A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems. (English) Zbl 1057.35029

Stokes and Navier-Stokes equations are considered in this paper especially from the numerical point of view. Numerical analysis of a discontinuous Galerkin method with nonoverlapping domain decomposition for steady incompressible Stokes and Navier-Stokes problems is investigated. As a discretization a conforming triangular finite element mesh is used in each subdomain. In each triangle the discretization velocity is a polynomial of degree \(k,\) \(\;k=1,\;2 \text{ or } 3\), while the discretized pressure is of degree \(k-1\). The variational formulation of the problem contains also a jump term on all triangle interfaces due to the domain decomposition method. An inf-sup condition is proved under some hypothesis for the domain decomposition with non-matching grids. Optimal a priori error estimates in the energy norm for the velocity field and the \(L_2\) norm for the pressure are derived first for Stokes and then also for Navier-Stokes problems.


35Q30 Navier-Stokes equations
76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
76D05 Navier-Stokes equations for incompressible viscous fluids
76M25 Other numerical methods (fluid mechanics) (MSC2010)
Full Text: DOI


[1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030
[2] Garth A. Baker, Wadi N. Jureidini, and Ohannes A. Karakashian, Piecewise solenoidal vector fields and the Stokes problem, SIAM J. Numer. Anal. 27 (1990), no. 6, 1466 – 1485. · Zbl 0719.76047
[3] R. Becker, P. Hansbo and R. Stenberg, A finite element method for domain decomposition with nonmatching grids, M2AN 37 (2003), pp. 209-225. · Zbl 1047.65099
[4] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058
[5] Bernardo Cockburn, Guido Kanschat, Dominik Schötzau, and Christoph Schwab, Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal. 40 (2002), no. 1, 319 – 343. · Zbl 1032.65127
[6] Michel Crouzeix and Richard S. Falk, Nonconforming finite elements for the Stokes problem, Math. Comp. 52 (1989), no. 186, 437 – 456. · Zbl 0685.76018
[7] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33 – 75. · Zbl 0302.65087
[8] Michel Fortin, An analysis of the convergence of mixed finite element methods, RAIRO Anal. Numér. 11 (1977), no. 4, 341 – 354, iii (English, with French summary). · Zbl 0373.65055
[9] M. Fortin and M. Soulie, A nonconforming piecewise quadratic finite element on triangles, Internat. J. Numer. Methods Engrg. 19 (1983), no. 4, 505 – 520. · Zbl 0514.73068
[10] V. Girault and J.-L. Lions, Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra, Port. Math. (N.S.) 58 (2001), no. 1, 25 – 57. · Zbl 0997.76043
[11] V. Girault, R. Glowinski, H. López, and J.-P. Vila, A boundary multiplier/fictitious domain method for the steady incompressible Navier-Stokes equations, Numer. Math. 88 (2001), no. 1, 75 – 103. · Zbl 1002.76067
[12] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. · Zbl 0585.65077
[13] V. Girault and R. L. Scott, A quasi-local interpolation operator preserving the discrete divergence, Calcolo 40 (2003), pp. 1-19. · Zbl 1072.65014
[14] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. · Zbl 0695.35060
[15] Paul Houston, Christoph Schwab, and Endre Süli, Discontinuous \?\?-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal. 39 (2002), no. 6, 2133 – 2163. · Zbl 1015.65067
[16] Ohannes A. Karakashian and Wadi N. Jureidini, A nonconforming finite element method for the stationary Navier-Stokes equations, SIAM J. Numer. Anal. 35 (1998), no. 1, 93 – 120. · Zbl 0933.76047
[17] P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89 – 123. Publication No. 33. · Zbl 0341.65076
[18] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). · Zbl 0212.43801
[19] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). · Zbl 0189.40603
[20] J. Tinsley Oden, Ivo Babuška, and Carlos Erik Baumann, A discontinuous \?\? finite element method for diffusion problems, J. Comput. Phys. 146 (1998), no. 2, 491 – 519. · Zbl 0926.65109
[21] Olivier Pironneau, Finite element methods for fluids, John Wiley & Sons, Ltd., Chichester; Masson, Paris, 1989. Translated from the French. · Zbl 0748.76003
[22] Béatrice Rivière, Mary F. Wheeler, and Vivette Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I, Comput. Geosci. 3 (1999), no. 3-4, 337 – 360 (2000). · Zbl 0951.65108
[23] Béatrice Rivière, Mary F. Wheeler, and Vivette Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal. 39 (2001), no. 3, 902 – 931. · Zbl 1010.65045
[24] B. Rivière and M. F. Wheeler, Nonconforming methods for transport with nonlinear reaction, Proceedings of the Joint Summer Research Conference on Fluid Flow and Transport in Porous Media (2001), Contemp. Math., vol. 295, Amer. Math. Soc., Providence, RI, 2002, pp. 421-432. · Zbl 1068.76053
[25] Roger Temam, Navier-Stokes equations, AMS Chelsea Publishing, Providence, RI, 2001. Theory and numerical analysis; Reprint of the 1984 edition. · Zbl 0981.35001
[26] Mary Fanett Wheeler, A priori \?\(_{2}\) error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723 – 759. · Zbl 0232.35060
[27] Mary Fanett Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1978), no. 1, 152 – 161. · Zbl 0384.65058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.