some three-dimensional finite elements for incompressible media. Comput. Methods Appl. Mech. Engrg.63, 261-269 (1987) · Zbl 0684.73036
[24]	Süli, E.: Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations. In: K.W. Morton, M.J. Baines (eds.), Numerical methods for fluid dynamics, pp. 439-448. Oxford University Press (1985)
[25]	Süli, E.: Convergence analysis of the Lagrange-Galerkin method for the Navier-Stokes equations. Oxford University Computing Laboratory Report. No 86/3 (1986)
[26]	Süli, E.: Stability and convergence of the Lagrange-Galerkin method with non-exact integration. In: J.R. Whiteman (ed.) The mathematics of finite elements and applications, pp. 435-442. New York: Academic Press (1988)
[27]	Temam, R.: Navier-Stokes Equations. Theory and numerical analysis. Amsterdam: North-Holland 1977 · Zbl 0383.35057
[28]	Zlámal, M.: Curved elements in the finite element method. I. SIAM J. Numer. Anal.10, 229-240 (1973) · Zbl 0285.65067