Heriot-Watt Mathematics Report Series
HWM09-15, 6 Jan 2010
Discontinuous Galerkin approximations for Volterra integral equations of the first kind
H Brunner, P J Davies and D B Duncan
Abstract
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 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) \ne 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.
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