Convergence of a mixed method for a semi-stationary compressible Stokes system

Karlsen, Kenneth; Karper, Trygve

doi:10.1090/S0025-5718-2010-02446-9

Convergence of a mixed method for a semi-stationary compressible Stokes system
HTML articles powered by AMS MathViewer

by Kenneth H. Karlsen and Trygve K. Karper PDF

Math. Comp. 80 (2011), 1459-1498 Request permission

Abstract:

We propose and analyze a finite element method for a semi- stationary Stokes system modeling compressible fluid flow subject to a Navier-slip boundary condition. The velocity (momentum) equation is approximated by a mixed finite element method using the lowest order Nédélec spaces of the first kind, while the continuity equation is approximated by a piecewise constant upwind discontinuous Galerkin method. Our main result states that the numerical method converges to a weak solution. The convergence proof consists of two main steps: (i) To establish strong spatial compactness of the velocity field, which is intricate since the element spaces are only $\operatorname {div}$ or $\operatorname {curl}$ conforming. (ii) To prove the strong convergence of the discontinuous Galerkin approximations, which is required in view of a nonlinear pressure function. Some tools involved in the analysis include a higher space-time integrability estimate for the discontinuous Galerkin approximations, an equation for the effective viscous flux, various renormalized formulations of the discontinuous Galerkin method, and weak convergence arguments.

References

Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1–155. MR 2269741, DOI 10.1017/S0962492906210018
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Multigrid in $H(\textrm {div})$ and $H(\textrm {curl})$, Numer. Math. 85 (2000), no. 2, 197–217. MR 1754719, DOI 10.1007/PL00005386
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
S. C. Brenner, J. Cui, F. Li, and L.-Y. Sung, A nonconforming finite element method for a two-dimensional curl-curl and grad-div problem, Numer. Math. 109 (2008), no. 4, 509–533. MR 2407321, DOI 10.1007/s00211-008-0149-7
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
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Gui-Qiang Chen, David Hoff, and Konstantina Trivisa, Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data, Comm. Partial Differential Equations 25 (2000), no. 11-12, 2233–2257. MR 1789926, DOI 10.1080/03605300008821583
Bernardo Cockburn, Discontinuous Galerkin methods for convection-dominated problems, High-order methods for computational physics, Lect. Notes Comput. Sci. Eng., vol. 9, Springer, Berlin, 1999, pp. 69–224. MR 1712278, DOI 10.1007/978-3-662-03882-6_{2}
Bernardo Cockburn and Chi-Wang Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput. 16 (2001), no. 3, 173–261. MR 1873283, DOI 10.1023/A:1012873910884
Snorre H. Christiansen, Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension, Numer. Math. 107 (2007), no. 1, 87–106. MR 2317829, DOI 10.1007/s00211-007-0081-2
Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547. MR 1022305, DOI 10.1007/BF01393835
Eduard Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 26, Oxford University Press, Oxford, 2004. MR 2040667
T. Gallouët, R. Herbin, and J.-C. Latché, A convergent finite element-finite volume scheme for the compressible Stokes problem. I. The isothermal case, Math. Comp. 78 (2009), no. 267, 1333–1352. MR 2501053, DOI 10.1090/S0025-5718-09-02216-9
Thierry Gallouët, Laura Gastaldo, Raphaele Herbin, and Jean-Claude Latché, An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations, M2AN Math. Model. Numer. Anal. 42 (2008), no. 2, 303–331. MR 2405150, DOI 10.1051/m2an:2008005
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
David Hoff, Global existence for $1$D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc. 303 (1987), no. 1, 169–181. MR 896014, DOI 10.1090/S0002-9947-1987-0896014-6
David Hoff, Discontinuous solutions of the Navier-Stokes equations for compressible flow, Arch. Rational Mech. Anal. 114 (1991), no. 1, 15–46. MR 1088275, DOI 10.1007/BF00375683
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) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89–123. MR 0658142
Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 1, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models; Oxford Science Publications. MR 1422251
Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press, Oxford University Press, New York, 1998. Compressible models; Oxford Science Publications. MR 1637634
Chun Liu and Noel J. Walkington, Convergence of numerical approximations of the incompressible Navier-Stokes equations with variable density and viscosity, SIAM J. Numer. Anal. 45 (2007), no. 3, 1287–1304. MR 2318813, DOI 10.1137/050629008
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
W. Reed and T. Hill. Triangular mesh methods for the neutron transport equation. Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
Friedrich Stummel, Basic compactness properties of nonconforming and hybrid finite element spaces, RAIRO Anal. Numér. 14 (1980), no. 1, 81–115 (English, with French summary). MR 566091
Richard S. Varga, Matrix iterative analysis, Second revised and expanded edition, Springer Series in Computational Mathematics, vol. 27, Springer-Verlag, Berlin, 2000. MR 1753713, DOI 10.1007/978-3-642-05156-2
Noel J. Walkington, Convergence of the discontinuous Galerkin method for discontinuous solutions, SIAM J. Numer. Anal. 42 (2005), no. 5, 1801–1817. MR 2139223, DOI 10.1137/S0036142902412233
Roger Zarnowski and David Hoff, A finite-difference scheme for the Navier-Stokes equations of one-dimensional, isentropic, compressible flow, SIAM J. Numer. Anal. 28 (1991), no. 1, 78–112. MR 1083325, DOI 10.1137/0728004
Jing Zhao and David Hoff, A convergent finite-difference scheme for the Navier-Stokes equations of one-dimensional, nonisentropic, compressible flow, SIAM J. Numer. Anal. 31 (1994), no. 5, 1289–1311. MR 1293516, DOI 10.1137/0731067
Jennifer Jing Zhao and David Hoff, Convergence and error bound analysis of a finite-difference scheme for the one-dimensional Navier-Stokes equations, Nonlinear evolutionary partial differential equations (Beijing, 1993) AMS/IP Stud. Adv. Math., vol. 3, Amer. Math. Soc., Providence, RI, 1997, pp. 625–631. MR 1468542, DOI 10.1090/amsip/003/65