Discontinuous Galerkin approximations for Volterra integral equations of the first kind

Motivated by the problem of developing accurate and stable time-stepping methods for the single-layer potential equation for acoustic scattering from a surface, we present new convergence results for piece-wise polynomial discontinuous Galerkin (DG) approximations of a first-kind Volterra integral equation of convolution kernel type, where the kernel K is smooth and satisfies K(0) ≠ 0. We show that an mth degree DG approximation exhibits global convergence of order m when m is odd and order m + 1 when m is even. There is local superconvergence of one order higher (i.e. order m + 1 when m is odd and m + 2 when m is even), but in the even order case, there is superconvergence only if the exact solution u of the equation satisfies u (m+1) (0) = 0. We also present numerical test results which show that these theoretical convergence rates are optimal.