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A splitting method using discontinuous Galerkin for the transient incompressibleNavier-Stokes equations

Published online by Cambridge University Press:  15 November 2005

Vivette Girault ,
Béatrice Rivière  and
Mary F. Wheeler
Vivette Girault
Affiliation:
Université Pierre et Marie Curie, Paris VI, Laboratoire Jacques-Louis Lions, , place Jussieu, 75252 Paris Cedex 05, France. girault@ann.jussieu.fr
Béatrice Rivière
Affiliation:
Department of Mathematics, University of Pittsburgh, 301 Thackeray, Pittsburgh, PA 15260, USA. riviere@math.pitt.edu
Mary F. Wheeler
Affiliation:
Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, University of Texas, 201 E. 24th St., Austin TX 78712, USA. mfw@ices.utexas.edu
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Abstract

In this paper we solve the time-dependent incompressible Navier-Stokesequations by splitting the non-linearity and incompressibility, andusing discontinuous or continuous finite element methods in space. Weprove optimal error estimates for the velocity and suboptimalestimates for the pressure. We present some numerical experiments.

Keywords

Operator splittingtime-dependent Navier-StokesSIPG.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 6 , November 2005 , pp. 1115 - 1147
Copyright
© EDP Sciences, SMAI, 2005

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References

R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
A.S. Almgren, J.B. Bell, P. Colella, L.H. Howell and M.L. Welcome, A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations. Technical Report LNBL-39075, UC-405 (1996).
Baumann, C.E. and Oden, J.T., A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 175 (1999) 311–341. CrossRef
Blasco, J. and Codina, R., Error estimates for an operator-splitting method for incompressible flows. Appl. Numer. Math. 51 (2004) 1–17. CrossRef
Blasco, J., Codina, R. and Huerta, A., A fractional-step method for the incompressible Navier-Stokes equations related to a predictor-multicorrector algorithm. Int. J. Numer. Meth. Fl. 28 (1997) 1391–1419. 3.0.CO;2-5>CrossRef
P.G. Ciarlet, The finite element methods for elliptic problems. North-Holland, Amsterdam (1978).
Chorin, A.J., Numerical solution of the Navier-Stokes equations. Math. Comp. 22 (1968) 745–762. CrossRef
Crouzeix, M. and Raviart, P.A., Conforming and non conforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. R3 (1973) 33–76.
Dawson, C. and J .Proft, Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4721–4746. CrossRef
C. Dawson, S. Sun and M. Wheeler, Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Engrg. (2003) 2565–2580.
Fernandez-Cara, E. and Beltram, M.M., The convergence of two numerical schemes for the Navier-Stokes equations. Numer. Math. 55 (1989) 33–60. CrossRef
Girault, V. and Lions, J.-L., Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra. Portugal. Math. 58 (2001) 25–57.
V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations. Lecture Notes in Math. 749, Springer-Verlag, Berlin, Heidelberg, New-York (1979).
Girault, V., Rivière, B. and Wheeler, M.F., A discontinuous Galerkin method with non-overlapping domain decomposition for the Stokes and Navier-Stokes problems. Math. Comp. 74 (2005) 53–84. CrossRef
R. Glowinski, Finite element methods for Incompressible Viscous Flows, in Numerical Methods for Fluids (Part 3), Handbook of Numerical Analysis, 9, Elsevier, North-Holland (2003).
P. Grisvard, Elliptic problems in nonsmooth domains, Pitman Monogr. Studies Pure Appl. Math. 24, Pitman, Boston, MA (1985).
Guermond, J.L. and Quartapelle, L., On the approximation of the unsteady Navier-Stokes equations by finite element projection methods. Numer. Math. 80 (1998) 207–238. CrossRef
Guermond, J.L. and Shen, J., Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal. 41 (2003) 112–134. CrossRef
Guermond, J.L. and Shen, J., A new class of truly consistent splitting schemes for incompressible flows. J. Comput. Phys. 192 (2003) 262–276. CrossRef
S. Kaya and B. Rivière, A discontinuous subgrid eddy viscosity method for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. (2005), to appear.
P. Lesaint and P.A. Raviart, On a finite element method for solving the neutron transport equation, in Mathematical Aspects of Finite Element Methods in Partial Differential Equations, C.A. de Boor Ed., Academic Press (1974) 89–123.
J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, I. Dunod, Paris (1968).
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969).
Quarteroni, A., Saleri, F. and Veneziani, A., Factorization methods for the numerical approximation of Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 188 (2000) 505–526. CrossRef
R. Rannacher, On Chorin's projection method for the incompressible Navier-Stokes equations, Navier-Stokes equations: Theory and Numerical Methods, R. Rautmann et al. Eds., Springer (1992).
Rivière, B., Wheeler, M.F. and Girault, V., Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I. Comput. Geosci. 3 (1999) 337–360. CrossRef
Temam, R., Sur l'approximation de la solution des equations de Navier-Stokes par la méthode des pas fractionnaires (I), (II). Arch. Rational Mech. Anal. 33 (1969) 377–385. CrossRef
R. Temam, Navier-Stokes equations. Theory and numerical analysis. North-Holland, Amsterdam (1979).
Turek, S., On discrete projection methods for the incompressible Navier-Stokes equations: an algorithmic approach. Comput. Methods Appl. Mech. Engrg. 143 (1997) 271–288. CrossRef
Wheeler, M.F., An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152–161. CrossRef
N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables. Springer-Verlag, New York (1971).