Convergence and superconvergence analysis of discontinuous Galerkin methods for index-2 integral-algebraic equations

The integral-algebraic equation (IAE) is a mixed system of first-kind and second-kind Volterra integral equations (VIEs). This paper mainly focuses on the discontinuous Galerkin (DG) method to solve index-2 IAEs. First, the convergence theory of perturbed DG methods for first-kind VIEs is established, and then used to derive the optimal convergence properties of DG methods for index-2 IAEs. It is shown that an $(m-1)$-th degree DG approximation exhibits global convergence of order~$m$ when~$m$ is odd, and of order~$m-1$ when~$m$ is even, for the first component~$x_1$ of the exact solution, corresponding to the second-kind VIE, whereas the convergence order is reduced by two for the second component~$x_2$ of the exact solution, corresponding to the first-kind VIE. Each component also exhibits local superconvergence of one order higher when~$m$ is even. When~$m$ is odd, superconvergence occurs only if $x_1$ satisfies $x_1^{(m)}(0)=0$. Moreover, with this condition, we can extend the local superconvergence result for~$x_2$ to global superconvergence when~$m$ is odd. Note that in the DG method for an index-1 IAE, generally, the global superconvergence of the exact solution component corresponding to the second-kind VIE can only be obtained by iteration. However, we can get superconvergence for all components of the exact solution of the index-2 IAE directly. Some numerical experiments are given to illustrate the obtained theoretical results.