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Simultaneous approximation terms for elastic wave equations on nonuniform grids

We consider the finite difference discretization of isotropic elastic wave equations on nonuniform grids. The intended applications are seismic studies, where heterogeneity of the earth media can lead to severe oversampling for simulations on uniform grids. To address this issue, we demonstrate how to properly couple two non-overlapping neighboring subdomains that are discretized uniformly, but with different grid spacings. Specifically, a numerical procedure is presented to impose the interface conditions weakly through carefully designed penalty terms, such that the overall semi-discretization conserves a discrete energy resembling the continuous energy possessed by the elastic wave system.