On Polynomial PMLs for Helmholtz Scattering Problems with High Wave-numbers

This paper presents a simple proof for the stability of circular perfectly matched layer (PML) methods for solving acoustic scattering problems in two and three dimensions. The medium function of PML allows arbitrary-order polynomials, and can be extended to general nondecreasing functions with a slight modification of the proof. In the regime of high wavenumbers, the inf-sup constant for the PML truncated problem is shown to be $\mathcal{O}(k^{−1}).$ Moreover, the PML solution converges to the exact solution exponentially, with a wavenumber-explicit rate, as either the thickness or medium property of PML increases. Numerical experiments are presented to verify the theories and performances of PML for variant polynomial degrees.