LAUSR

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 $${\mathcal {O}}(n^{-r})$$O(n-r) in $$L^2$$L2-norm and order $${\mathcal {O}}(n^{-r+\frac{1}{2}})$$O(n-r+12) in infinity norm, and the iterated Legendre Galerkin solution converges with the order $${\mathcal {O}}(n^{-2r})$$O(n-2r) in both $$L^2$$L2-norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order $${\mathcal {O}}(n^{-r })$$O(n-r) in both $$L^2$$L2-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 $${\mathcal {O}}(n^{-3r})$$O(n-3r) in both infinity and $$L^2$$L2 norm. Numerical examples are given to illustrate the theoretical results.