Approximation of nonlinear spectral problems in a Hilbert space
type: Item
instance of: scholarly article
title: Approximation of nonlinear spectral problems in a Hilbert space
zbMATH Open document ID: 1328.65131
DOI: 10.1134/S0012266115070113
published in: Differential Equations
publication date: 2015-10-14T00:00:00Z
review text: The article deals with the following eigenvalue problems \[ a(\lambda,u,v) = \lambda b(\lambda,u,v), \qquad 0 < \lambda < \infty, \;u,v \in V,\tag{1} \] in a Hilbert space \(V\). Here \(a: \;\Lambda \times V \times V \to {\mathbb R}\) is a symmetrical bounded positive definite form, \(b: \;\Lambda \times V \times V \to {\mathbb R}\) a compact symmetrical bounded positive definite form. It is assumed that the Rayleigh ratio \[ R(\mu,v) = \frac{a(\mu,v,v)}{b(\mu,v,v)} \] is a nonincreasing function from \(\lambda\). For a fixed \(\mu \in (0,\infty)\) each problem (1) has a countable set of eigenvalues \(\gamma_1(\mu) \leq \ldots \leq \gamma_n(\mu) \leq \ldots\), \(\gamma_n(\mu) \to \infty\) of finite multiplicity. In the first part of the article, the author proves that problem (1) has a sequence of eigenvalues \(\lambda_k\) of finite multiplicity; each of these eigenvalues is a solution of the scalar equation \(\mu = \gamma_i(\mu)\). In the second part of the article, the approximation of problem (1) by finite-dimensional problems \[ a(\lambda,u,v) = \lambda b(\lambda,u,v), \qquad 0 < \lambda < \infty, \;u,v \in V_h,\tag{2} \] is studied, where \(V_h\) is a family of finite-dimensional subspaces \(V\) satisfying the limit density condition \[ \varepsilon_h(v) = \bigg(\inf_{v_h \in V_h}\;\|v - v_h\|\bigg)\bigg|_{h \to 0} \to 0. \] The author proves the solvability of problem (2) and the convergence, as \(h \to 0\), of its solutions to solutions of (1). Moreover, estimates of type \[ 0 \leq \lambda_k^h - \lambda_k \leq c(\varepsilon^h)^2, \qquad \varepsilon^h = \sup \;\{\varepsilon_h(u):\;u \in V_h,\;\|u\| = 1\}, \] are obtained. It should be noted that some results of this article are well known in terms of self adjoint operators in Hilbert spaces.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J10
zbMATH DE Number: 6493791
zbMATH Keywords: nonlinear eigenvalue problems for quadratic forms
zbMATH Keywords: Rayleigh-Ritz method
zbMATH Keywords: finite-dimensional approximations of linear problems in Hilbert spaces
zbMATH Keywords: symmetric quadratic forms
zbMATH Keywords: positive definite quadratic forms
zbMATH Keywords: compact quadratic forms
zbMATH Keywords: convergence of approximations
zbMATH Keywords: error estimates for approximations
author: S. I. Solov'ev
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
cites work: Q3126801
cites work: Finite element methods (Part 1)
cites work: Q3358152
cites work: Q3823096
cites work: Q3960064
cites work: Numerical study of stability and nonlinear eigenvalue problems
cites work: Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes
cites work: Q4697780
cites work: Investigation of a spectral problem for the Helmholtz operator in the plane
cites work: The bisection method for symmetrie eigenvalue problems with a parameter entering nonlinearly
cites work: Preconditioned iterative methods for a class of nonlinear eigenvalue problems
cites work: Error of the Bubnov-Galerkin method with perturbations for symmetric spectral problems with nonlinear entrance of the parameter
cites work: The finite-element method for symmetric nonlinear eigenvalue problems
cites work: Q3621775
cites work: Q4889598
cites work: Superconvergence of finite element approximations to eigenspaces
cites work: A class of high-resolution algorithms for incompressible flows
cites work: Approximation of sign-indefinite spectral problems
cites work: Efficient GPU implementation of the linearly interpolated bounce-back boundary condition
cites work: Approximation of differential eigenvalue problems with a nonlinear dependence on the parameter
description: scientific article; zbMATH DE number 6493791