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The stress-strain constitutive law for viscoelastic materials such as softtissues, metals at high temperature, and polymers, can be written as a Volterraintegral equation of the second kind with a \emph{fading memory} kernel. Thisintegral relationship yields current stress for a given strain history and canbe used in the momentum balance law to derive a mathematical model for theresulting deformation. We consider such a dynamic linear viscoelastic modelproblem resulting from using a \textit{Dirichlet-Prony} series of decayingexponentials to provide the fading memory in the Volterra kernel. We introducetwo types of \textit{internal variable} to replace the Volterra integral with asystem of auxiliary ordinary differential equations and then use a spatiallydiscontinuous symmetric interior penalty Galerkin (SIPG) finite element methodand -- in time -- a Crank-Nicolson method to formulate the fully discreteproblems: one for each type of internal variable. We present \textit{a priori}stability and error analyses without using Gr\"onwall's inequality, and withthe result that the constants in our estimates grow linearly with time ratherthan exponentially. In this sense the schemes are therefore suited tosimulating long time viscoelastic response and this (to our knowledge) is thefirst time that such high quality estimates have been presented for SIPG finiteelement approximation of dynamic viscoelasticty problems. We also carry out anumber of numerical experiments using the FEniCS environment (e.g.\url{https://fenicsproject.org}) and explain how the codes can be obtained andthe results reproduced.

A Priori Analysis of a Symmetric Interior Penalty Discontinuous Galerkin Finite Element Method for a Dynamic Linear Viscoelasticity Model