A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation

• Bermúdez et al.'s type A PML damping function performs best for kδPML≪1kδPML≪1.
• Extensive numerical experiments reveal an optimal range of PML thicknesses, IoptIopt.
• For kδPML∈IoptkδPML∈Iopt the error is small and insensitive to change.
• Choosing kδPMLkδPML from anywhere within IoptIopt yields a parameter-free PML method.

This paper presents a parameter-free perfectly matched layer (PML) method for the finite-element-based solution of the Helmholtz equation. We employ one of Bermúdez et al.'s unbounded absorbing functions for the complex coordinate mapping underlying the PML. With this choice, the only free parameter that controls the accuracy of the numerical solution for a fixed numerical cost (characterised by the number of elements in the bulk and the PML regions) is the thickness of the perfectly matched layer, δPMLδPML. We show that, for the case of planar waves, the absorbing function performs best for PMLs whose thickness is much smaller than the wavelength. We then perform extensive numerical experiments to explore its performance for non-planar waves, considering domain shapes with smooth and polygonal boundaries, different solution types (smooth and singular), and a wide range of wavenumbers, k  , to identify an optimal range for the normalised PML thickness, kδPMLkδPML, such that, within this range, the error introduced by the presence of the PML is consistently small and insensitive to change. This implies that if the PML thickness is chosen from within this range no further PML optimisation is required, i.e. the method is parameter-free. We characterise the dependence of the error on the discretisation parameters and establish the conditions under which the convergence of the solution under mesh refinement is controlled exclusively by the discretisation of the bulk mesh.