The author studies the stability and pointwise error estimates in the -norm for finite element approximations to the parabolic integro-differential equation
where denotes a general (Volterra) integro-differential operator on a Hilbert space; typically,
with linear and elliptic of second order, and is a linear differential operator of order not exceeding two. The derivation of sharp error estimates is based on a certain adjoint equation whose solution may be viewed as a regularized Green’s function associated with the Ritz-Volterra operator [compare an earlier paper by Y. P. Lin, V. Thomée and L. B. Wahlbin, SIAM J. Numer. Anal. 28, No. 4, 1047-1070 (1991; Zbl 0728.65117)]. The results are applied to a number of concrete problems: parabolic integro-differential equations, Sobolev’s equation, and a diffusion equation with a nonlocal boundary condition. There are no numerical examples.