Spectral Approximation of Linear Operators (Françoise Chatelin)

SIAM Review ◽  
1985 ◽  
Vol 27 (3) ◽  
pp. 459-459
Author(s):  
P. M. Anselone
Keyword(s):  
Linear Operators ◽  
Spectral Approximation

scholarly journals The Convergence of the Legendre–Galerkin Spectral Method for Constructing Atmospheric Acoustic Normal Modes

2020 ◽  
Vol 28 (03) ◽  
pp. 2050002
Author(s):  
Richard B. Evans
Keyword(s):  
Legendre Polynomials ◽  
Normal Modes ◽  
Sobolev Norm ◽  
Linear Operators ◽  
Basis Sets ◽  
Spectral Approximation ◽  
Spectral Approximations ◽  
Potential Applications ◽  
Galerkin Spectral Method ◽  
Theoretical Results

The asymptotic rate of convergence of the Legendre–Galerkin spectral approximation to an atmospheric acoustic eigenvalue problem is established, as the dimension of the approximating subspace approaches infinity. Convergence is in the [Formula: see text] Sobolev norm and is based on the existing theory [F. Chatelin, Spectral Approximations of Linear Operators (SIAM, 2011)]. The assumption is made that the eigenvalues are simple. Numerical results that help interpret the theory are presented. Eigenvalues corresponding to acoustic modes with smaller [Formula: see text] norms are especially accurately approximated, even with lower dimensioned basis sets of Legendre polynomials. The deficiencies in the potential applications of the theoretical results are noted in connection with the numerical examples.

scholarly journals Spectral approximation theorems for bounded linear operators

1973 ◽  
Vol 8 (2) ◽  
pp. 279-287 ◽  
Author(s):  
W.S. Lo
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Eigenvalue Problem ◽  
Linear Operators ◽  
Spectral Approximation ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Approximation Theorems ◽  
Generalized Eigenvectors

In this paper we present some approximation theorems for the eigenvalue problem of a compact linear operator defined on a Banach space. In particular we examine: criteria for the existence and convergence of approximate eigenvectors and generalized eigenvectors; relations between the dimensions of the eigenmanifolds and generalized eigenmanifolds of the operator and those of the approximate operators.

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