On the numerical solution of double-periodic elliptic eigenvalue problems
type: Item
instance of: scholarly article
title: On the numerical solution of double-periodic elliptic eigenvalue problems
zbMATH Open document ID: 0685.65094
DOI: 10.1007/BF02241855
published in: Computing
publication date: 1989-01-01T00:00:00Z
review text: The computation of solutions \(u=u(x,\lambda)\) of the eigenvalue problem \(\Delta u+\lambda (u+f(u))=0,\) \(\lambda >0,\) is considered which are \(2\pi\)-periodic in x and \(2\pi\) /\(\alpha\)-periodic in \(\lambda\) with prescribed \(\alpha\), notable \(\alpha =3^{1/2}.\) Subspaces V with the desired symmetry properties are constructed and a Ritz method is applied for the discretization in V. The resulting finite-dimensional bifurcation problem is solved by an algorithm of \textit{H. B. Keller}, \textit{W. F. Langford} [Arch. Rat. Mech. Analysis 48, 83-158 (1972; Zbl 0249.47058)]. If \(f: R\to R,\) \(f(0)=f'(0)=0,\) is entire or a polynomial and V an algebra, then the Ritz approximation is stable with respect to perturbations with reduced symmetry. Some examples illustrate the procedure.
Mathematics Subject Classification ID: 65N25
Mathematics Subject Classification ID: 65H17
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 4123224
zbMATH Keywords: double-periodic elliptic eigenvalue problems
zbMATH Keywords: numerical examples
zbMATH Keywords: Ritz method
zbMATH Keywords: bifurcation problem
author: Eckart W. Gekeler
reviewed by: Werner C. Rheinboldt
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02241855
OpenAlex ID: W1485388721
description: scientific article; zbMATH DE number 4123224