## Stable Spectral Methods with No Spurious Eigenvalues |

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### Contents

Introduction | 1 |

Spectral Approximations | 27 |

Stability analysis and results for the different methods | 49 |

Copyright | |

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### Common terms and phrases

12 digits 2nd kind 3rd kind approach Approximation error Bessel polynomials calculations chapter characteristic polynomial Chebyshev Galerkin method Chebyshev polynomials Chebyshev weight Chebyshev-Tau method coefficients compute corresponding differentiation implementation eigenvalue problem equal to zero equivalent exact eigenvalues expansion functions expression following differential equation fourth order problem Galerkin method Gegenbauer polynomials Gegenbauer-Inviscid Galerkin method Gegenbauer-Tau method given Gottlieb and Orszag heat equation integration Invisc Inviscid-Galerkin Jacobi polynomials least negative eigenvalue Legendre polynomials lemma Lundblahd Navier-Stokes equations no-slip boundary conditions Orr-Sommerfeld orthogonal polynomials Phi formulation Plugging pn(x polynomial approximation polynomial of degree polynomial solution positive pair proof prove real and negative real negative eigenvalues right hand side roundoff errors satisfies the following satisfy the boundary second derivative solve spectral methods stable polynomials Stokes equations Tau method terms of Chebyshev test functions theorem three methods Tn(x uN(x versus the exact Waleffe weight function Weighted Residuals XD2u Zebib