Brunner, H. and Davies, P.J. and Duncan, D.B. (2009) Discontinuous Galerkin approximations for Volterra integral equations of the first kind. IMA Journal of Numerical Analysis, 29 (4). pp. 856-881. ISSN 0272-4979
Motivated by the problem of developing accurate and stable time-stepping methods for the single layer potential equation for accoustic scattering from a surface, we present new convergence results for piecewise 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) 6= 0. We show that a m-th 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.
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