Discontinuous Galerkin approximations of the Stokes and Navier-Stokes equations

Chrysafinos, Konstantinos; Walkington, Noel

doi:10.1090/S0025-5718-10-02348-3

Discontinuous Galerkin approximations of the Stokes and Navier-Stokes equations
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by Konstantinos Chrysafinos and Noel J. Walkington PDF

Math. Comp. 79 (2010), 2135-2167 Request permission

Abstract:

Numerical schemes to compute approximate solutions of the evolutionary Stokes and Navier-Stokes equations are studied. The schemes are discontinuous in time and conforming in space and of arbitrarily high order. Fully-discrete error estimates are derived and dependence of the viscosity constant is carefully tracked. It is shown that the errors are bounded by projection errors of the exact solution which exhibit optimal rates when the solutions are smooth.

References

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
Garth A. Baker, Vassilios A. Dougalis, and Ohannes A. Karakashian, On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations, Math. Comp. 39 (1982), no. 160, 339–375. MR 669634, DOI 10.1090/S0025-5718-1982-0669634-0
Randolph E. Bank and Rafael F. Santos, Analysis of some moving space-time finite element methods, SIAM J. Numer. Anal. 30 (1993), no. 1, 1–18. MR 1202654, DOI 10.1137/0730001
Markus Bause, On optimal convergence rates for higher-order Navier-Stokes approximations. I. Error estimates for the spatial discretization, IMA J. Numer. Anal. 25 (2005), no. 4, 812–841. MR 2170525, DOI 10.1093/imanum/dri019
F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with French summary). MR 365287
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
K. Chrysafinos and Noel J. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 44 (2006), no. 1, 349–366. MR 2217386, DOI 10.1137/030602289
K. Chrysafinos and Noel J. Walkington, Error estimates for discontinuous Galerkin approximations of implicit parabolic equations, SIAM J. Numer. Anal. 43 (2006), no. 6, 2478–2499. MR 2206444, DOI 10.1137/040602948
Konstantinos Chrysafinos and Noel J. Walkington, Lagrangian and moving mesh methods for the convection diffusion equation, M2AN Math. Model. Numer. Anal. 42 (2008), no. 1, 25–55. MR 2387421, DOI 10.1051/m2an:2007053
Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132, DOI 10.1137/1.9780898719208
Bernardo Cockburn, George E. Karniadakis, and Chi-Wang Shu, The development of discontinuous Galerkin methods, Discontinuous Galerkin methods (Newport, RI, 1999) Lect. Notes Comput. Sci. Eng., vol. 11, Springer, Berlin, 2000, pp. 3–50. MR 1842161, DOI 10.1007/978-3-642-59721-3_{1}
Peter Constantin and Ciprian Foias, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. MR 972259, DOI 10.7208/chicago/9780226764320.001.0001
M. Delfour, W. Hager, and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations, Math. Comp. 36 (1981), no. 154, 455–473. MR 606506, DOI 10.1090/S0025-5718-1981-0606506-0
Todd F. Dupont and Yingjie Liu, Symmetric error estimates for moving mesh Galerkin methods for advection-diffusion equations, SIAM J. Numer. Anal. 40 (2002), no. 3, 914–927. MR 1949398, DOI 10.1137/S0036142900380431
Kenneth Eriksson, Claes Johnson, and Vidar Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 4, 611–643 (English, with French summary). MR 826227, DOI 10.1051/m2an/1985190406111
C. Ross Ethier and D. A. Steinman Exact fully 3D Navier-Stokes solutions for benchmarking IJNME, 19(5), July 2009, pp 369–375.
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
John G. Heywood and Rolf Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982), no. 2, 275–311. MR 650052, DOI 10.1137/0719018
John G. Heywood and Rolf Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. III. Smoothing property and higher order error estimates for spatial discretization, SIAM J. Numer. Anal. 25 (1988), no. 3, 489–512. MR 942204, DOI 10.1137/0725032
John G. Heywood and Rolf Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization, SIAM J. Numer. Anal. 27 (1990), no. 2, 353–384. MR 1043610, DOI 10.1137/0727022
L. S. Hou, Error estimates for semidiscrete finite element approximations of the Stokes equations under minimal regularity assumptions, J. Sci. Comput. 16 (2001), no. 3, 287–317. MR 1873285, DOI 10.1023/A:1012869611793
Pierre Jamet, Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain, SIAM J. Numer. Anal. 15 (1978), no. 5, 912–928. MR 507554, DOI 10.1137/0715059
Claes Johnson, Numerical solution of partial differential equations by the finite element method, Studentlitteratur, Lund, 1987. MR 911477
Th. D. Katsaounis, Fully discrete nonconforming finite element schemes for the nonstationary Navier-Stokes equations, East-West J. Numer. Math. 6 (1998), no. 4, 273–298. MR 1667308
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
Yingjie Liu, Randolph E. Bank, Todd F. Dupont, Sonia Garcia, and Rafael F. Santos, Symmetric error estimates for moving mesh mixed methods for advection-diffusion equations, SIAM J. Numer. Anal. 40 (2002), no. 6, 2270–2291 (2003). MR 1974185, DOI 10.1137/S003614290038073X
O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math. 38 (1981/82), no. 3, 309–332. MR 654100, DOI 10.1007/BF01396435
Shapiro, A The use of an exact solution of the Navier-Stokes equations in a validation test of a three-dimensional non-hydrostatic numerical model, Mon. Wea. Rev., 121, 1993, pp 2420–2425.
Endre Süli, Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math. 53 (1988), no. 4, 459–483. MR 951325, DOI 10.1007/BF01396329
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
Roger Temam, Navier-Stokes equations and nonlinear functional analysis, 2nd ed., CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. MR 1318914, DOI 10.1137/1.9781611970050
Vidar Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR 1479170, DOI 10.1007/978-3-662-03359-3
C. Y. Wang, Exact solutions of the unsteady Navier-Stokes equations, AMR 42 (1989), no. 11, S269–S282. Mechanics Pan-America 1989 (Rio de Janeiro, 1989). MR 1037589, DOI 10.1115/1.3152400
C. Y. Wang, Exact solutions of the steady-state Navier-Stokes equations, Annual review of fluid mechanics, Vol. 23, Annual Reviews, Palo Alto, CA, 1991, pp. 159–177. MR 1090331
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
N. J. Walkington, Compactness properties of the DG and CG time stepping schemes for parabolic equations, submitted to SINUM (June 2008), preprint available at http://www.math.cmu.edu/noelw.