ON THE CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS FOR INTEGRAL-ALGEBRAIC EQUATIONS OF INDEX 1

The integral-algebraic equation (IAE) of index 1 is a mixed system of first-kind and second-kind Volterra integral equations (VIEs). In this paper, the discontinuous Galerkin (DG) method is proposed to solve the index-1 IAE, and the optimal global convergence order is obtained. The iterated DG method is introduced in order to improve the numerical accuracy, and the global superconvergence of the iterated DG solution is derived. However, due to the lack of the local superconvergence of the DG residual for first-kind VIEs, there is no local superconvergence for the mixed IAE system of first-kind and second-kind VIEs, and the numerical experiments also verify this. Some numerical experiments are given to illustrate the obtained theoretical results.