  A discontinuous Petrov-Galerkin (DPG) method is used to solve the
time-harmonic equations of linear viscoelasticity. It is based on a "broken"
primal variational formulation, which is very similar to the classical primal
variational formulation used in Galerkin methods, but has additional
"interface" variables at the boundaries of the mesh elements. Both the
classical and broken formulations are proved to be well-posed in the
infinite-dimensional setting, and the resulting discretization is proved to be
stable. A full $hp$-convergence analysis is also included, and the analysis is
verified using computational simulations. The method is particularly useful as
it carries its own natural arbitrary-$p$ a posteriori error estimator, which is
fundamental for solving problems with localized solution features. This proves
to be useful when validating calibration models of dynamic mechanical analysis
(DMA) experiments. Indeed, different DMA experiments of epoxy and silicone
resins were successfully validated to within $5\%$ of the quantity of interest
using the numerical method.
