  We explain how to use smooth bivariate splines of arbitrary degree to solve
the exterior Helmholtz equation based on a Perfectly Matched Layer (PML)
technique. In a previous study (cf. [26]), it was shown that bivariate spline
functions of high degree can approximate the solution of the bounded domain
Helmholtz equation with an impedance boundary condition for large wave numbers
$k\sim 1000$. In this paper, we extend this study to the case of the Helmholtz
equation in an unbounded domain. The PML is constructed using a complex
stretching of the coordinates in a rectangular domain, resulting in a weighted
Helmholtz equation with Dirichlet boundary conditions. The PML weights are also
approximated by using spline functions. The computational algorithm developed
in [26] is used to approximate the solution of the resulting weighted Helmholtz
equation. Numerical results show that the PML formulation for the unbounded
domain Helmholtz equation using bivariate splines is very effective over a
range of examples.
