Petrov-galerkin methods for nonlinear volterra integro-differential equations

Tao Lin; Yanping Lin; Ping Luo; Ming Rao; Shuhua Zhang

Petrov-galerkin methods for nonlinear volterra integro-differential equations

Tao Lin, Yanping Lin, Ping Luo, Ming Rao, Shuhua Zhang

Research output: Journal article publication › Journal article › Academic research › peer-review

Abstract

This paper presents a class of Petrov-Galerkin finite element (PGFE) methods for the initial-value problem for nonlinear Volterra integro-differential equations: y′(t) = f(t, y(t)) + ∫0t k(t, s, y(s))ds, t ∈ I := [0, T], y(0) = 0. These methods have global optimal convergence rates, and have certain global and local super-convergence features. Several post-processing techniques are proposed to obtain globally super-convergent approximations. As by products, these super-convergent approximations can be used as efficient a-posteriori error estimators. Numerical examples are provided to illustrate properties of these methods.

Original language	English
Pages (from-to)	405-426
Number of pages	22
Journal	Dynamics of Continuous, Discrete and Impulsive Systems Series B: Application and Algorithm
Volume	8
Issue number	3
Publication status	Published - 1 Sept 2001
Externally published	Yes

Keywords

A-posteriori error estimators
Interpolation post-processing
Optimal error estimates
Petrov-Galerkin finite element methods
Volterra integro-differential equations

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Applied Mathematics

Cite this

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title = "Petrov-galerkin methods for nonlinear volterra integro-differential equations",

abstract = "This paper presents a class of Petrov-Galerkin finite element (PGFE) methods for the initial-value problem for nonlinear Volterra integro-differential equations: y′(t) = f(t, y(t)) + ∫0t k(t, s, y(s))ds, t ∈ I := [0, T], y(0) = 0. These methods have global optimal convergence rates, and have certain global and local super-convergence features. Several post-processing techniques are proposed to obtain globally super-convergent approximations. As by products, these super-convergent approximations can be used as efficient a-posteriori error estimators. Numerical examples are provided to illustrate properties of these methods.",

keywords = "A-posteriori error estimators, Interpolation post-processing, Optimal error estimates, Petrov-Galerkin finite element methods, Volterra integro-differential equations",

author = "Tao Lin and Yanping Lin and Ping Luo and Ming Rao and Shuhua Zhang",

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TY - JOUR

T1 - Petrov-galerkin methods for nonlinear volterra integro-differential equations

AU - Lin, Tao

AU - Lin, Yanping

AU - Luo, Ping

AU - Rao, Ming

AU - Zhang, Shuhua

PY - 2001/9/1

Y1 - 2001/9/1

N2 - This paper presents a class of Petrov-Galerkin finite element (PGFE) methods for the initial-value problem for nonlinear Volterra integro-differential equations: y′(t) = f(t, y(t)) + ∫0t k(t, s, y(s))ds, t ∈ I := [0, T], y(0) = 0. These methods have global optimal convergence rates, and have certain global and local super-convergence features. Several post-processing techniques are proposed to obtain globally super-convergent approximations. As by products, these super-convergent approximations can be used as efficient a-posteriori error estimators. Numerical examples are provided to illustrate properties of these methods.

AB - This paper presents a class of Petrov-Galerkin finite element (PGFE) methods for the initial-value problem for nonlinear Volterra integro-differential equations: y′(t) = f(t, y(t)) + ∫0t k(t, s, y(s))ds, t ∈ I := [0, T], y(0) = 0. These methods have global optimal convergence rates, and have certain global and local super-convergence features. Several post-processing techniques are proposed to obtain globally super-convergent approximations. As by products, these super-convergent approximations can be used as efficient a-posteriori error estimators. Numerical examples are provided to illustrate properties of these methods.

KW - A-posteriori error estimators

KW - Interpolation post-processing

KW - Optimal error estimates

KW - Petrov-Galerkin finite element methods

KW - Volterra integro-differential equations

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M3 - Journal article

SN - 1492-8760

VL - 8

SP - 405

EP - 426

JO - Dynamics of Continuous, Discrete and Impulsive Systems Series B: Application and Algorithm

JF - Dynamics of Continuous, Discrete and Impulsive Systems Series B: Application and Algorithm

IS - 3

ER -

Petrov-galerkin methods for nonlinear volterra integro-differential equations

Abstract

Keywords

ASJC Scopus subject areas

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