Error analysis of the spectral element method with Gauss-Lobatto-Legendre points for the acoustic wave equation in heterogeneous media

We present the error analysis of a high-order method for the two-dimensional acoustic wave equation in the particular case of constant compressibility and variable density. The domain discretization is based on the spectral element method with Gauss-Lobatto-Legendre (GLL) collocation points, whereas the time discretization is based on the explicit leapfrog scheme. As usual, GLL points are also employed in the numerical quadrature, so that the mass matrix is diagonal and the resulting algebraic scheme is explicit in time. The analysis provides an a priori estimate which depends on the time step, the element length, and the polynomial degree, generalizing several known results for the wave equation in homogeneous media. Numerical examples illustrate the validity of the estimate under certain regularity assumptions and provide expected error estimates when the medium is discontinuous.