Spectral Methods for Non-Standard Eigenvalue Problems. Fluid and Structural Mechanics and Beyond

1 General Formulation of Spectral Approximation
1.1 The Tau and Galerkin Spectral Methods
1.2 Theoretical Foundation of Spectral Collocation

2 Tau and Galerkin Methods for Fourth Order GEPs
2.1 The ChT Method
2.2 The ChG Method
2.3 ChT Methods for GEPs with k Dependent Boundary Conditions
2.3.1 Second Order S-L Problems with Parameter Dependent Boundary Conditions
2.3.2 The Stability of some Elastic Systems
2.3.3 A Modified ChT Method for a Particular O-S Problem

3 The Chebyshev Collocation Method
3.1 ChC Method Versus ChG and ChT Methods in Solving Fourth Order GEPs
3.2 The Viola’s Eigenvalue Problem
3.3 Linear Hydrodynamic Stability of Thermal Convection with Variable Gravity Field
3.4 Linear Hydrodynamic Stability of EHD Convection Between Two Parallel Walls
3.5 Multiparameter Mathieu’s Problem
3.6 Improvements Induced by JD Methods

4 The Laguerre Collocation Method
4.1 LC Solutions to a Third Order Linear Boundary Value Problem on the Half-Line
4.2 The Falkner-Skan Problem
4.3 The Laguerre Differentiation Matrices
4.4 The LC algorithm
4.5 Numerical Solutions to Falkner-Skan Problem
4.6 Second Order Nonlinear Singular Boundary Value Problems on the Half-Line
4.7 Second Order Eigenvalue Problems on Half-Line
4.8 Fourth Order Eigenvalue Problems on Half-Lin
4.9 The Movement of a Pile

5 Conclusions and Further Developments
5.1 Lessons Learned Along the Way
5.2 Further Developments

The chapter contains first the general formulation of the spectral approximation as a weighted residual method, i.e., the projection and interpolation operators, test and trial (shape) functions etc. Then, the functional framework of the tau and Galerkin methods based on Chebyshev polynomials is provided, mainly discussing the projection operators. The Chebyshev collocation is introduced in some details. The Chebyshev-Gauss quadrature formulas are reviewed, and the interpolation operator along with the collocation (otherwise called pseudospectral) differentiation matrices are considered. On a regular second order S-L problem, some remarks on the behavior of solutions of Chebyshev collocation, Chebyshev tau and of a Galerkin type method with respect to their order of convergence are conducted.

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The chapter is devoted to the efficient implementation of Chebyshev collocation method. First, the performances of the method in solving fourth order GEPs are compared with those of ChT and ChG counterparts. Then, ChC method is used to solve some genuinely high order, i.e., larger than two, and/or singularly perturbed eigenvalue problems. Two of them, of sixth and eighth order represent linear hydrodynamic stability problems. Also some fourth order problems with variable coefficients (tensile instabilities of thin annular plates etc.) are successfully considered. In order to reduce the high order problems to systems of second order equations supplied with Dirichlet boundary conditions we introduce a so called “\(D^{(2)}\)” strategy or factorization. Using this strategy with \(N=2^10\) a conjecture with respect to the first eigenvalue of the Viola’s problem is stated. This is a fourth order singularly perturbed eigenvalue problem. A special attention is paid to the well known Mathieu’s system as a MEP. A lot of eigenmodes and eigenfrequencies corresponding to various geometries of the vibrating elliptic membrane problem, in which this system is originated, are displayed. In order to avoid spurious eigenvalues (at infinity) and to improve the computation of a specified region of the spectrum, mainly in case of large problems, some Jacobi Davidson type methods are used. Making use of the pseudospectrum of a singular GEP we comment on the backward stability and the order of convergence of JD and Arnoldi methods in computing the first two eigenvalues.

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