Numer Algor (2009) 52:69–88 DOI 10.1007/s11075-008-9258-8 ORIGINAL PAPER Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation William McLean · Kassem Mustapha Received: 16 September 2008 / Accepted: 2 December 2008 / Published online: 19 December 2008 © Springer Science + Business Media, LLC 2008 Abstract We employ a piecewise-constant, discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Denoting the maximum time step by k, we prove an a priori error bound of order k under realistic assumptions on the regularity of the solution. We also show that a spatial dis- cretization using continuous, piecewise-linear finite elements leads to an addi- tional error term of order h 2 max(1, log k −1 ). Some simple numerical examples illustrate this convergence behaviour in practice. Keywords Non-uniform time steps · Memory term · Finite elements 1 Introduction We study the initial-boundary value problem ∂ t u + ∂ −α t Au = f for 0 < t < T , with u = u 0 at t = 0, (1) in the case −1 <α< 0, where the fractional power of ∂ t is interpreted in the Riemann–Liouville sense and typically A = −∇ 2 on a bounded domain. We thank the University of New South Wales for financial support provided by a Faculty Research Grant. W. McLean (B ) School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia e-mail: w.mclean@unsw.edu.au K. Mustapha Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia e-mail: kassem@kfupm.edu.sa