Superconvergence analysis on discontinuous Galerkin methods for second-kind Volterra integral equations

In this paper, the superconvergence properties of discontinuous Galerkin (DG) approximations for second-kind Volterra integral equations (VIEs) with smooth kernels are rigorously analyzed. It is theoretically demonstrated that, within each subinterval, the Gauss points are exactly the optimal local superconvergence points for the DG solution. Based on this critical property, a simple yet efficient Lagrange-type interpolation postprocessing technique is developed to enhance the global convergence order. More importantly, the proposed approach achieves the same superconvergence order as the classical iterated postprocessing methods but completely avoids the computational cost of additional local integrations. Numerical experiments are provided to illustrate the theoretical results.