Collocation methods for cordial Volterra integro-differential equations

This paper is concerned with collocation methods for cordial Volterra integro-differential equations (CVIDEs) with noncompact cordial operators. The existence, uniqueness and regularity of the exact solutions to CVIDEs are discussed, and a resolvent representation of the derivative of the exact solution is obtained. We approximate the exact solution by collocation in the space of continuous piecewise polynomials of degree m. The solvability of the collocation equations is proved for sufficiently small meshes diameter. It is shown that, if the solution is sufficiently smooth, the collocation solutions are convergent with global order of convergence m. Using an approach based on the resolvent formula, we prove that global superconvergence of order m+1 is attained with iterated collocation based on some special points. Some numerical examples are provided to verify the convergence results.