×

zbMATH — the first resource for mathematics

Spectral approximation of the periodic nonperiodic Navier-Stokes equations. (English) Zbl 0583.65085
In order to approximate the Navier-Stokes equations with periodic boundary conditions in two directions and a no-slip boundary condition in the third direction by spectral methods, we justify by theoretical arguments an appropriate choice of discrete spaces for the velocity and the pressure. The compatibility between these two spaces is checked via an inf-sup condition. We analyze a spectral and a collocation pseudo- spectral method for the Stokes problem and a collocation pseudo-spectral method for the Navier-Stokes equations. We derive error bounds of spectral type, i.e. which behave like \(M^{-\sigma}\), where M depends on the number of degrees of freedom of the method and \(\sigma\) represents the regularity of the data.

MSC:
65Z05 Applications to the sciences
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
65N15 Error bounds for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Adams, R.A.: Sobolev Spaces. New York: Academic Press 1975 · Zbl 0314.46030
[2] Berg, J., Löfström, J.: Interpolation Spaces: An Introduction, Berlin, Heidelberg, New York: Springer 1976
[3] Bernardi, C., Maday, Y., Métivet, B.: Calcul de la pression dans la résolution spectrale du problème de Stokes. La Recherche Aérospatiale1, 1-21 (1987) · Zbl 0642.76037
[4] Brachet, M.E., Meiron, D.I., Orszag, S.A., Nickel, B.G., Morf, R.H., Frisch, U.: Small-Scale Structure of Taylor Green Vortex. J. Fluid Mech.130, 411-452 (1983) · Zbl 0517.76033 · doi:10.1017/S0022112083001159
[5] Brezzi, F.: On the Existence, Uniqueness and Approximation of Saddle-Point Problems Arising from Lagrange Multipliers. RAIRO Anal. Numér.8, 129-151 (1974) · Zbl 0338.90047
[6] Brezzi, F., Rappaz, J., Raviart, P.-A.: Finite-Dimensional Approximation of Nonlinear Problems. Part I: Branches of Nonsingular Solutions. Numer. Math.36, 1-25 (1980) · Zbl 0488.65021 · doi:10.1007/BF01395985
[7] Canuto, C., Maday, Y., Quarteroni, A.: Analysis of the Combined Finite Element and Fourier Interpolation. Numer. Math.39, 205-220 (1982) · Zbl 0496.42002 · doi:10.1007/BF01408694
[8] Canuto, C., Quarteroni, A.: Approximation Results for orthogonal Polynomials in Sobolev Spaces. Math. Comput.38, 67-86 (1982) · Zbl 0567.41008 · doi:10.1090/S0025-5718-1982-0637287-3
[9] Canuto, C., Sacchi-Landriani, G.: Analysis of the Kleiser-Schumann Method. Numer. Math.50, 217-243 (1986) · Zbl 0623.76019 · doi:10.1007/BF01390431
[10] Crouzeix, M.: Approximation de problèmes faiblement non linéaires. Université de Rennes (1983)
[11] Dang, K., Roy, P.: Numerical Simulation of Homogeneous Turbulence. In: Proceedings of Workshop on Macroscopic Modelling of Turbulent Flows and Fluid Mixtures. (O. Pironneau ed.), Berlin, Heidelberg, new York: Springer 1985
[12] Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. New York: Academic Press 1984 · Zbl 0537.65020
[13] Ferziger, J.H., Reynolds, W.C.: Reports TF5 to 21, Dept. of Mechanical Engineering, Stanford University (1975-1985)
[14] Girault, V., Raviart, P.-A. Finite Element Approximation of the Navier-Stokes Equations. Berlin, Heidelberg, New York, Tokyo: Springer 1986 · Zbl 0585.65077
[15] Hochstrasser, U.W.: Orthogonal Polynomials, Chapter 22. In: Handbook of Mathematical Functions. (M. Abramowitz and I.A. Stegun eds), New York: Dover Publications 1970
[16] Huberson, S., Morchoisne, Y.: Large Eddy Simulation by Spectral Methods or by Multi-Level Particles Method. Proceedings of the 6th AIAA Computational Fluid Dynamics Conference, Danvers (MA) 1983
[17] Kim, J., Moin, P.: Application of a Fractional-Step Method to Incompressible Navier-Stokes Equation, NASA Technical Memorandum 85898 (1984) · Zbl 0582.76038
