scholarly journals Spectral pollution and how to avoid it

2009 ◽  
Vol 100 (3) ◽  
pp. 864-900 ◽  
Author(s):  
Mathieu Lewin ◽  
Éric Séré
Keyword(s):  
Spectral Pollution

On spectral pollution

1990 ◽  
Vol 59 (2) ◽  
pp. 199-216 ◽  
Author(s):  
X. Llobet ◽  
K. Appert ◽  
A. Bondeson ◽  
J. Vaclavik
Keyword(s):  
Spectral Pollution

scholarly journals Essential numerical ranges for linear operator pencils

2019 ◽  
Vol 40 (4) ◽  
pp. 2256-2308
Author(s):  
Sabine Bögli ◽  
Marco Marletta
Keyword(s):  
Linear Operator ◽  
Numerical Range ◽  
Projection Methods ◽  
Operator Pencil ◽  
Operator Matrices ◽  
New Concepts ◽  
Block Operator ◽  
Block Operator Matrices ◽  
Spectral Pollution ◽  
Operator Pencils

Abstract We introduce concepts of essential numerical range for the linear operator pencil $\lambda \mapsto A-\lambda B$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=\lambda x$ into the pencil problem $BTx=\lambda Bx$ for suitable choices of $B$, we can obtain nonconvex spectral enclosures for $T$ and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of nonselfadjoint Schrödinger operators which has not been possible to treat with existing methods.

scholarly journals Analysis of FEAST Spectral Approximations Using the DPG Discretization

2019 ◽  
Vol 19 (2) ◽  
pp. 251-266 ◽  
Author(s):  
Jay Gopalakrishnan ◽  
Luka Grubišić ◽  
Jeffrey Ovall ◽  
Benjamin Parker
Keyword(s):  
Differential Operator ◽  
Optical Fiber ◽  
Numerical Experiments ◽  
Partial Differential Operator ◽  
Partial Differential ◽  
Subspace Iteration ◽  
Theoretical Assumptions ◽  
Spectral Approximations ◽  
Spectral Pollution ◽  
Core Modes

AbstractA filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as “FEAST”, has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent of the partial differential operator is approximated by the discontinuous Petrov–Galerkin (DPG) method, it is shown that there is no spectral pollution. The theory also provides bounds on the discretization errors in the spectral approximations. Numerical experiments for simple operators illustrate the theory and also indicate the value of the algorithm beyond the confines of the theoretical assumptions. The utility of the algorithm is illustrated by applying it to compute guided transverse core modes of a realistic optical fiber.

scholarly journals Generalized Weyl theorems and spectral pollution in the Galerkin method

2012 ◽  
pp. 329-354 ◽  
Author(s):  
Lyonell Boulton ◽  
Nabile Boussaïd ◽  
Mathieu Lewin
Keyword(s):  
Galerkin Method ◽  
The Galerkin Method ◽  
Spectral Pollution

scholarly journals Periodic Schrödinger Operators with Local Defects and Spectral Pollution

2012 ◽  
Vol 50 (6) ◽  
pp. 3016-3035 ◽  
Author(s):  
Eric Cancès ◽  
Virginie Ehrlacher ◽  
Yvon Maday
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Local Defects ◽  
Spectral Pollution

scholarly journals On the convergence of second-order spectra and multiplicity

2010 ◽  
Vol 467 (2125) ◽  
pp. 264-284 ◽  
Author(s):  
Lyonell Boulton ◽  
Michael Strauss
Keyword(s):  
Hilbert Space ◽  
Finite Element ◽  
Numerical Experiments ◽  
Adjoint Operator ◽  
Second Order ◽  
Precise Information ◽  
Order Spectrum ◽  
The Galerkin Method ◽  
Isolated Eigenvalues ◽  
Spectral Pollution

The notion of second-order relative spectrum of a self-adjoint operator acting on a Hilbert space has been studied recently in connection with the phenomenon of spectral pollution in the Galerkin method. In this paper we examine how the second-order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of the underlying operator. Our theoretical findings are supported by various numerical experiments on the computation of guaranteed eigenvalue inclusions via finite element bases.

scholarly journals A class of strongly stable approximation for unbounded operators

2019 ◽  
pp. 218-234
Author(s):  
Ammar Khellaf ◽  
Sarra Benarab ◽  
Hamza Guebbai ◽  
Wassim Merchela
Keyword(s):  
Sufficient Conditions ◽  
Theoretical Background ◽  
Spectral Approximation ◽  
Unbounded Operators ◽  
Numerical Application ◽  
Spectrum Method ◽  
Pollution Problem ◽  
Generalized Spectrum ◽  
Spectral Pollution ◽  
Discretization Process

We derive new sufficient conditions to solve the spectral pollution problem by using the generalized spectrum method. This problem arises in the spectral approximation when the approximate matrix may possess eigenvalues which are unrelated to any spectral properties of the original unbounded operator. We develop the theoretical background of the generalized spectrum method as well as illustrate its effectiveness with the spectral pollution. As a numerical application, we will treat the Schr¨odinger’s operator where the discretization process based upon the Kantorovich’s projection.

Sign in / Sign up

Export Citation Format

Share Document