zbMATH — the first resource for mathematics

Computational aspects of the spectral Galerkin FEM for the Orr-Sommerfeld equation. (English) Zbl 0979.76048
Most of previous spectral techniques applied for solving the Orr-Sommerfeld equation (OSE) employed tau discretization and Chebyshev polynomials. The use of tau discretization is accompanied by spurious eigenvalues not related to OSE and by the appearance of singular matrix in generalized eigenvalue problem. Here, making use of a variational formulation of OSE, the author performs a spectral discretization by means of Galerkin method. By adopting integrated Legendre polynomials as basis functions, the boundary conditions are satisfied at any spectral order, and non-singular matrices \(\mathbf A\) and \(\mathbf B\) are obtained in the equation \({\mathbf A}{\mathbf x}=\lambda{\mathbf B}{\mathbf x}\). For plane Poiseuille flow, the stiffness and the mass matrices are sparse with bandwidths 7 and 5 respectively, and the entries can be calculated explicitly for any polynomial flow profile \(U\). According to the convergence results, no spurious eigenvalue has been found. Numerical experiments with spectral orders up to \(p = 600\) illustrate this analysis.

76M10 Finite element methods applied to problems in fluid mechanics
76M22 Spectral methods applied to problems in fluid mechanics
76E05 Parallel shear flows in hydrodynamic stability
PDF BibTeX Cite
Full Text: DOI
[1] Orszag, J. Fluid Mech. 50 pp 689– (1971) · Zbl 0237.76027
[2] ?Calculating large spectra in hydrodynamic stability: a p-FEM approach to solve the Orr-Sommerfeld equation?, Diploma Thesis, Seminar for Applied Mathematics, Swiss Federal Institute of Technology Z?rich, 1998.
[3] and ?A spectral Galerkin method for hydrodynamic stability problems?, Research Report No. 98-06, Seminar for Applied Mathematics, Swiss Federal Institute of Technology Z?rich, 1998.
[4] and Hydrodynamic Stability, Cambridge University Press, Cambridge, 1981.
[5] Str?mungslehre2nd edn, Springer, Berlin, 1989.
[6] Fluid- und Thermodynamik, Springer, Berlin, 1995.
[7] and Numerical Aspects of Spectral Methods: Theory and Applications, SIAM-CB\IS, Philadelphia, 1972.
[8] Gary, J. Comput. Phys. 5 pp 169– (1970) · Zbl 0191.16603
[9] Gardner, J. Comput. Phys. 80 pp 169– (1970)
[10] McFadden, J. Comput. Phys. 91 pp 228– (1990) · Zbl 0717.65063
[11] Lindsay, Int. J. Numer. Methods Fluids 15 pp 1277– (1992) · Zbl 0762.76082
[12] Dongarra, Appl. Numer. Math. 22 pp 399– (1996) · Zbl 0867.76025
[13] p- and hp-Finite Element Methods, Oxford University Press, Oxford, 1998.
[14] and Handbook of Mathematical Functions, Dover, New York, 1971.
[15] Numerische Mathematik, 2nd edn, Teubner, Stuttgart, 1988. · Zbl 0669.65002
[16] Thomas, Phys. Rev. 91 pp 780– (1953) · Zbl 0051.17303
[17] and Matrix Computation, The Johns Hopkins University Press, Baltimore, MD, 1996.
[18] Abdullah, Math. Models Methods Appl. Sci. 1 pp 153– (1991) · Zbl 0742.76031
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.