### 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 language | English |
---|---|

Pages (from-to) | 213-224 |

Number of pages | 12 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 29 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2008 Jan |

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### 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

*Numerical Functional Analysis and Optimization*,

*29*(1-2), 213-224. https://doi.org/10.1080/01630560701766700

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

Research output: Contribution to journal › Article

*Numerical Functional Analysis and Optimization*, vol. 29, no. 1-2, pp. 213-224. https://doi.org/10.1080/01630560701766700

}

TY - JOUR

T1 - Discretization principles for linear two-point boundary value problems, II

AU - Yamamoto, Tetsuro

AU - Oishi, Shinichi

AU - Fang, Qing

PY - 2008/1

Y1 - 2008/1

N2 - 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.

AB - 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.

KW - Discretization principles

KW - Finite difference methods

KW - Finite element methods

KW - Two-point boundary value problems

UR - http://www.scopus.com/inward/record.url?scp=40049102202&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=40049102202&partnerID=8YFLogxK

U2 - 10.1080/01630560701766700

DO - 10.1080/01630560701766700

M3 - Article

AN - SCOPUS:40049102202

VL - 29

SP - 213

EP - 224

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 1-2

ER -