A splitting method using discontinuous Galerkin for the transient incompressible Navier-Stokes equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 6, p. 1115-1147
In this paper we solve the time-dependent incompressible Navier-Stokes equations by splitting the non-linearity and incompressibility, and using discontinuous or continuous finite element methods in space. We prove optimal error estimates for the velocity and suboptimal estimates for the pressure. We present some numerical experiments.
DOI : https://doi.org/10.1051/m2an:2005048
Classification:  65M12,  65M15,  65M60
@article{M2AN_2005__39_6_1115_0,
     author = {Girault, Vivette and Rivi\`ere, B\'eatrice and Wheeler, Mary F.},
     title = {A splitting method using discontinuous Galerkin for the transient incompressible Navier-Stokes equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {6},
     year = {2005},
     pages = {1115-1147},
     doi = {10.1051/m2an:2005048},
     zbl = {1085.76037},
     mrnumber = {2195907},
     language = {en},
     url = {http://http://www.numdam.org/item/M2AN_2005__39_6_1115_0}
}
Girault, Vivette; Rivière, Béatrice; Wheeler, Mary F. A splitting method using discontinuous Galerkin for the transient incompressible Navier-Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 6, pp. 1115-1147. doi : 10.1051/m2an:2005048. http://www.numdam.org/item/M2AN_2005__39_6_1115_0/

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030

[2] 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). | Zbl 0933.76055

[3] C.E. Baumann and J.T. Oden, A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 175 (1999) 311-341. | Zbl 0924.76051

[4] J. Blasco and R. Codina, Error estimates for an operator-splitting method for incompressible flows. Appl. Numer. Math. 51 (2004) 1-17. | Zbl 1126.76339

[5] J. Blasco, R. Codina and A. Huerta, 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. | Zbl 0935.76041

[6] P.G. Ciarlet, The finite element methods for elliptic problems. North-Holland, Amsterdam (1978). | MR 1115235 | Zbl 0999.65129

[7] A.J. Chorin, Numerical solution of the Navier-Stokes equations. Math. Comp. 22 (1968) 745-762. | Zbl 0198.50103

[8] M. Crouzeix and P.A. Raviart, Conforming and non conforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. R3 (1973) 33-76. | Numdam | Zbl 0302.65087

[9] C. Dawson and J .Proft, Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4721-4746. | Zbl 1015.76046

[10] C. Dawson, S. Sun and M. Wheeler, Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Engrg. (2003) 2565-2580. | Zbl 1067.76565

[11] E. Fernandez-Cara and M.M. Beltram, The convergence of two numerical schemes for the Navier-Stokes equations. Numer. Math. 55 (1989) 33-60. | Zbl 0645.76032

[12] V. Girault and J.-L. Lions, Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra. Portugal. Math. 58 (2001) 25-57. | Zbl 0997.76043

[13] 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). | MR 851383 | Zbl 0413.65081

[14] V. Girault, B. Rivière and M.F. Wheeler, A discontinuous Galerkin method with non-overlapping domain decomposition for the Stokes and Navier-Stokes problems. Math. Comp. 74 (2005) 53-84. | Zbl 1057.35029

[15] 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). | MR 2009826 | Zbl 1040.76001

[16] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman Monogr. Studies Pure Appl. Math. 24, Pitman, Boston, MA (1985). | MR 775683 | Zbl 0695.35060

[17] J.L. Guermond and L. Quartapelle, On the approximation of the unsteady Navier-Stokes equations by finite element projection methods. Numer. Math. 80 (1998) 207-238. | Zbl 0914.76051

[18] J.L. Guermond and J. Shen, Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal. 41 (2003) 112-134. | Zbl 1130.76395

[19] J.L. Guermond and J. Shen, A new class of truly consistent splitting schemes for incompressible flows. J. Comput. Phys. 192 (2003) 262-276. | Zbl 1032.76529

[20] 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. | MR 2182140 | Zbl 1096.76026

[21] 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. | Zbl 0341.65076

[22] J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, I. Dunod, Paris (1968). | Zbl 0165.10801

[23] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969). | MR 259693 | Zbl 0189.40603

[24] A. Quarteroni, F. Saleri and A. Veneziani, Factorization methods for the numerical approximation of Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 188 (2000) 505-526. | Zbl 0976.76044

[25] 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). | Zbl 0769.76053

[26] B. Rivière, M.F. Wheeler and V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I. Comput. Geosci. 3 (1999) 337-360. | Zbl 0951.65108

[27] R. Temam, 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. | Zbl 0207.16904

[28] R. Temam, Navier-Stokes equations. Theory and numerical analysis. North-Holland, Amsterdam (1979). | MR 603444 | Zbl 0426.35003

[29] S. Turek, On discrete projection methods for the incompressible Navier-Stokes equations: an algorithmic approach. Comput. Methods Appl. Mech. Engrg. 143 (1997) 271-288. | Zbl 0898.76069

[30] M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152-161. | Zbl 0384.65058

[31] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables. Springer-Verlag, New York (1971). | MR 307493 | Zbl 0209.47103