Methodes d 'elements spectra ox pour des problemes hybrides duaux de second ordre axisymetriques
Application a l'algorithme de projection de Goda
Keywords:
axisymmetric, mixed spectral elements, projection, Darcy, Navier-StokesAbstract
The subject of this paper consists in studying the second order elliptic equations under the hybrid dual formulation, in an axisymmetric context and their discretization by spectral methods. Next, we present a spectral approximation, based on staggered grids, without spurious modes and we give optimal error estimates in mono- and multi-domain configurations. We describe the implementation of this method and carry out a numerical discussion to quantify its efficiency. Then, we use this technique for the approximation of the projection step of second order Goda algorithm applied to the numerical simulation of the unsteady Stokes equations and we prove unconditonal stability of the resulting scheme. Finally, some numerical experiments are provided showing the ability of such a technique to simulate a axisymetric Navier-Stokes flow.
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[BAB 87] BABUVSKA 1., SURI M., « The optimal Convergence Rate on the p-Version of the
Finite Element Method "• SIAM J. Numer. Anal., vol. 24, 1987, p. 750-776.
[BER 91] BERNARDI C., MADAY Y., « Polynomial Approximation of Some Singular Functions
"• Applicable Analysis: an International Journal, vol. 10, 1991, p. 1-32.
[BER 92] BERNARDI C., MADAY Y., Approximations spectrales de problemes aux limites
elliptiques, Mathematiques et Applications, Paris, 1992.
[BOL 83] BOLAND J., NICOLAIDES R., « Stability of Finite Element under Divergence
Constraints "• SIAM 1. Num. Anal., vol. 20, 1983, p. 722-731.
[BOU 94] BOUKIR K., ~ Methode en temps d'ordre eleve par decomposition d'operateurs.
Application aux equations de Navier-Stokes •. These de doctoral, Universite Pierre et
Marie Curie, 1994.
[BRE 74] BREZZI F., « On the Existence, Uniqueness and Approximation of Saddle Point
Problems Arizing from Lagrange Multipliers •, R.A.I.R.O. Anal. Numer., vol. 8 R2, 1974,
p. 129-151.
[BRE 91] BREZZI F., FORTIN M., Mixed and Hibryd Finite Element Methods, Springer Series
in Computational Mathematics, Springer Verlag, New York, 1991.
[C.B 92] C. BERNARDI M. D., MADAY Y.," Polynomials in Weighted Sobolev Spaces: Basics
and Trace Liftings "• rapport de recherche n° R92039, 1992, Laboratoire d' Analyse
Numerique Universite Pierre et Marie Curie, Paris 6.
[C.B 94] C. BERNARDI M. D., MADAY Y., « Numerical Analysis and Spectral Methods in
Axisymmetric Domains, Part 1: Functional Prerequisite "· rapport de recherche n° R94008,
, Laboratoire d' Analyse Numerique Universite Pierre et Marie Curie, Paris 6.
[C.B 99] C. BERNARDI M. D., MADAY Y., Spectral Methods for Axisymmetric Domains,
Series in Applied Mathematics, Paris, 1999.
[C.C 87] C. CANUTO M.Y. HUSSAIN! A. Q., ZANG T., Spectral Methods in Fluid Dynamics,
Springer-Verlag, 1987.
[CHO 68] CHORIN A., « Numerical Simulation of the Navier-Stokes Equations »,
Math. Comput., vol. 22, 1968, p. 745-762.
[DAU 87] DAUTRAY R., LIONS J.-L., Analyse mathematique et calcul nwnerique pour les
sciences etles techniques, Masson, Paris, 1987.
[DAU 88] DAUGE M., Elliptic Boundary Value Problems on Corner Domains, Lecture Notes
in Mathematic, Paris, 1988.
[GIR 81] GIRAULT V., RAVIART P.-A., Finite Elements Approximation for the Navier-Stokes
Equations, Lectures Notes in Mathematics. A. Do1d and B. Eckman eds, Paris, 1981.
[GIR 86] GIRAULT V., RAVIART P.-A., Finite Element Methods for the Navier-Stokes Equations,
Theory and Algorithms, Springer-Verlag, Paris, 1986.
