On a mixed Galerkin method for semi-explicit index-1 integro-differential algebraic equations

The semi-explicit index-1 integro-differential algebraic equation (IDAE) is a coupled system of Volterra integro-differential equations (VIDEs) and second-kind Volterra integral equations (VIEs). The existence, uniqueness and regularity of the exact solution are analyzed in detail. A numerical scheme mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) method is proposed for the IDAE with the VIDE part approximated by CG schemes and the VIE part approximated by DG schemes. First, the global convergence order of the numerical solution is obtained, which is optimal for the VIE part, but not for the VIDE part. To improve the numerical accuracy, the iterated DG method is introduced for the VIE part. By virtue of the iterated DG method, the optimal global convergence is obtained for the VIDE part, and the global and local superconvergence results are gained for the new combination of numerical schemes with CG and iterated DG methods. Some numerical experiments are given to illustrate the theoretical results.