Previous |  Up |  Next

Article

Title: Stability of a finite element method for 3D exterior stationary Navier-Stokes flows (English)
Author: Deuring, Paul
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 52
Issue: 1
Year: 2007
Pages: 59-94
Summary lang: English
.
Category: math
.
Summary: We consider numerical approximations of stationary incompressible Navier-Stokes flows in 3D exterior domains, with nonzero velocity at infinity. It is shown that a P1-P1 stabilized finite element method proposed by C. Rebollo: A term by term stabilization algorithm for finite element solution of incompressible flow problems, Numer. Math. 79 (1998), 283–319, is stable when applied to a Navier-Stokes flow in a truncated exterior domain with a pointwise boundary condition on the artificial boundary. (English)
Keyword: stationary incompressible Navier-Stokes flows
Keyword: exterior domains
Keyword: stabilized finite element methods
Keyword: stability estimates
MSC: 35Q30
MSC: 65N30
MSC: 76D05
MSC: 76M10
idZBL: Zbl 1164.35455
idMR: MR2293528
DOI: 10.1007/s10492-007-0003-8
.
Date available: 2009-09-22T18:28:24Z
Last updated: 2020-07-02
Stable URL: http://hdl.handle.net/10338.dmlcz/134663
.
Reference: [1] R. A.  Adams: Sobolev Spaces.Academic Press, New York, 1975. Zbl 0314.46030, MR 0450957
Reference: [2] F.  Alouges, J.  Laminie, and S. M.  Mefire: Exponential meshes and three-dimensional computation of a magnetic field.Numer. Methods Partial Differ. Equations 19 (2003), 592–637. MR 1996222, 10.1002/num.10064
Reference: [3] K. I.  Babenko, M. M.  Vasil’ev: On the asymptotic behavior of a steady flow of viscous fluid at some distance from an immersed body.J.  Appl. Math. Mech. 37 (1973), 651–665. MR 0347214, 10.1016/0021-8928(73)90115-9
Reference: [4] S.  Bönisch, V.  Heuveline, and P.  Wittwer: Adaptive boundary conditions for exterior flow problems.J.  Math. Fluid Mech. 7 (2005), 85–107. MR 2127743, 10.1007/s00021-004-0108-8
Reference: [5] S. C.  Brenner, L. R.  Scott: The Mathematical Theory of Finite Element Methods, 2nd edition.Springer-Verlag, New York, 2002. MR 1894376
Reference: [6] F.  Brezzi, M.  Fortin: Mixed and Hybrid Finite Element Methods.Springer-Verlag, New York, 1991. MR 1115205
Reference: [7] C.-H.  Bruneau: Boundary conditions on artificial frontiers for incompressible and compressible Navier-Stokes equations.M2AN, Math. Model. Numer. Anal. 34 (2000), 303–314. Zbl 0954.76014, MR 1765661, 10.1051/m2an:2000142
Reference: [8] C.-H.  Bruneau, P.  Fabrie: New efficient boundary conditions for incompressible Navier-Stokes equations: a well-posedness result.M2AN, Math. Model. Numer. Anal. 30 (1996), 815–840. MR 1423081, 10.1051/m2an/1996300708151
Reference: [9] C.  Calgaro, P.  Deuring, and D.  Jennequin: A preconditioner for generalized saddle point problems: application to 3D  stationary Navier-Stokes equations.Numer. Methods Partial Differ. Equations 22 (2006), 1289-1313. MR 2257634, 10.1002/num.20154
Reference: [10] P.  Deuring: Finite element methods for the Stokes system in three-dimensional exterior domains.Math. Methods Appl. Sci. 20 (1997), 245–269. Zbl 0870.76041, MR 1430495, 10.1002/(SICI)1099-1476(199702)20:3<245::AID-MMA856>3.0.CO;2-F
Reference: [11] P.  Deuring: A stable mixed finite element method on truncated exterior domains.M2AN, Math. Model. Numer. Anal. 32 (1998), 283–305. Zbl 0904.65108, MR 1627147, 10.1051/m2an/1998320302831
Reference: [12] P.  Deuring: Approximating exterior flows by flows on truncated exterior domains: piecewise polygonial artificial boundaries.In: Elliptic and Parabolic problems. Proceedings of the 4th  European Conference, Rolduc and Gaeta, 2001, J.  Bemelmans (ed.), World Scientific, Singapore, 2002, pp. 364–376. MR 1937556
Reference: [13] P.  Deuring: Exterior stationary Navier-Stokes flows in  3D with non-zero velocity at infinity: asymptotic behaviour of the second derivatives of the velocity.Commun. Partial Differ. Equations 30 (2005), 987–1020. MR 2180292, 10.1081/PDE-200064436
Reference: [14] P.  Deuring: A finite element method for 3D  exterior Oseen flows: error estimates.Submitted. Zbl 1148.35062
