Geometric meshes in collocation methods for Volterra integral equations with proportional delays

摘要

In this paper we analyse the local superconvergence properties of iterated piecewise polynomial collocation solutions for linear second-kind Volterra integral equations with (vanishing) proportional delay qt (0 < q < 1). It is shown that on suitable geometric meshes depending on q, collocation at the Gauss points leads to almost optimal superconvergence at the mesh points. This contrasts with collocation on uniform meshes where the problem regarding the attainable order of local superconvergence remains open.