# zbMATH — the first resource for mathematics

Finite element methods for the Stokes system in three-dimensional exterior domains. (English) Zbl 0870.76041
The article treats the question of how to numerically solve the Dirichlet problem for the Stokes system in the exterior of a three-dimensional bounded Lipschitz domain. At the first step, the solution is approximated by functions solving the Stokes system in a truncated domain and satisfying a suitable artificial boundary condition on the outer boundary of this truncated domain. At the second step, this new problem is approximately solved in finite element spaces related to a graded mesh. The difference between this finite element approximation and the exact solution of the exterior Stokes problem is estimated in the norm of suitable unweighted $$L^2$$-Sobolev spaces.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76D07 Stokes and related (Oseen, etc.) flows 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text:
##### References:
 [1] Agmon, Comm. Pure Appl. Math. 17 pp 35– (1964) [2] and , The Mathematical Theory of Finite Element Methods, Springer, New York, 1994. · doi:10.1007/978-1-4757-4338-8 [3] Cattabriga, Rend. Sem. Mat. Univ. Padova 31 pp 308– (1961) [4] Dauge, SIAM J. Math. Anal. 20 pp 74– (1989) [5] Deuring, Math. Meth. in the Appl. Sci. 13 pp 335– (1990) [6] Deuring, Math. Meth. in the Appl. Sci. 13 pp 323– (1990) [7] Math. Meth. in the Appl. Sci. 14 pp 445– (1991) [8] Deuring, J. Appl. Math. Phys. (ZAMP) 41 pp 829– (1990) [9] Deuring, Commun. Part. Diff. Equ. 16 pp 1513– (1991) [10] Deuring, Proc. Royal Soc. Edinburgh 122A pp 1– (1992) · Zbl 0780.35074 · doi:10.1017/S0308210500020916 [11] The Stokes System in an Infinite Cone, Akademie Verlag, Berlin, 1994. · Zbl 0799.35177 [12] ’Stable mixed finite element methods in truncated exterior domains’, submitted. [13] Deuring, Math. Nachr. 157 pp 277– (1992) [14] Deuring, Bayreuth. Math. Schr. 28 pp 1– (1989) [15] Deuring, Math. Nachr. 171 pp 111– (1995) [16] Deuring, Bayreuth. Math. Schr. 17 pp 1– (1988) [17] Fabes, Duke Math. J. 57 pp 769– (1988) [18] and , Function Spaces, Noordhoff, Leyden, 1977. [19] An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Volume I. Linearized Steady Problems, Springer, New York, 1994. [20] and , Finite Element Methods for Navier-Stokes Equations, Springer, Berlin, 1986. · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5 [21] Numerical Methods for Nonlinear Variational Problems, Springer, New York, 1984. · doi:10.1007/978-3-662-12613-4 [22] Goldstein, Math. Comp. 36 pp 387– (1981) [23] Goldstein, SIAM J. Numer. Anal. 30 pp 159– (1993) [24] Guirguis, SIAM J. Numer. Anal. 24 pp 310– (1987) [25] Guirguis, RAIRO Modél. Math. Anal. Numér. 21 pp 445– (1987) [26] Han, J. Comput. Math. 3 pp 179– (1985) [27] Han, Math. Comp. 59 pp 21– (1992) [28] Hebeker, Numer. Methods Partial Differential Equations 2 pp 273– (1986) [29] Kozlov, J. Reine Angew. Math. 456 pp 65– (1994) [30] Kozono, Indiana Univ. Math. J. 40 pp 1– (1991) [31] and , ’Approximation of exterior problems. Optimal conditions for the Laplacian’, Preprint, Universität-GH Paderborn, 1995. [32] Sequeira, Math. Meth. in the Appl. Sci. 5 pp 356– (1983) [33] Sequeira, Math. Meth. in the Appl. Sci. 8 pp 117– (1986) [34] and , ’A new approach to the Helmholtz decomposition and the Neumann problem in Lq-spaces for bounded and exterior domains’, in: Series on Advances in Mathematics for Applied Science, Vol. II. pp. 1-35, World Scientific, Singapore, 1992. [35] Navier-Stokes Equations, North-Holland, Amsterdam, Singapore, 1977. [36] The Stokes Equations, Akademie Verlag, Berlin, 1994. [37] ’Abschätzungen für das Neumann-Problem und die Helmholtz-Zerlegung von Lp”, Nachr. Akad. d. Wis. Göttingen, II. Math.-Phys. Klasse. Jahrgang 1990, No. 2. [38] ’Vorlesung über das Außenraumproblem für die instationären Gleichungen von Navier-Stokes’, Vorlesungsreihe No. 11 (Rudolph-Lipschitz-Vorlesung) SFB 256, Bonn, 1990. [39] Wiegner, Math. Meth. in the Appl. Sci. 16 pp 877– (1993)
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.