Celebratio Mathematica — Calderón

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J. E. Santos, J. Douglas, Jr., and A. P. Calderón: “Finite element methods for a composite model in elastodynamics,” SIAM J. Numer. Anal. 25 : 3 (1988), pp. 513–532. MR 942205 Zbl 0684.73034 article

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.

Jim Douglas, Jr.	Related
Juan Enrique Santos	Related