Absorbing boundary conditions for wave transmissions

We study wave transmission through infinite media. From the computational point of view, an infinite domain is replaced by artificially chosen bounded domains. In the first part of this work we concentrate on the scalar wave equation in a half plane and in a rectangular domain with second order absorbing boundary conditions imposed on these artificial boundaries. Since the boundary conditions are of the same order as the differential equation inside the domain, no traditional finite element method has been applicable to our problems. We first seek higher order energies for our initial-boundary value problem, which are natural in the sense that these energies do not increase in time. Next, we use these higher order energies to reduce the original second order problems to first order symmetric hyperbolic systems which are also dissipative. We then propose weak formulations, and then finite element methods are investigated. Stability and error estimates are given for Galerkin methods. We conclude that finite element methods are now applicable for our model problems. In the second part of this work we summarize some related works. Continuous-time and discrete-time Galerkin methods are studied for a model derived by Santos, Douglas, Corbero, and Lovera to describe wave propagations in a porous medium saturated by a two-phase immiscible fluid. The model is then applied to an acoustic well-logging, and finite element methods are investigated. We also propose that a slip boundary condition should be used on the solid surface for a two-phase flow model. Finally, we summarize our recent on-going results on the frequency domain treatments of waves, show existence and uniqueness results, and give a numerical analysis of the frequency domain problem. Also, applications of the frequency domain treatments of waves have been made to several problems