Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for the Elastic Wave Problem in Multicontinuum Media

This paper considers the process of elastic wave propagation in heterogeneous high-contrast media. We use the multicontinuum homogenization method to derive a general multicontinuum elastic wave model and a particular dual-continuum isotropic case. For this dual-continuum case, we present the internally penalized discontinuous Galerkin (IPDG) method on a fine grid that can capture the complexities of the multicontinuum medium but entails high computational costs. We propose a coarse-grid approximation based on the GMsDGM to reduce the required computational resources. In this multiscale approach, we construct a multiscale space by solving local spectral problems on each coarse element. We present the numerical results for a two-dimensional model problem in a heterogeneous medium. For checking the proposed multiscale approach, we compute the relative errors between the reference fine-grid solution and the multiscale solution with different numbers of multiscale basis functions. The results demonstrate that the proposed method can provide high accuracy using a few degrees of freedom. Moreover, the combined multicontinuum and multiscale modeling approach provides a practical and efficient framework for solving problems associated with multiscale elastic wave propagation problems.