Computational aspects of the spectral Galerkin FEM for the Orr-Sommerfeld equation

Abstract

Since Orszag's paper [‘Accurate solution of the Orr–Sommerfeld stability equation’, J. Fluid Mech., 50, 689–703 (1971)], most of the subsequent spectral techniques for solving the Orr–Sommerfeld equation (OSE) employed the Tau discretization and Chebyshev polynomials. The use of the Tau discretization appears to be accompanied by so-called spurious eigenvalues not related to the OSE and a singular matrix B in the generalized eigenvalue problem. Starting from a variational formulation of the OSE, a spectral discretization is performed using a Galerkin method. By adopting integrated Legendre polynomials as basis functions, the boundary conditions can be satisfied exactly for any spectral order and the non-singular matrices A and B are obtained in Ax=λBx. 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 (thus avoiding quadrature errors) for any polynomial flow profile U. According to the convergence results [Hancke, ‘Calculating large spectra in hydrodynamic stability: a p FEM approach to solve the Orr–Sommerfeld equation’, Diploma Thesis, Swiss Federal Institute of Technology Zürich, Seminar for Applied Mathematics, 1998; Hancke, Melenk and Schwab, ‘A spectral Galerkin method for hydrodynamic stability problems’, Research Report No. 98-06, Seminar for Applied Mathematics, Swiss Federal Institute of Technology, Zürich], no spurious eigenvalue has been found. Numerical experiments with spectral orders up to p=600 illustrate the results. Copyright © 2000 John Wiley & Sons, Ltd.