# The perfectly matched layer (PML): theory and practice

Real world wave propagation problems are often formulated in large or unbounded domains, while numerical simulations must be restricted to smaller computational domains, by introducing artificial boundaries. An effective and reliable domain truncation scheme becomes essential, since it enables efficient and high fidelity numerical simulations. The perfectly matched layer (PML) has emerged as an efficient and robust technology to simulate the absorption of waves in many applications. However, previous attempts to effectively include the PML in many modern numerical methods, such as the discontinuous Galerkin or the finite element method, proved to be a nightmare for practitioners. Exponential and/or linear growth is often seen in numerical simulations.

In this talk we will discuss the well-posedness and stability of the PML initial boundary value problem (IBVP). In particular, we will perform a spectral analysis of the integro-differential operator corresponding to the PML-IBVP, and derive general solutions of the PML-IBVP in the Fourier- Laplace domain. We will extend the analysis to discrete approximations and prove numerical stability.

Finally, I will present a large scale numerical simulation of a geophysical wave propagation problem, involving the scattering and interactions of acousto-elastic waves, in a complicated Earth model, with geologically constrained complex free-surface topography.