Indian Journal of Science and Technology

Abstract

We presented the results of polynomial piece-wise approximations of discontinuous Galerkin (DG) for a Volterra integral equation of the first kind with kernel convolution where the kernel K is smooth and applies to the K(0) ≠ 0 condition. We show that a DG approximation of m-th degree results in the total convergence of m degree if m is an odd number and when m is an even number, it gives a m + 1 degree. There is also a local hyper-convergence of one level higher order (For example, when m is odd, it is of m + 1 order and when m is even, it is of m + 1 order). But in the cases with even order, the hyper-convergence exists only when the exact solution u of the equation applies to u(m+1) (0) = 0. We have also provided the numerical tests results that show that the theoretical convergence is optimal.
Keywords: Approximation Error, Convolution Kernel, Discontinuous Galerkin Approximations, Global Convergence, Local Superconvergence