A Priori Analysis of Discontinuous Galerkin Finite Element Method for Dynamic Viscoelastic Models.

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/}.