Discontinuous Galerkin Methods for Auto-Convolution Volterra Integral Equations

The discontinuous Galerkin (DG) method is employed to solve the autoconvolution Volterra integral equations (AVIEs). The solvability of the DG method is discussed, then it is proved that the quadrature DG (QDG) method obtained from the DG method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the piecewise discontinuous polynomial collocation method. The uniform boundedness of the DG solution is provided by defining a discrete weighted exponential norm, and the optimal global convergence order of the DG solution is obtained. In order to improve the numerical accuracy, the iterated DG method is introduced. By virtue of a projection operator, the optimal m+1 superconvergence order of the iterated DG solution is gained, as well as 2m local superconvergence order at mesh points. It is noting that both the global and local superconvergence are obtained under the same regularity assumption as that for the convergence, other than the collocation method, one has to improve the regularity of the exact solution to obtain the superconvergence of the iterated collocation method. Some numerical experiments are given to illustrate the theoretical results.