The perfectly matched layer for the ultra weak variational formulation of the 3D Helmholtz equation

We investigate the feasibility of using the perfectly matched layer (PML) as an absorbing boundary condition for the ultra weak variational formulation (UWVF) of the 3D Helmholtz equation. The PML is derived using complex stretching of the spatial variables. This leads to a modified Helmholtz equation for which the UWVF can be derived. In the standard discrete UWVF, the approximating subspace is constructed from local solutions of the Helmholtz equation. In previous studies plane wave basis functions have been advocated because they simplify the building of the UWVF matrices. For the PML domain we propose a special set of plane wave basis functions which allow fast computations and efficiently reduce spurious numerical reflections. The method is validated by numerical experiments. In comparison to a low-order absorbing boundary condition, the PML shows superior performance