Geometric meshes and their application to Volterra integro-differential equations with singularities

In this paper we introduce a new kind of mesh, a 'geometric mesh', and discuss the corresponding beta-polynomial collocation method for Volterra integro-differential equations with weakly singular kernels. It will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and such meshes. The advantage of geometric meshes is that the cost of computing the beta-polynomial collocation approximations can be decreased greatly.