  Deformations of viscoelastic materials such as soft tissues, metals at high
temperature, and polymers can be described as Volterra integral equations of
the second kind. We consider the viscoelasticity model problem involving with
\textit{Dirichlet Prony} series kernel, which resulting constitutive relation
with exponentially decaying faded memory. We introduce \textit{internal
variables} to replace Volterra integral to avoid the use of numerical
integration for the convolution. We can deal with the fading memory by solving
auxiliary ordinary differential equation systems govern by internal variables.
We use a spatially discontinuous Galerkin finite element method and a finite
difference method in time to formulate the fully discrete problem. We present
\textit{a priori} analysis for long time viscoelastic response without
Gr\"onwall's inequality. At the end, we carry out a number of numerical
experiments based on using the FEniCS environment,
\texttt{https://fenicsproject.org/}.
