Superconvergence in collocation methods for Volterra integral equations with vanishing delays

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In this paper, we investigate the optimal (global and local) convergence orders of the (iterated) collocation solutions for second-kind Volterra integral equations with vanishing delays on quasi-geometric meshes. It turns out that the classical global convergence results still hold under certain regularity conditions of the given functions. In particular, it is shown that the optimal local superconvergence order p=2m can be attained if collocation is at the m Gauss(–Legendre) points, which contrasts with collocations both on uniform meshes and on geometric meshes. Numerical experiments are performed to confirm our theoretical results.

论文关键词:Volterra integral equation,Vanishing delay,Collocation method,Superconvergence,Quasi-geometric mesh

论文评审过程:Received 26 September 2015, Revised 3 June 2016, Available online 21 June 2016, Version of Record 5 July 2016.

论文官网地址:https://doi.org/10.1016/j.cam.2016.06.010