Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations

Di Pietro, Daniele; Ern, Alexandre

doi:10.1090/S0025-5718-10-02333-1

Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations
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by Daniele A. Di Pietro and Alexandre Ern PDF

Math. Comp. 79 (2010), 1303-1330 Request permission

Abstract:

Two discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes: (i) a discrete counterpart of the continuous Sobolev embeddings, in both Hilbertian and non-Hilbertian settings; (ii) a compactness result for bounded sequences in a suitable Discontinuous Galerkin norm, together with a weak convergence property for some discrete gradients. The proofs rely on techniques inspired by the Finite Volume literature, which differ from those commonly used in Finite Element analysis. The discrete functional analysis tools are used to prove the convergence of Discontinuous Galerkin approximations of the steady incompressible Navier–Stokes equations. Two discrete convective trilinear forms are proposed, a nonconservative one relying on Temam’s device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure.

References

Douglas N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742–760. MR 664882, DOI 10.1137/0719052
Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. MR 1885715, DOI 10.1137/S0036142901384162
F. Bassi, A. Crivellini, D. A. Di Pietro, and S. Rebay, An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations, J. Comput. Phys. 218 (2006), no. 2, 794–815. MR 2269385, DOI 10.1016/j.jcp.2006.03.006
F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti, and M. Savini, A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, Proceedings of the $2^{\mathrm {nd}}$ European Conference on Turbomachinery Fluid Dynamics and Thermodynamics (R. Decuypere and G. Dibelius, eds.), 1997, pp. 99–109.
Susanne C. Brenner, Poincaré-Friedrichs inequalities for piecewise $H^1$ functions, SIAM J. Numer. Anal. 41 (2003), no. 1, 306–324. MR 1974504, DOI 10.1137/S0036142902401311
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376, DOI 10.1007/978-1-4757-3658-8
Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
F. Brezzi, G. Manzini, D. Marini, P. Pietra, and A. Russo, Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differential Equations 16 (2000), no. 4, 365–378. MR 1765651, DOI 10.1002/1098-2426(200007)16:4<365::AID-NUM2>3.0.CO;2-Y
A. Buffa and C. Ortner, Compact embeddings of broken Sobolev spaces and applications, IMA J. Numer. Anal. (2009), To appear.
Bernardo Cockburn, Guido Kanschat, and Dominik Schotzau, A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comp. 74 (2005), no. 251, 1067–1095. MR 2136994, DOI 10.1090/S0025-5718-04-01718-1
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. MR 1921922, DOI 10.1137/S0036142900380121
Bernardo Cockburn and Chi-Wang Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2440–2463. MR 1655854, DOI 10.1137/S0036142997316712
Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
Daniele A. Di Pietro, Analysis of a discontinuous Galerkin approximation of the Stokes problem based on an artificial compressibility flux, Internat. J. Numer. Methods Fluids 55 (2007), no. 8, 793–813. MR 2359551, DOI 10.1002/fld.1495
Daniele A. Di Pietro, Alexandre Ern, and Jean-Luc Guermond, Discontinuous Galerkin methods for anisotropic semidefinite diffusion with advection, SIAM J. Numer. Anal. 46 (2008), no. 2, 805–831. MR 2383212, DOI 10.1137/060676106
Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory, SIAM J. Numer. Anal. 44 (2006), no. 2, 753–778. MR 2218968, DOI 10.1137/050624133
Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs’ systems. II. Second-order elliptic PDEs, SIAM J. Numer. Anal. 44 (2006), no. 6, 2363–2388. MR 2272598, DOI 10.1137/05063831X
Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs’ systems. III. Multifield theories with partial coercivity, SIAM J. Numer. Anal. 46 (2008), no. 2, 776–804. MR 2383211, DOI 10.1137/060664045
R. Eymard, T. Gallouët, M. Ghilani, and R. Herbin, Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes, IMA J. Numer. Anal. 18 (1998), no. 4, 563–594. MR 1681074, DOI 10.1093/imanum/18.4.563
Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1086/phos.67.4.188705
—, Discretization schemes for heterogeneous and anisotropic diffusion problems on general nonconforming meshes, Available online as HAL report 00203269, January 2008, Submitted for publication.
R. Eymard, T. Gallouët, R. Herbin, and J.-C. Latché, Analysis tools for finite volume schemes, Acta Math. Univ. Comenian. (N.S.) 76 (2007), no. 1, 111–136. MR 2331058
R. Eymard, R. Herbin, and J. C. Latché, Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2D or 3D meshes, SIAM J. Numer. Anal. 45 (2007), no. 1, 1–36. MR 2285842, DOI 10.1137/040613081
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. MR 851383, DOI 10.1007/978-3-642-61623-5
Vivette Girault, Béatrice Rivière, and Mary F. Wheeler, A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems, Math. Comp. 74 (2005), no. 249, 53–84. MR 2085402, DOI 10.1090/S0025-5718-04-01652-7
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. MR 1618436, DOI 10.1137/S0036142996297199
L. I. G. Kovasznay, Laminar flow behind two-dimensional grid, Proc. Cambridge Philos. Soc. 44 (1948), 58–62. MR 24282
A. Lasis and E. Süli, Poincaré-type inequalities for broken Sobolev spaces, Tech. Report 03/10, Oxford University Computing Laboratory, Oxford, England, 2003.
Adrian Lew, Patrizio Neff, Deborah Sulsky, and Michael Ortiz, Optimal BV estimates for a discontinuous Galerkin method for linear elasticity, AMRX Appl. Math. Res. Express 3 (2004), 73–106. MR 2091832, DOI 10.1155/S1687120004020052
Igor Mozolevski, Endre Süli, and Paulo Rafael Bösing, Discontinuous Galerkin finite element approximation of the two-dimensional Navier-Stokes equations in stream-function formulation, Comm. Numer. Methods Engrg. 23 (2007), no. 6, 447–459. MR 2329348, DOI 10.1002/cnm.944
J. Nečas, Equations aux dérivées partielles, Presses de l’Université de Montréal, Montréal, Canada, 1965.
Roger Temam, Navier-Stokes equations, Revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979. Theory and numerical analysis; With an appendix by F. Thomasset. MR 603444