MATHEMATICS OF COMPUTATION Volume 65, Number 213 January 1996, Pages 1–17 NONSMOOTH DATA ERROR ESTIMATES FOR APPROXIMATIONS OF AN EVOLUTION EQUATION WITH A POSITIVE-TYPE MEMORY TERM CH. LUBICH, I.H. SLOAN, AND V. THOM ´ EE Abstract. 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 first- and 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. 1. Introduction We shall consider initial value problems of the form u t +  t 0 β(t − s)Au(s) ds = f (t) for t> 0, (1.1) u(0) = u 0 . Here, u t = ∂u/∂t and A is a selfadjoint positive definite second-order elliptic par- tial differential operator in Ω ⊂ R d , with Dirichlet boundary conditions, or, more generally, a positive definite linear operator in a real Hilbert space H. The kernel β is assumed to be real-valued and positive definite, i.e., for each T> 0 the kernel β belongs to L 1 (0,T ) and satisfies  T 0 ϕ(t)  t 0 β(t − s)ϕ(s) dsdt ≥ 0 for all ϕ ∈ C[0,T ]. (1.2) As is easily seen by an energy argument, the positive definiteness of β and A implies stability for the solution of (1.1), or ‖u(t)‖≤‖u 0 ‖ +2  t 0 ‖f (s)‖ ds for t> 0, (1.3) where ‖·‖ denotes the norm in the Hilbert space H. Such problems, or nonlinear versions thereof, are used to model viscoelasticity and heat conduction in materials with memory, cf. the references in [3]. When β is Received by the editor August 30, 1994. 1991 Mathematics Subject Classification. Primary 45K05, 65M60, 65D32. Key words and phrases. Evolution equation, memory term, nonsmooth data, convolution quad- rature. c 1996 American Mathematical Society 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use