On the treatment of exterior domains for the time-harmonic equations of stellar oscillations
In a recent article we started to analyze the time-harmonic equations of stellar oscillations. As a first step we considered bounded domains together with an essential boundary condition and established the well-posedness of the equation. In this article we consider the physical relevant case of the domain being $\mathbb{R}^3$. We discuss the treatment of the exterior domain, and show how to couple the two parts to obtain a well-posedness result. Further, for the Cowling approximation (which neglects the Eulerian perturbation of gravity) we derive a scalar equation in the atmosphere, couple it to the vectorial interior equation, and prove the well-posedness of the new system. This coupled system has the big advantages that it simplifies the construction of approximating transparent boundary conditions and leads to significant less degrees of freedom for discretizations.