Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations

Shu Hua Zhang; Tao Lin; Yan Ping Lin; Ming Rao

Applications of Mathematics (2000)

  • Volume: 45, Issue: 4, page 241-263
  • ISSN: 0862-7940

Abstract

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We present two defect correction schemes to accelerate the Petrov-Galerkin finite element methods [19] for nonlinear Volterra integro-differential equations. Using asymptotic expansions of the errors, we show that the defect correction schemes can yield higher order approximations to either the exact solution or its derivative. One of these schemes even does not impose any extra regularity requirement on the exact solution. As by-products, all of these higher order numerical methods can also be used to forma posteriori error estimators for accessing actual errors of the Petrov-Galerkin finite element solutions. Numerical examples are also provided to illustrate the theoretical results obtained in this paper.

How to cite

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Zhang, Shu Hua, et al. "Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations." Applications of Mathematics 45.4 (2000): 241-263. <http://eudml.org/doc/33058>.

@article{Zhang2000,
abstract = {We present two defect correction schemes to accelerate the Petrov-Galerkin finite element methods [19] for nonlinear Volterra integro-differential equations. Using asymptotic expansions of the errors, we show that the defect correction schemes can yield higher order approximations to either the exact solution or its derivative. One of these schemes even does not impose any extra regularity requirement on the exact solution. As by-products, all of these higher order numerical methods can also be used to forma posteriori error estimators for accessing actual errors of the Petrov-Galerkin finite element solutions. Numerical examples are also provided to illustrate the theoretical results obtained in this paper.},
author = {Zhang, Shu Hua, Lin, Tao, Lin, Yan Ping, Rao, Ming},
journal = {Applications of Mathematics},
keywords = {Volterra integro-differential equations; Petrov-Galerkin methods; asymptotic expansions; defect correction; a posteriori error estimators; Volterra integro-differential equations; Petrov-Galerkin methods; asymptotic expansions; defect correction; a posteriori error estimators},
language = {eng},
number = {4},
pages = {241-263},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations},
url = {http://eudml.org/doc/33058},
volume = {45},
year = {2000},
}

TY - JOUR
AU - Zhang, Shu Hua
AU - Lin, Tao
AU - Lin, Yan Ping
AU - Rao, Ming
TI - Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations
JO - Applications of Mathematics
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 45
IS - 4
SP - 241
EP - 263
AB - We present two defect correction schemes to accelerate the Petrov-Galerkin finite element methods [19] for nonlinear Volterra integro-differential equations. Using asymptotic expansions of the errors, we show that the defect correction schemes can yield higher order approximations to either the exact solution or its derivative. One of these schemes even does not impose any extra regularity requirement on the exact solution. As by-products, all of these higher order numerical methods can also be used to forma posteriori error estimators for accessing actual errors of the Petrov-Galerkin finite element solutions. Numerical examples are also provided to illustrate the theoretical results obtained in this paper.
LA - eng
KW - Volterra integro-differential equations; Petrov-Galerkin methods; asymptotic expansions; defect correction; a posteriori error estimators; Volterra integro-differential equations; Petrov-Galerkin methods; asymptotic expansions; defect correction; a posteriori error estimators
UR - http://eudml.org/doc/33058
ER -

References

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