Advances in Computational Mathematics

, Volume 39, Issue 3–4, pp 611–644

Multilevel correction for collocation solutions of Volterra integral equations with proportional delays



In this paper, we propose a convergence acceleration method for collocation solutions of the linear second-kind Volterra integral equations with proportional delay qt\((0<q<1)\). This convergence acceleration method called multilevel correction method is based on a kind of hybrid mesh, which can be viewed as a combination between the geometric meshes and the uniform meshes. It will be shown that, when the collocation solutions are continuous piecewise polynomials whose degrees are less than or equal to \({m} (m \leqslant 2)\), the global accuracy of k level corrected approximation is \(O(N^{-(2m(k+1)-\varepsilon)})\), where N is the number of the nodes, and \(\varepsilon\) is an arbitrary small positive number.


Delay integral equation Geometric mesh Collocation method Superconvergence High order interpolation operator Multilevel correction Hybrid meshes 

Mathematics Subject Classifications (2010)

65R20 34K28 


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematical and System Sciences, Chinese Academy of SciencesBeijingChina

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