On Convergence and Superconvergence of Discontinuous Galerkin Method for Semi-Explicit Index-1 Integro-Differential Algebraic Equations

This paper mainly focuses on the discontinuous Galerkin (DG) method for solving the semi-explicit index-1 integro-differential algebraic equation (IDAE), which is a coupled system of Volterra integro-differential equations (VIDEs) and second-kind Volterra integral equations (VIEs). The DG approach is applied to both the VIDE and VIE components of the system. The global convergence respectively in the $L^2$-norm and $L^\infty$-norm is established, and the local superconvergence for VIDE component is obtained. Furthermore, numerical examples are presented to validate the theoretical convergence and superconvergence results.