Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem

Abstract

Abstract. We consider the application of a perfectly matched layer (PML) technique to approximate solutions to the elastic wave scattering problem in the frequency domain. The PML is viewed as a complex coordinate shift in spherical coordinates which leads to a variable complex coefficient equation for the displacement vector posed on an infinite domain (the complement of the scatterer). The rapid decay of the PML solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation (for the truncated problem).We prove existence and uniqueness of the solutions to the infinite domain and truncated domain PML equations (provided that the truncated domain is sufficiently large). We also show exponential convergence of the solution of the truncated PML problem to the solution of the original scattering problem in the region of interest. We then analyze a Galerkin numerical approximation to the truncated PML problem and prove that it is well posed provided that the PML damping parameter and mesh size are small enough. Finally, computational results illustrating the efficiency of the finite element PML approximation are presented.

On the use of perfectly matched layers in the presence of long or backward propagating guided elastic waves

“…References pertaining to PML for time-harmonic elastic wave propagation problems are, for instance, [22,23,8,24]. Here, only the most straighforward kind of PML, that is, with constant coefficients, is considered, but the whole reasoning still applies if these coefficients are complex-valued functions of the space variables satisfying the conditions stated below.…”

“…. , N ε − 1, it is a simple matter, using (23), to see that the equations (20) are equivalent to the linear system (24).…”

“…. , N ε − 1, knowing that these are solution to the linear system (24). To compute the entries of the matrix (resp.…”

Analysis of a Cartesian PML approximation to acoustic scattering problems in R2

“…Because of the exponential decay of PML solutions of the unbounded domain problem, it is natural to truncate the problem to a bounded domain with a convenient boundary condition, e.g., a homogeneous Dirichlet boundary condition, at the outer boundary of the computational domain. This method has been applied to the approximation of solutions to Maxwell's equations [5,6,9,10,13], elasticity problems [11,21,20] and acoustic resonances [22,23,27] as well as acoustic scattering problems [9,29].…”

A note on the existence and uniqueness of solutions of frequency domain elastic wave problems: A priori estimates in H1

“…The Kupradze-Sommerfeld radiation condition, though, provides additional numerical modeling difficulties. In a subsequent paper [3], we shall use the existence and uniqueness results of this paper as one step in the analysis of numerical approximations based on the "so-called" perfectly matched layer (PML). PML represents an efficient way to develop approximate boundary conditions for this problem and avoids the computational splitting of the solution.…”