Numerical solutions for second-kind Volterra integral equations by Galerkin methods

Shu Hua Zhang; Yan Ping Lin; Ming Rao

Applications of Mathematics (2000)

  • Volume: 45, Issue: 1, page 19-39
  • ISSN: 0862-7940

Abstract

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In this paper, we study the global convergence for the numerical solutions of nonlinear Volterra integral equations of the second kind by means of Galerkin finite element methods. Global superconvergence properties are discussed by iterated finite element methods and interpolated finite element methods. Local superconvergence and iterative correction schemes are also considered by iterated finite element methods. We improve the corresponding results obtained by collocation methods in the recent papers [6] and [9] by H. Brunner, Q. Lin and N. Yan. Moreover, using an interpolation post-processing technique, we obtain a global superconvergence of the O ( h 2 r ) -convergence rate in the piecewise-polynomial space of degree not exceeding ( r - 1 ) . As a by-product of our results, all these higher order numerical methods can also provide an a posteriori error estimator, which gives critical and useful information in the code development.

How to cite

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Zhang, Shu Hua, Lin, Yan Ping, and Rao, Ming. "Numerical solutions for second-kind Volterra integral equations by Galerkin methods." Applications of Mathematics 45.1 (2000): 19-39. <http://eudml.org/doc/33047>.

@article{Zhang2000,
abstract = {In this paper, we study the global convergence for the numerical solutions of nonlinear Volterra integral equations of the second kind by means of Galerkin finite element methods. Global superconvergence properties are discussed by iterated finite element methods and interpolated finite element methods. Local superconvergence and iterative correction schemes are also considered by iterated finite element methods. We improve the corresponding results obtained by collocation methods in the recent papers [6] and [9] by H. Brunner, Q. Lin and N. Yan. Moreover, using an interpolation post-processing technique, we obtain a global superconvergence of the $O(h^\{2r\})$-convergence rate in the piecewise-polynomial space of degree not exceeding $(r-1)$. As a by-product of our results, all these higher order numerical methods can also provide an a posteriori error estimator, which gives critical and useful information in the code development.},
author = {Zhang, Shu Hua, Lin, Yan Ping, Rao, Ming},
journal = {Applications of Mathematics},
keywords = {Volterra integral equations; Galerkin methods; convergence and superconvergence; interpolation post-processing; iterative correction; a posteriori error estimators; Volterra integral equations; Galerkin methods; convergence and superconvergence; interpolation post-processing; iterative correction; a posteriori error estimators},
language = {eng},
number = {1},
pages = {19-39},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Numerical solutions for second-kind Volterra integral equations by Galerkin methods},
url = {http://eudml.org/doc/33047},
volume = {45},
year = {2000},
}

TY - JOUR
AU - Zhang, Shu Hua
AU - Lin, Yan Ping
AU - Rao, Ming
TI - Numerical solutions for second-kind Volterra integral equations by Galerkin methods
JO - Applications of Mathematics
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 45
IS - 1
SP - 19
EP - 39
AB - In this paper, we study the global convergence for the numerical solutions of nonlinear Volterra integral equations of the second kind by means of Galerkin finite element methods. Global superconvergence properties are discussed by iterated finite element methods and interpolated finite element methods. Local superconvergence and iterative correction schemes are also considered by iterated finite element methods. We improve the corresponding results obtained by collocation methods in the recent papers [6] and [9] by H. Brunner, Q. Lin and N. Yan. Moreover, using an interpolation post-processing technique, we obtain a global superconvergence of the $O(h^{2r})$-convergence rate in the piecewise-polynomial space of degree not exceeding $(r-1)$. As a by-product of our results, all these higher order numerical methods can also provide an a posteriori error estimator, which gives critical and useful information in the code development.
LA - eng
KW - Volterra integral equations; Galerkin methods; convergence and superconvergence; interpolation post-processing; iterative correction; a posteriori error estimators; Volterra integral equations; Galerkin methods; convergence and superconvergence; interpolation post-processing; iterative correction; a posteriori error estimators
UR - http://eudml.org/doc/33047
ER -

References

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