We present a stable discontinuous Galerkin (DG) method with a perfectly
matched layer (PML) for three and two space dimensional linear elastodynamics,
in velocity-stress formulation, subject to well-posed linear boundary
conditions. First, we consider the elastodynamics equation, in a cuboidal
domain, and derive an unsplit PML truncating the domain using complex
coordinate stretching. Leveraging the hyperbolic structure of the underlying
system, we construct continuous energy estimates, in the time domain for the
elastic wave equation, and in the Laplace space for a sequence of PML model
problems, with variations in one, two and three space dimensions, respectively.
They correspond to PMLs normal to boundary faces, along edges and in corners.
Second, we develop a DG numerical method for the linear elastodynamics equation
using physically motivated numerical flux and penalty parameters, which are
compatible with all well-posed, internal and external, boundary conditions.
When the PML damping vanishes, by construction, our choice of penalty
parameters yield an upwind scheme and a discrete energy estimate analogous to
the continuous energy estimate. Third, to ensure numerical stability of the
discretization when PML damping is present, it is necessary to extend the
numerical DG fluxes, and the numerical inter-element and boundary procedures,
to the PML auxiliary differential equations. This is crucial for deriving
discrete energy estimates analogous to the continuous energy estimates. By
combining the DG spatial approximation with the high order ADER time stepping
scheme and the accuracy of the PML we obtain an arbitrarily accurate wave
propagation solver in the time domain. Numerical experiments are presented in
two and three space dimensions corroborating the theoretical results