Superconvergence results for linear second-kind Volterra integral equations

Original Research
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Abstract

In this paper, Galerkin method is applied to approximate the solution of Volterra integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We prove that the approximate solutions of the Galerkin method converge to the exact solution with the order \({\mathcal {O}}(h^{r}),\) whereas the iterated Galerkin solutions converge with the order \({\mathcal {O}}(h^{2r})\) in infinity norm, where h is the norm of the partition and r is the smoothness of the kernel. We also consider the multi-Galerkin method and its iterated version, and we prove that the iterated multi-Galerkin solution converges with the order \({\mathcal {O}}(h^{3r})\) in infinity norm. Numerical examples are given to illustrate the theoretical results.

Keywords

Volterra integral equations Smooth kernels Galerkin method Multi-Galerkin method Piecewise polynomials Superconvergence rates 

Mathematics Subject Classification

45B05 45G10 65R20 

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Copyright information

© Korean Society for Computational and Applied Mathematics 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

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