Discrete superconvergence of collocation solutions for first-kind volterra integral equations

Hui Liang; Hermann BRUNNER

doi:10.1216/JIE-2012-24-3-359

Discrete superconvergence of collocation solutions for first-kind volterra integral equations

Hui Liang^*
, Hermann BRUNNER

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

5 Citations (Scopus)

Abstract

It is known that collocation solutions for firstkind Volterra integral equations based on (discontinuous or continuous) piecewise polynomials cannot exhibit local superconvergence at the points of a uniform mesh. In this paper we present a complete analysis of local superconvergence of such collocation solutions for first-kind Volterra integral equations at non-mesh points. In particular, we discuss (i) the existence of superconvergence points for prescribed collocation points; (ii) the existence of collocation points for prescribed superconvergence points. Numerous examples illustrate the theory.

Original language	English
Pages (from-to)	359-391
Number of pages	33
Journal	Journal of Integral Equations and Applications
Volume	24
Issue number	3
DOIs	https://doi.org/10.1216/JIE-2012-24-3-359
Publication status	Published - 2012

User-Defined Keywords

Collocation solutions
First-kind volterra integral equations
Piecewise polynomials
Superconvergence at non-mesh points

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10.1216/JIE-2012-24-3-359

Cite this

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title = "Discrete superconvergence of collocation solutions for first-kind volterra integral equations",

abstract = "It is known that collocation solutions for firstkind Volterra integral equations based on (discontinuous or continuous) piecewise polynomials cannot exhibit local superconvergence at the points of a uniform mesh. In this paper we present a complete analysis of local superconvergence of such collocation solutions for first-kind Volterra integral equations at non-mesh points. In particular, we discuss (i) the existence of superconvergence points for prescribed collocation points; (ii) the existence of collocation points for prescribed superconvergence points. Numerous examples illustrate the theory.",

keywords = "Collocation solutions, First-kind volterra integral equations, Piecewise polynomials, Superconvergence at non-mesh points",

author = "Hui Liang and Hermann BRUNNER",

note = "The first author{\textquoteright}s research is supported by the National Nature Science Foundation of China (No. 11101130), the Heilongjiang University Science Funds for Young Scholar (No. QL201004) and the Research Fund of the Heilongjiang Provincial Education Department (No. 12511414). She gratefully acknowledges the financial support and the hospitality extended to her by HKBU{\textquoteright}s Department of Mathematics. The work of the second author was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant No. 9406) and by the Hong Kong Research Grants Council (RGC Grant HKBU 200207).",

year = "2012",

doi = "10.1216/JIE-2012-24-3-359",

language = "English",

volume = "24",

pages = "359--391",

journal = "Journal of Integral Equations and Applications",

issn = "0897-3962",

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T1 - Discrete superconvergence of collocation solutions for first-kind volterra integral equations

AU - Liang, Hui

AU - BRUNNER, Hermann

N1 - The first author’s research is supported by the National Nature Science Foundation of China (No. 11101130), the Heilongjiang University Science Funds for Young Scholar (No. QL201004) and the Research Fund of the Heilongjiang Provincial Education Department (No. 12511414). She gratefully acknowledges the financial support and the hospitality extended to her by HKBU’s Department of Mathematics. The work of the second author was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant No. 9406) and by the Hong Kong Research Grants Council (RGC Grant HKBU 200207).

PY - 2012

Y1 - 2012

N2 - It is known that collocation solutions for firstkind Volterra integral equations based on (discontinuous or continuous) piecewise polynomials cannot exhibit local superconvergence at the points of a uniform mesh. In this paper we present a complete analysis of local superconvergence of such collocation solutions for first-kind Volterra integral equations at non-mesh points. In particular, we discuss (i) the existence of superconvergence points for prescribed collocation points; (ii) the existence of collocation points for prescribed superconvergence points. Numerous examples illustrate the theory.

AB - It is known that collocation solutions for firstkind Volterra integral equations based on (discontinuous or continuous) piecewise polynomials cannot exhibit local superconvergence at the points of a uniform mesh. In this paper we present a complete analysis of local superconvergence of such collocation solutions for first-kind Volterra integral equations at non-mesh points. In particular, we discuss (i) the existence of superconvergence points for prescribed collocation points; (ii) the existence of collocation points for prescribed superconvergence points. Numerous examples illustrate the theory.

KW - Collocation solutions

KW - First-kind volterra integral equations

KW - Piecewise polynomials

KW - Superconvergence at non-mesh points

UR - https://www.scopus.com/pages/publications/84872483187

U2 - 10.1216/JIE-2012-24-3-359

DO - 10.1216/JIE-2012-24-3-359

M3 - Journal article

AN - SCOPUS:84872483187

SN - 0897-3962

VL - 24

SP - 359

EP - 391

JO - Journal of Integral Equations and Applications

JF - Journal of Integral Equations and Applications

IS - 3

ER -

Discrete superconvergence of collocation solutions for first-kind volterra integral equations

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