Some gradient methods in the eigenvalue problem for polynomial operator bundles consisting of self-adjoint operators

1993 ◽  
Vol 64 (5) ◽  
pp. 1164-1167
Author(s):  
B. M. Podlevskii
Keyword(s):  
Eigenvalue Problem ◽  
Gradient Methods ◽  
Polynomial Operator ◽  
Adjoint Operators

scholarly journals A virtual element method for the transmission eigenvalue problem

2018 ◽  
Vol 28 (14) ◽  
pp. 2803-2831 ◽  
Author(s):  
David Mora ◽  
Iván Velásquez
Keyword(s):  
Eigenvalue Problem ◽  
Optimal Order ◽  
Virtual Element Method ◽  
Order Error ◽  
Transmission Eigenvalue ◽  
Fourth Order Eigenvalue Problem ◽  
Transmission Eigenvalue Problem ◽  
Virtual Element ◽  
Adjoint Operators ◽  
Element Method

In this paper, we analyze a Virtual Element Method (VEM) for solving a non-self-adjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. We write a variational formulation and propose a [Formula: see text]-conforming discretization by means of the VEM. We use the classical approximation theory for compact non-self-adjoint operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes.

scholarly journals RIGHT SEMIDEFINITE EIGENVALUE PROBLEMS

2003 ◽  
Vol 46 (3) ◽  
pp. 561-573 ◽  
Author(s):  
Paul Binding ◽  
Hans Volkmer
Keyword(s):  
Hilbert Space ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Positive Semidefinite ◽  
Subject Classification ◽  
Secondary 34B24 ◽  
Compact Resolvent ◽  
Primary 47A75 ◽  
Adjoint Operators

AbstractThe relationships between various notions of completeness of eigenvectors and root vectors of the eigenvalue problem $Af=\lambda Bf$ are investigated. Here $A$ and $B$ are self-adjoint operators in Hilbert space with $B$ bounded and positive semidefinite, and with $A$ having compact resolvent.AMS 2000 Mathematics subject classification: Primary 47A75. Secondary 34B24; 35P10

Finite element approximation of eigenvalue problems

Acta Numerica ◽  
2010 ◽  
Vol 19 ◽  
pp. 1-120 ◽  
Author(s):  
Daniele Boffi
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Differential Forms ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Galerkin Approximations ◽  
Laplace Eigenvalue ◽  
Main Emphasis ◽  
Adjoint Operators

We discuss the finite element approximation of eigenvalue problems associated with compact operators. While the main emphasis is on symmetric problems, some comments are present for non-self-adjoint operators as well. The topics covered include standard Galerkin approximations, non-conforming approximations, and approximation of eigenvalue problems in mixed form. Some applications of the theory are presented and, in particular, the approximation of the Maxwell eigenvalue problem is discussed in detail. The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations. Several examples and numerical computations complete the paper, ranging from very basic exercises to more significant applications of the developed theory.

An abstract multiparameter spectral theory

1982 ◽  
Vol 92 (3-4) ◽  
pp. 193-204 ◽  
Author(s):  
P. A. Binding ◽  
A. Källström ◽  
B. D. Sleeman
Keyword(s):  
Eigenvalue Problem ◽  
Tensor Product ◽  
Spectral Theory ◽  
Hilbert Spaces ◽  
Product Space ◽  
Infinite Dimensions ◽  
Tensor Product Space ◽  
Image Position ◽  
Adjoint Operators

SynopsisWe consider the eigenvalue problemfor self-adjoint operators Ai and Bij on separable Hilbert spaces Hi. It is assumed that and Bij are bounded with compact. Various properties of the eigentuples λi, and xi are deduced under a “definiteness condition” weaker than those used by previous authors, at least in infinite dimensions. In particular, a Parseval relation and eigenvector expansion are derived in a suitably constructed tensor product space.

VI.—On Solvability of Some Two-Parameter Eigenvalue Problems in Hilbert Space

1968 ◽  
Vol 68 (1) ◽  
pp. 83-93
Author(s):  
Z. Bohte
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Radius Of Convergence ◽  
Finite Radius ◽  
Image Position ◽  
Adjoint Operators ◽  
Two Parameter

SynopsisThis paper studies two particular cases of the general 2-parameter eigenvalue problem namelywhere A, B, B1, B2, C, C1, C2 are self-adjoint operators in Hilbert space, all except A being bounded. The disposable parameters λ and μ have to be determined so that the equations have non-trivial solutions x, y.On the assumption that the solution is known for ∊ = o, solutions are constructed in the form of series for λ, μ, x, y as power series in ∊ with finite radius of convergence.

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