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Keywords:
Stokes equations; nonstandard boundary conditions; finite element method; approximation of boundary
Stokes equations; nonstandard boundary conditions; finite element method; approximation of boundary
Summary:
We investigate a finite element discretization of the Stokes equations with nonstandard boundary conditions, defined in a bounded three-dimensional domain with a curved, piecewise smooth boundary. For tetrahedral triangulations of this domain we prove, under general assumptions on the discrete problem and without any additional regularity assumptions on the weak solution, that the discrete solutions converge to the weak solution. Examples of appropriate finite element spaces are given.
References:
[1] F. Brezzi, M. Fortin: Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, 1991. MR 1115205
[2] P. G. Ciarlet: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, v. II – Finite Element Methods (Part 1), P. G. Ciarlet, J. L. Lions (eds.), North-Holland, Amsterdam, 1991, pp. 17–351. MR 1115237 | Zbl 0875.65086
[3] P. G. Ciarlet, P.-A. Raviart: The combined effect of curved boundaries and numerical integration in isoparametric finite element methods. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. K. Aziz (ed.), Academic Press, New York, 1972, pp. 409–474. MR 0421108
[4] G. J. Fix, M. D. Gunzburger and J. S. Peterson: On finite element approximations of problems having inhomogeneous essential boundary conditions. Comput. Math. Appl. 9 (1983), 687–700. MR 0726817
[5] V. Girault, P.-A. Raviart: Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin, 1986. MR 0851383
[6] P. Knobloch: Solvability and Finite Element Discretization of a Mathematical Model Related to Czochralski Crystal Growth. PhD Thesis, Preprint MBI-96-5, Otto-von-Guericke-Universität, Magdeburg, 1996. Zbl 0865.65094
[7] P. Knobloch: Variational crimes in a finite element discretization of 3D Stokes equations with nonstandard boundary conditions. East-West J. Numer. Math. 7 (1999), 133–158. MR 1699239 | Zbl 0958.76043
[8] J. Nečas: Les Méthodes Directes en Théorie des Équations Elliptiques. Academia, Praha, 1967. MR 0227584
[9] E. M. Stein: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, 1970. MR 0290095 | Zbl 0207.13501
[10] G. Strang, G. J. Fix: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, New Jersey, 1973. MR 0443377
[11] A. Ženíšek: How to avoid the use of Green’s theorem in the Ciarlet-Raviart theory of variational crimes. RAIRO, Modelisation Math. Anal. Numer. 21 (1987), 171–191. DOI 10.1051/m2an/1987210101711 | MR 0882690
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