*(English)*Zbl 0879.65097

The author studies the stability and pointwise error estimates in the ${L}^{\infty}$-norm for finite element approximations to the parabolic integro-differential equation

where $V\left(t\right)$ denotes a general (Volterra) integro-differential operator on a Hilbert space; typically,

with $A\left(t\right)$ linear and elliptic of second order, and $B(t,\tau )$ is a linear differential operator of order not exceeding two. The derivation of sharp ${L}^{\infty}$ 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.

##### MSC:

65R20 | Integral equations (numerical methods) |

45K05 | Integro-partial differential equations |

45N05 | Abstract integral equations, integral equations in abstract spaces |