A Spectral Element Approximation To The Galbrun Equation

The Galbrun equation, a nonstandard wave equation, was established by H. Gal-brun in the early 1930s. It is often used to study sound propagation in ?ows and hasbeen investigated in many ways. One of the important aspects of these investigationsis the development of the numerical methods, most of which are focused on using thefinite element method. A few works have been done on the spectral element method.In this paper, we consider the initial- and Dirichlet boundary-value problem for thegeneralized Galbrun equation. Precisely, we develop a numerical method based on theclassical Newmark's schema in time and a spectral element discretization in space. Forthe continuous problem, we study the existence, uniqueness and stability properties ofthe solution. For the semi-discrete approximation, we prove the stability of the schemeand derive an error estimate. From the estimate, it is seen that the accuracy of thespectral element approximation is spectral. For the full-discrete approximation, thestability of the numerical solution is obtained. Finally, several numerical examples areprovided to confirm the theoretical analysis.