| Title:
|
Numerical solutions for second-kind Volterra integral equations by Galerkin methods (English) |
| Author:
|
Zhang, Shuhua |
| Author:
|
Lin, Yanping |
| Author:
|
Rao, Ming |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
45 |
| Issue:
|
1 |
| Year:
|
2000 |
| Pages:
|
19-39 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Volterra integral equations |
| Keyword:
|
Galerkin methods |
| Keyword:
|
convergence and superconvergence |
| Keyword:
|
interpolation post-processing |
| Keyword:
|
iterative correction |
| Keyword:
|
a posteriori error estimators |
| MSC:
|
45L05 |
| MSC:
|
65B05 |
| MSC:
|
65N30 |
| MSC:
|
65R20 |
| idZBL:
|
Zbl 1058.65148 |
| idMR:
|
MR1738894 |
| DOI:
|
10.1023/A:1022284616125 |
| . |
| Date available:
|
2009-09-22T18:02:29Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134427 |
| . |
| Reference:
|
[1] K. Atkinson, J. Flores: The discrete collocation method for nonlinear integral equations.IMA J. Numer. Anal. 13 (1993), 195–213. MR 1210822, 10.1093/imanum/13.2.195 |
| Reference:
|
[2] H. Brunner: Iterated collocation methods and their discretization for Volterra integral equations.SIAM J. Numer. Anal. 21 (1984), 1132–1145. MR 0765511, 10.1137/0721070 |
| Reference:
|
[3] H. Brunner: The approximate solution of Volterra equations with nonsmooth solutions.Utilitas Math. 27 (1985), 57–95. Zbl 0563.65077, MR 0804372 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[8] H. Brunner, A. Pedas, G. Vainikko: The piecewise polynomial collocation methods for nonlinear weakly singular Volterra equations.Research Reports A 392, Institute of Mathematics, Helsinki University of Technology, 1997. |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
[15] Q. Lin, S. Zhang, N. Yan: An acceleration method for integral equations by using interpolation post-processing.Advances in Comput. Math. 9 (1998), 117–129. MR 1662762, 10.1023/A:1018925103993 |
| Reference:
|
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| . |
