Global convergence and local superconvergence of first-kind Volterra integral equation approximations (Q2902205)

The results of the paper are concerning the convergence and superconvergence analysis of three discontinuous piecewise polynomial methods for the approximation of the solution of first kind Volterra integral equations with convolution kernel: discontinuous collocation, discontinuous Galerkin (DG), and quadrature discontinuous Galerkin (QDG) methods. In order to simplify the convergence analysis for collocation methods, the authors introduce new polynomial basis functions: the derivatives of the Lagrange polynomials of degree \(m + 1\). This choice leads to a new result about the superconvergence of discontinuous collocation methods.NEWLINENEWLINE Moreover, it is proved that a QDG method with degree of precision equal to \(m\) and \(m + 1\) quadrature points is equivalent to a collocation scheme such that its convergence analysis reduces it to those of the collocation method. The convergence of QDG methods is separately investigated for odd and even \(m\). As a consequence, in the even case the superconvergence of the Gauss-Legendre QDG method is exactly determined. Finally, it is proved that if the degree of precision of a quadrature rule is at least \(2m+2\), then the convergence and superconvergence behaviour of the QDG scheme is the same as of the DG scheme. The optimality of the theoretical convergence rates is confirmed by some numerical tests.