Discontinuous Galerkin Methods for Auto-Convolution Volterra Integral Equations (Advances in Applied Mathematics and Mechanics, 2025)
The discontinuous Galerkin (DG) method is employed to solve the auto-convolution Volterra integral equations (AVIEs). The solvability of the DG methodis discussed, then it is proved that the quadrature DG (QDG) method obtained fromthe DG method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the piecewise discontinuous polynomial collocationmethod. The uniform boundedness of the DG solution is provided by defining a discrete weighted exponential norm, and the optimal global convergence order of theDG solution is obtained. In order to improve the numerical accuracy, the iterated DGmethod 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 superconvergenceare obtained under the same regularity assumption as that for the convergence, otherthan the collocation method, one has to improve the regularity of the exact solutionto obtain the superconvergence of the iterated collocation method. Some numericalexperiments are given to illustrate the theoretical results.