Finite Element Methods for a Composite Model in Elastodynamics
Juan Enrique Santos, Jim Jr. Douglas, A. P. Calderón · SIAM Journal on Numerical Analysis · 1988
The propagation of waves through a composite isotropic nonhomogeneous elastic system $\Omega $ consisting of an elastic solid $\Omega _s $ with an imbedded fluid-saturated porous medium $\Omega _p $ is analyzed. The linear elastodynamic equation for $\Omega _s $ and Biot’s low-frequency dynamic equations for $\Omega _p $ are chosen to describe the propagation, with appropriate boundary conditions at the interface between the media and along the artificial boundaries of the model. First the existence and uniqueness of the solution are analyzed and then the continuous and discrete-time Galerkin procedures for obtaining approximate solutions are defined. Standard finite element subspaces of $H^1 (\Omega )$ are used to approximate the displacement vector components for the solid part of $\Omega $, while mixed finite element subspaces of $H(\operatorname{div},\Omega _p )$ are employed to approximate the vector displacement for $\Omega _p $. Convergence of optimal order is derived for the methods under certain smoothness assumptions for the solution of the differential problem.