Superconvergence results of Legendre spectral projection methods for Volterra integral equations of second kind

In this paper, Legendre spectral projection methods are applied for the Volterra integral equations of second kind with a smooth kernel. We prove that the approximate solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the order $O\left(n^{-r}\right)$ in $L^2$-norm and order $O\left(n^{-r+\frac{1}{2}}\right)$ in infinity norm, and the iterated Legendre Galerkin solution converges with the order $O\left(n^{-2r}\right)$ in both $L^2$-norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order $O\left(n^{-r}\right)$ in both $L^2$-norm and infinity norm, n being the highest degree of Legendre polynomials employed in the approximation and r being the smoothness of the kernels. We have also considered multi-Galerkin method and its iterated version, and prove that the iterated multi-Galerkin solution converges with the order $O\left(n^{-3r}\right)$ in both infinity and $L^2$ norm. Numerical examples are given to illustrate the theoretical results.