  Consider an exterior problem of the three-dimensional elastic wave equation,
which models the scattering of a time-harmonic plane wave by a rigid obstacle.
The scattering problem is reformulated into a boundary value problem by
introducing a transparent boundary condition. Given the incident field, the
direct problem is to determine the displacement of the wave field from the
known obstacle; the inverse problem is to determine the obstacle's surface from
the measurement of the displacement on an artificial boundary enclosing the
obstacle. In this paper, we consider both the direct and inverse problems. The
direct problem is shown to have a unique weak solution by examining its
variational formulation. The domain derivative is studied and a frequency
continuation method is developed for the inverse problem. Numerical experiments
are presented to demonstrate the effectiveness of the proposed method.

Analysis of PDEs