[GOD 79] GODA K., « A Multistep Technique with Implicit Difference Schemes for Calculating
vo and Three dimensional Cavity Flows "• 1. Compu. Physics, vol. 30, 1979,
p. 76-95.
[GUE 94] GUERMOND J.-L., « Remarques sur les methodes de projection pour !'approximation
des equations de Navier-Stokes "• Numer. Math., vol. 67, 1994, p. 467-473.
[JEN 90] JENSEN S., VOGELIUS M.," Divergence Stability in Connection with the p- Version
of the Finite Element Method "·Model. Math. et Anal. Numer., vol. 24, 1990, p. 737-767.
[KAN 86] KAN J. V., « A Second Order Accurate Pressure-Correction Scheme for Viscous
Incompressible Flow "• SIAM J. Sci. Compu., vol. 7, 1986, p. 870-891.
[M.A 94] M. AZAIEZ C. B., GRUNDMANN M., ~ Spectral MethJd Applied to Porous Media
»,East-West J. Numer. Math., vol. 2, 1994, p. 91-105.
[M.A 95] M. AZAIEZ M. D., MADAY Y., Methodes spectrales et des elements spectraux,
Editions Didactique INRIA, Paris, 1995.
[M.A 98] M. AZAIEZ F. BEN BELGACEM M.G., KHALLOUF H.,~ Staggered Grids Hybrid
Dual Spectral Element Method. Application to the High-Order Time Splitting Schemes for
Navier-Stokes problem », Computer Methods of Appl. Mech. Engrg., vol. 166, 1998,
p. 183-199.
[MAD 92] MADAY Y., PAVONI D., « Spectral Approximation of Axisymmetric Stokes
Flow. », rapport de recherche n° R92034, 1992, Laboratoire d' Analyse Numerique Universite
Pierre et Marie Curie, Paris 6.
[MAD 94] MADAY Y., PATERA A., • Spectral Element Method for Navier-Stokes Equations
», in Noor( ed.) State-of-the-Art-Surveys in Computationnal Mechanics, vol. 139,
, p. 71-143.
[RAY 77] RAVIART P.-A., THOMAS 1.-M., • A Mixed Finite Element Method for Second
Order Elliptic Problems », Mathematical Aspects of fmite Element Methos, Roma, 1975,
, Springer-Verlag, p. 292-315.
[R.E 64] R. E. LYNCH 1. R. R., THOMAS D. H., • Direct solution of partial difference equations
by tensor product methods "• Numer. Math., vol. 6, 1964, p. 185-199.
[ROB 91] ROBERTS 1. E., THOMAS 1.-M., « Mixed and Hybrid Methods Numerical Analysis
"• Handbook of Numerical Analysis, Roma, 1975, 1991, Elsevier Science Publishers
B.V., North Holland, p. 1523-1639.
(SHE 92] SHEN J., • On Error Estimation of the Projection Method for Unsteady NavierStokes
Equations: First Order Schemes », SIAM. J. Numer. Anal., vol. 29, 1992, p. 57-77.
(SHE 94] SHEN 1 ., • Remarks on the Pressure Error Estimates for the Projection Methods »,
Numer. Math., vol. 67, 1994, p. 513-520.
[SHE 98] SHEN 1 ., LOPEZ 1. M., « An Efficient Spectral Projection Method for the NavierStokes
Equations in Cylindrical Geometries. I. Axisymmetric Cases », J. Compu. Physics,
vol. 139, 1998, p. 306-326.
[TEM 68] TEMAM R., « Une methode d'approximation de Ia solution des equations de
Navier-Stokes », Bull. Soc. Math. France, vol. 98, 1968, p. 115-152.
[THO 77] THOMAS 1.-M., ~ Sur !'analyse numerique des methodes d'elements finis hybrides
et mixtes »,These d'etat, Universite Pierre et Marie Curie, 1977.
[TRI 78] TRIEBEL H., Interpolation Theory, Function spaces, Differential operators, North
Holand, 2nd edition, 1978.
