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

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