  We show that even though the Discontinuous Galerkin Spectral Element Method
is stable for hyperbolic boundary-value problems, and the overset domain
problem is well-posed in an appropriate norm, the energy of the approximation
is bounded by data only for fixed polynomial order and time. In the absence of
dissipation, coupling of the overlapping domains is destabilizing by allowing
positive eigenvalues in the system to be integrated in time. This coupling can
be stabilized in one space dimension by using the upwind numerical flux. To
help provide additional dissipation, we introduce a novel penalty method that
applies dissipation at arbitrary points within the overlap region and depends
only on the difference between the solutions. We present numerical experiments
in one space dimension to illustrate the implementation of the well-posed
penalty formulation, and show spectral convergence of the approximations when
dissipation is applied.
