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

Tetsuro Yamamoto, Shinichi 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

    Fingerprint

    Two-point Boundary Value Problem
    Boundary value problems
    Discretization
    Difference equations
    Finite difference method
    Boundary Value Problem
    Finite Difference Equation
    Diagonal matrix
    Tridiagonal matrix
    Piecewise Linear
    Unique Solution
    Difference Method
    Finite Difference
    Partition

    Keywords

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

    ASJC Scopus subject areas

    • Applied Mathematics
    • Control and Optimization

    Cite this

    Discretization principles for linear two-point boundary value problems, II. / Yamamoto, Tetsuro; Oishi, Shinichi; Fang, Qing.

    In: Numerical Functional Analysis and Optimization, Vol. 29, No. 1-2, 01.2008, p. 213-224.

    Research output: Contribution to journalArticle

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