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Spectral Element Discretization of the Stokes Equations in Deformed Axisymmetric Geometries

Published online by Cambridge University Press:  03 June 2015

Zakaria Belhachmi  and
Andreas Karageorghis
Zakaria Belhachmi*
Affiliation:
Laboratoire de Mathématiques, Informatique et Applications, EA CNRS, Université de Haute Alsace, Rue des Frères Lumières, 68096 Mulhouse, France
Andreas Karageorghis*
Affiliation:
Department of Mathematics and Statistics, University of Cyprus, P.O.Box 20537, 1678 Nicosia, Cyprus
*
Corresponding author. URL: http://www.math.uha.fr/belhachmi/ Email: [email protected]
Email: [email protected]
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Abstract

In this paper, we study the numerical solution of the Stokes system in deformed axisymmetric geometries. In the azimuthal direction the discretization is carried out by using truncated Fourier series, thus reducing the dimension of the problem. The resulting two-dimensional problems are discretized using the spectral element method which is based on the variational formulation in primitive variables. The meridian domain is subdivided into elements, in each of which the solution is approximated by truncated polynomial series. The results of numerical experiments for several geometries are presented.

Keywords

65N3565N55Spectral element methodStokes equationsvariational formulationdeformed geometriesFourier expansion
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 4 , August 2011 , pp. 448 - 469
Copyright
Copyright © Global-Science Press 2011

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References

[1] Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1993.Google Scholar
[2] Belhachmi, Z., Bernardi, C., Deparis, S. and Hecht, F., A truncated Fourier/finite element discretization of the Stokes equations in an axisymmetric domain, Math. Models Methods Appl. Sci., 16 (2006), pp. 233–263.CrossRefGoogle Scholar
[3] Belhachmi, Z., Bernardi, C., Deparis, S. and Hecht, F., An efficient discretization of the Navier-Stokes equations in an axisymmetric domain, part 1: the discrete problem and its numerical analysis, J. Sci. Comput., 27 (2006), pp. 97–110.CrossRefGoogle Scholar
[4] Belhachmi, Z., Bernardi, C. and Karageorghis, A., Spectral element discretization of the circular driven cavity, part II: the bilaplacian equation, SIAM J. Numer. Anal., 38 (2001), pp. 1926–1960.CrossRefGoogle Scholar
[5] Belhachmi, Z., Bernardi, C. and Karageorghis, A., Spectral element discretization of the circular driven cavity, part III: the Stokes equations in primitive variables, J. Math. Fluid. Mech., 5 (2003), pp. 24–69.Google Scholar
[6] Belhachmi, Z. and Karageorghis, A., A spectral mortar element discretization of the Poisson equation with mixed boundary conditions, J. Sci. Comput., 30 (2007), pp. 275–299.CrossRefGoogle Scholar
[7] Belhachmi, Z. and Karageorghis, A., Spectral element discretization of the Stokes equations in complex geometries: implementational details, Technical Report TR-06-2009, Department of Mathematics and Statistics, University of Cyprus, 2009.Google Scholar
[8] Belgacem, F. Ben, The mortar finite element method with Lagrange multipliers, Numer. Math., 84 (1999), pp. 173–197.CrossRefGoogle Scholar
[9] Bernardi, C., Dauge, M., Maday, Y. and Azaïez, M., Spectral Methods for Axisym-metric Domains, Series in Applied Mathematics 3, Gauthier-Villars & North-Holland, Paris, 1999.Google Scholar
[10] Bernardi, C. and Maday, Y., Approximations Spectrales de Problémes aux Limites Elliptiques, Springer-Verlag, Paris, 1992.Google Scholar
[11] Bernardi, C. and Maday, Y., Spectral methods, in Handbook of Numerical Analysis Ciarlet, V, P. G. and Lions, J.-L. eds, North-Holland, pp. 209–485, 1997.Google Scholar
[12] Ciarlet, P. G., Introduction à l’Analyse Numérique Matricielle et à L’Optimisation, Collection “Mathématiques Appliquées pour la Mîtrise”, Masson, Paris, 1982.Google Scholar
[13] Girault, V. and Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag, Berlin, 1986.Google Scholar
[14] Lions, J.-L. and Magenes, E., Problémes aux Limites Non Homogénes et Applications, Dunod, Paris, 1968.Google Scholar
[15] Maday, Y. and Ronquist, E. M., Optimal error analysis of spectral methods with emphasis on non-constants and deformed geometries, Comput. Methods Appl. Mech. Engrg., 80 (1990), pp. 91–115.CrossRefGoogle Scholar