Reference: [15] P.  Deuring, S.  Kračmar: Artificial boundary conditions for the Oseen system in 3D  exterior domains.Analysis 20 (2000), 65–90. MR 1757070, 10.1524/anly.2000.20.1.65
Reference: [16] P.  Deuring, S.  Kračmar: Exterior stationary Navier-Stokes flows in  3D with non-zero velocity at infinity: approximation by flows in bounded domains.Math. Nachr. 269–270 (2004), 86–115. MR 2074775, 10.1002/mana.200310167
Reference: [17] R.  Farwig: A variational approach in weighted Sobolev spaces to the operator $- \Delta + \partial / \partial x_1$ in exterior domains of  $\mathbb{R}^3$.Math.  Z. 210 (1992), 449–464. MR 1171183
Reference: [18] R.  Farwig: The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces.Math.  Z. 211 (1992), 409–447. MR 1190220, 10.1007/BF02571437
Reference: [19] M.  Feistauer, C.  Schwab: On coupled problems for viscous flows in exterior domains.Math. Models Methods Appl. Sci. 8 (1998), 658–684. MR 1634842, 10.1142/S0218202598000305
Reference: [20] M.  Feistauer, C.  Schwab: Coupled problems for viscous incompressible flow in exterior domains.In: Applied Nonlinear Analysis, A. Sequeira (ed.), Kluwer/Plenum, New York, 1999, pp. 97–116. MR 1727443
Reference: [21] M.  Feistauer, C.  Schwab: Coupling of an interior Navier-Stokes problem with an exterior Oseen problem.J.  Math. Fluid Mech. 3 (2001), 1–17. MR 1830652, 10.1007/PL00000961
Reference: [22] R.  Finn: On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems.Arch. Ration. Mech. Anal. 19 (1965), 363–406. Zbl 0149.44606, MR 0182816, 10.1007/BF00253485
Reference: [23] G. P.  Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol.  I. Linearized Steady Problems (rev. ed.).Springer-Verlag, New York, 1998. MR 1284205
Reference: [24] G. P.  Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol.  II. Nonlinear Steady Problems.Springer-Verlag, New York, 1994. Zbl 0949.35005, MR 1284206
Reference: [25] V.  Girault, P.-A.  Raviart: Finite Element Methods for Navier-Stokes Equations.Springer-Verlag, Berlin, 1986. MR 0851383
Reference: [26] C.  I.  Goldstein: The finite element method with nonuniform mesh sizes for unbounded domains.Math. Comput. 36 (1981), 387–404. Zbl 0467.65058, MR 0606503, 10.1090/S0025-5718-1981-0606503-5
Reference: [27] C. I.  Goldstein: Multigrid methods for elliptic problems in unbounded domains.SIAM J.  Numer. Anal. 30 (1993), 159–183. Zbl 0772.65075, MR 1202661, 10.1137/0730008
Reference: [28] P.  Grisvard: Elliptic Problems in Nonsmooth Domains.Pitman, Boston, 1985. Zbl 0695.35060, MR 0775683
Reference: [29] G. H.  Guirguis: On the coupling of boundary integral and finite element methods for the exterior Stokes problem in  3D.SIAM J.  Numer. Anal. 24 (1987), 310–322. MR 0881366, 10.1137/0724023
Reference: [30] G. H.  Guirguis, M. D.  Gunzburger: On the approximation of the exterior Stokes problem in three dimensions.M2AN, Math. Model. Numer. Anal. 21 (1987), 445–464. MR 0908240, 10.1051/m2an/1987210304451
Reference: [31] M. D.  Gunzburger: Finite Element Methods for Viscous Incompressible Flows.Academic Press, Boston, 1989. Zbl 0697.76031, MR 1017032
Reference: [32] L.  Halpern, M.  Schatzman: Artificial boundary conditions for incompressible viscous flows.SIAM J.  Math. Anal. 20 (1989), 308–353. MR 0982662, 10.1137/0520021
Reference: [33] Yinnian He: Coupling boundary integral and finite element methods for the Oseen coupled problem.Comput. Math. Appl. 44 (2002), 1413–1429. Zbl 1037.76039, MR 1938777, 10.1016/S0898-1221(02)00266-3
Reference: [34] J. G.  Heywood, R.  Rannacher, and S.  Turek: Artificial boundaries and flux and pressure conditions for incompressible Navier-Stokes equations.Int. J.  Numer. Methods Fluids 22 (1996), 325–352. MR 1380844, 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y
Reference: [35] S.  Kračmar, J.  Neustupa: Global existence of weak solutions of a nonsteady variational inequality of the Navier-Stokes type with mixed boundary conditions.In: Proceedings of the International Symposium on Numerical Analysis (ISNA’92), Charles University, Prague, 1993, pp. 156–177.
