Discontinuous Galerkin approximations for Volterra integral equations of the first kind

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

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Abstract

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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