[18] Kleiser, L., Schumann, U.: Treatment of Incompressibility and Boundary Conditions in 3-D Numerical Spectral Simulations of Plane Channel Flows. Proceedings of the Third GAMM Conference on Numerical Methods in Fluid Mechanics. Braunschweig: Vieweg 1980 · Zbl 0463.76020
[19] Kleiser, L., Schumann, U.: Spectral Simulations of the Laminar-Turbulent Transition Precess in Plane Poiseuille Flow, Proceedings of the Symposium on Spectral Methods for Partial Differential Equations. (R. Voigt ed.), SIAM-CBMS 1983
[20] Lequéré, P., Alziary de Roquefort, T.: Sur une méthode spectrale semi-implicite pour la résolution des équations de Navier-Stokes d’un écoulement bidimensionnel visqueux incompressible. C.R. Acad. Sci., Paris S II,294, 941-944 (1982) · Zbl 0489.76037
[21] Lions, J.-L. Magenes, E.: Problèmes aux limites non homogènes et applications, Volume 1. Dunod 1968 · Zbl 0165.10801
[22] Lions, J.-L. Magenes, E.: Problèmes aux limites non homogènes et applications, Volume 2. Dunod 1968 · Zbl 0165.10801
[23] Maday, Y.: Analysis of Spectral Operators in One-Dimensional Domains. Icase Report no 85-17 (submitted to Math. Comput.)
[24] Maday, Y., Quarteroni, A.: Spectral and Pseudo-Spectral Approximations of Navier-Stokes Equations. SIAM J. Numer. Anal.19, 761-780 (1982) · Zbl 0503.76035 · doi:10.1137/0719053
[25] Malik, M.R., Zang, T.A., Hussaini, M.Y.: A Spectral Collocation Method for the Navier-Stokes Equations. Icase Report no 84-19 (1984) · Zbl 0573.76036
[26] Marcus, P.S.: Simulation of Taylor Couette Flow, Parts I and II. J. Fluid Mech.146, 45-113 (1984) · Zbl 0561.76037 · doi:10.1017/S0022112084001762
[27] Métivet, B.: Résolution des équations de Navier-Stokes par méthodes spectrales. Thèse, Université P. et M. Curie 1987
[28] Moin, P., Kim, J.: On the Numerical Solution of Time-Dependent Viscous Incompressible Fluid Flows Involving Solid Boundaries. J. Comput. Phys.35, 381-392 (1980) · Zbl 0425.76027 · doi:10.1016/0021-9991(80)90076-5
[29] Montigny-Rannou, F.: Influence of Compatibility Conditions in Numerical Simulation of Inhomogeneous Incompressible Flows. Proceedings of the 5th GAMM Conference, Rome 1983 · Zbl 0552.76036
[30] Morchoisne, Y.: Résolution des équations de Navier-Stokes par une méthode spectrale de sousdomaines. Comptes-Rendus du 3ème Congrès Intern. sur les Méthodes Numériques de l’Ingénieur. (P. Lascaux ed.), Paris 1983
[31] Moser, R.D., Moin, P., Leonard, A.: A Spectral Numerical Method for the Navier-Stokes Equations with Applications to Taylor-Couette Flow. J. Comput. Phys.52, 524-544 (1983) · Zbl 0529.76034 · doi:10.1016/0021-9991(83)90006-2
[32] Orszag, S.A., Kells, L.C.: Transition to Turbulence in Plane Poiseuille and Plane Couette Flow, Part I. J. Fluid Mech.96, 159-205 (1980) · Zbl 0418.76036 · doi:10.1017/S0022112080002066
[33] Orszag, S.A., Patterson, G.S.: Numerical Simulation of the Three-Dimensional Homogeneous Isotropic Turbulence. Physical Review Letters28, no 2 (1972) · Zbl 0227.76080
[34] Patera, A.T.: A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion. J. Comput. Phys.54, 468-488 (1984) · Zbl 0535.76035 · doi:10.1016/0021-9991(84)90128-1
[35] Patera, A.T., Orszag, S.A.: Secondary Instability of Wall Bounded Shear Flows, J. Fluid Mech.128, 347-385 (1983) · Zbl 0556.76039 · doi:10.1017/S0022112083000518
[36] Quarteroni, A.: Blending Fourier and Chebyshev Interpolation (submitted to J. Approx. Theory) · Zbl 0645.41002
[37] Rogallo, R.S.: Numerical Experiments in Homogeneous Turbulence. NASA Technical Memorandum 81315 (1981)
[38] Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis. Amsterdam: Norht-Holland 1977 · Zbl 0383.35057
[39] Verfürth, R.: Error Estimates for a Mixed Finite Element Approximation of the Stokes Equations. RAIRO Anal. Numér.18, 175-182 (1984) · Zbl 0557.76037
[40] Zang, T.A., Hussaini, M.Y.: Fourier-Legendre Spectral Methods for Incompressible Channel Flow. Proceedings of the 9th Intern. Conf. on Numerical Methods in Fluid Dynamics, Saclay 1984 · Zbl 0592.76100
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.