Reference: [36] S.  Kračmar, J.  Neustupa: A weak solvability of a steady variational inequality of the Navier-Stokes type with mixed boundary conditions.Nonlinear Anal., Theory Methods Appl. 47 (2001), 4169–4180. MR 1972357, 10.1016/S0362-546X(01)00534-X
Reference: [37] P.  Kučera: Solutions of the Navier-Stokes equations with mixed boundary conditions in a bounded domain.In: Analysis, Numerics and Applications of Differential and Integral Equations. Pitman Research Notes in Mathematics Series  379, M. Bach (ed.), Addison Wesley, London, 1998, pp. 127–131. MR 1606691
Reference: [38] P.  Kučera: A structure of the set of critical points to the Navier-Stokes equations with mixed boundary conditions.In: Navier-Stokes Equations: Theory and Numerical Methods. Pitman Research Notes in Mathematics Series  388, R. Salvi (ed.), Addison Wesley, London, 1998, pp. 201–205. MR 1773598
Reference: [39] P.  Kučera, Z.  Skalák: Local solutions to the Navier-Stokes equations with mixed boundary conditions.Acta Appl. Math. 54 (1998), 275–288. MR 1671783, 10.1023/A:1006185601807
Reference: [40] S. A.  Nazarov, M.  Specovius-Neugenbauer: Approximation of exterior problems. Optimal conditions for the Laplacian.Analysis 16 (1996), 305–324. MR 1429456, 10.1524/anly.1996.16.4.305
Reference: [41] S. A.  Nazarov, M.  Specovius-Neugenbauer: Approximation of exterior boundary value problems for the Stokes system.Asymptotic Anal. 14 (1997), 233–255. MR 1458705
Reference: [42] S.  A.  Nazarov, M.  Specovius-Neugebauer: Nonlinear artificial boundary conditions with pointwise error estimates for the exterior three dimensional Navier-Stokes problem.Math. Nachr. 252 (2003), 86–105. MR 1903042, 10.1002/mana.200310039
Reference: [43] J.  Nečas: Les méthodes directes en théorie des équations elliptiques.Masson, Paris, 1967. MR 0227584
Reference: [44] K.  Nishida: Numerical method for Oseen’s linearized equations in three-dimensional exterior domains.J.  Comput. Appl. Math. 152 (2003), 405–409. Zbl 1059.76055, MR 1991305, 10.1016/S0377-0427(02)00770-7
Reference: [45] A.  Quarteroni, A.  Valli: Numerical Approximation of Partial Differential Equations.Springer-Verlag, New York, 1994. MR 1299729
Reference: [46] T. C.  Rebollo: A term by term stabilization algorithm for finite element solution of incompressible flow problems.Numer. Math. 79 (1998), 283–319. Zbl 0910.76033, MR 1622522, 10.1007/s002110050341
Reference: [47] A.  Sequeira: The coupling of boundary integral and finite element methods for the bidimensional exterior steady Stokes problem.Math. Methods Appl. Sci. 5 (1983), 356–375. Zbl 0521.76034, MR 0716661, 10.1002/mma.1670050124
Reference: [48] A.  Sequeira: On the computer implementation of a coupled boundary and finite element method for the bidimensional exterior steady Stokes problem.Math. Methods Appl. Sci. 8 (1986), 117–133. Zbl 0619.76039, MR 0833255, 10.1002/mma.1670080109
Reference: [49] S. V.  Tsynkov: Numerical solution of problems on unbounded domains. A review.Appl. Numer. Math. 27 (1998), 465–532. Zbl 0939.76077, MR 1644674, 10.1016/S0168-9274(98)00025-7
.

Files

Files Size Format View
AplMat_52-2007-1_3.pdf 505.2Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo