Discretization principles for linear two-point boundary value problems, II

Tetsuro Yamamoto, Shin'Ichi Oishi, Qing Fang

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Consider the boundary value problem Lu-(pu')'+qu'+ru=f, a≤x≤b, u(a)=u(b)=0. Let HνAνU=f and [image omitted] be its finite difference equations and piecewise linear finite element equations on partitions [image omitted], ν=1, 2,... with [image omitted], [image omitted] as ν, where Hν are nνnν diagonal matrices and Aν as well as [image omitted] are nνnν tridiagonal. It is shown that the following three conditions are equivalent: (i) The boundary value problem has a unique solution uC2[a, b]. (ii) For sufficiently large νν0, the inverse [image omitted] exists and [image omitted], i, j with a constant M0 independent of hν. (iii) For sufficiently large ν[image omitted], [image omitted] exists and [image omitted], i, j with a constant [image omitted] independent of hν. It is also shown by a numerical example that the finite difference method with uniform nodes xi+1=xi+h, 0≤i≤n, h=(b-a)/(n+1) applied to the boundary value problem with no solution gives a ghost solution for every n.

Original languageEnglish
Pages (from-to)213-224
Number of pages12
JournalNumerical Functional Analysis and Optimization
Volume29
Issue number1-2
DOIs
Publication statusPublished - 2008 Jan 1

Keywords

  • Discretization principles
  • Finite difference methods
  • Finite element methods
  • Two-point boundary value problems

ASJC Scopus subject areas

  • Analysis
  • Signal Processing
  • Computer Science Applications
  • Control and Optimization

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