  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.
