Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term

We study the numerical approximation of an integro-differential equation which is intermediate between the heat and wave equations. The proposed discretization uses convolution quadrature based on the firstand second-order backward difference methods in time, and piecewise linear finite elements in space. Optimal-order error bounds in terms of the initial data and the inhomogeneity are shown for positive times, without assumptions of spatial regularity of the data.