A numerical method to verify the elliptic eigenvalue problems including a uniqueness property
type: Item
instance of: scholarly article
title: A numerical method to verify the elliptic eigenvalue problems including a uniqueness property
zbMATH Open document ID: 0941.65108
published in: Computing
publication date: 2000-02-01T00:00:00Z
review text: A verification method proposed by \textit{K. Nagatou}, \textit{N. Yamamoto} and \textit{M. T. Nakao} [An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness. Numer. Funct. Anal. Optim. 20, No. 5-6, 543-565 (1959)] in the context of nonlinear boundary value problems, is extended to elliptic eigenvalue problems \(-\Delta u+ qu=\lambda u\) in \(\Omega\), \(u=0\) on \(\partial\Omega\) under the normalization \(\int_\Omega u^2dx= 1\). Here, \(\Delta\) denotes the Laplace operator on a bounded convex domain \(\Omega\subset \mathbb{R}^2\). Using accurate finite element approximations to eigensolutions \((u,\lambda)\in H^1_0(\Omega)\times \mathbb{R}\) together with certain a posteriori estimates, Banach's fixed point theorem can be applied, provided that some verification condition is satisfied. In this way the author obtains inclusion sets for locally unique solutions \((u,\lambda)\) and separately for \(\lambda\) as well. Two examples are treated and numerical results are presented.
Mathematics Subject Classification ID: 65N25
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 65G20
Mathematics Subject Classification ID: 65N30
zbMATH DE Number: 1383911
zbMATH Keywords: verification method
zbMATH Keywords: elliptic eigenvalue problems
zbMATH Keywords: finite element
zbMATH Keywords: inclusion sets
zbMATH Keywords: numerical results
author: Kaori Nagatou
reviewed by: Waldemar Velte
describes a project that uses: PROFIL/BIAS
MaRDI profile type: Publication
full work available at URL: s006070050054
OpenAlex ID: W2063697047
DOI: 10.1007/S006070050054
description: scientific article; zbMATH DE number 1383911