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.

翻译：即使间断伽辽金谱元方法对双曲边值问题是稳定的，且叠合区域问题在适当范数下是适定的，但仅当多项式阶数和时间固定时，逼近的能量才受到数据的约束。在无耗散情况下，通过允许系统中正特征值随时间积分，重叠区域的耦合会导致失稳。通过使用迎风数值通量，这种耦合在一维空间中可被稳定化。为提供额外耗散，我们提出一种新型惩罚方法，其在重叠区域内任意点施加耗散，且仅依赖于解之间的差异。我们通过一维数值实验验证了适定惩罚公式的实现，并展示了施加耗散时逼近的谱收敛性。