  We propose and rigorously analyse semi- and fully discrete discontinuous
Galerkin methods for an initial and boundary value problem describing inertial
viscoelasticity in terms of elastic and viscoelastic stress components, and
with mixed boundary conditions. The arbitrary-order spatial discretisation
imposes strongly the symmetry of the stress tensor, and it is combined with a
Newmark trapezoidal rule as time-advancing scheme. We establish stability and
convergence properties, and the theoretical findings are further confirmed via
illustrative numerical simulations in 2D and 3D.
