  We study a second order hyperbolic initial-boundary value partial
differential equation with memory, that results in an integro-differential
equation with a convolution kernel. The kernel is assumed to be either smooth
or no worse than weakly singular, that arise, e.g. in linear and fractional
order viscoelasticity. Existence and uniqueness of the spatial local and global
Galerkin approximation of the problem is proved by means of Picard iteration.
Then spatial finite element approximation of the problem is formulated, and
optimal order a priori error estimates are proved by energy method. The
required regularity of the solution, for optimal order convergence, is the same
as minimum regularity of the solution for second order hyperbolic partial
differential equations.
