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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.

MSC:
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
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