Solving Time-Harmonic Scattering Problems Based on the Pole Condition II: Convergence of the PML Method

Convergence of the Uniaxial Perfectly Matched Layer Method for Time-Harmonic Scattering Problems in Two-Layered Media

“…Lassas and Somersalo [16], [17], Hohage et al [14] studied the acoustic scattering problems for circular and smooth PML layers. It is proved in [14,16,17] that the PML solution converges exponentially to the solution of the original scattering problem as the thickness of the PML layer tends to infinity.…”

“…Lassas and Somersalo [16], [17], Hohage et al [14] studied the acoustic scattering problems for circular and smooth PML layers. It is proved in [14,16,17] that the PML solution converges exponentially to the solution of the original scattering problem as the thickness of the PML layer tends to infinity. In the practical application of PML methods, the adaptive PML method was proposed in Chen and Wu [4] for a scattering problem by periodic structures (the grating problem), in Chen and Liu [5] for the acoustic scattering problem, and in Chen and Chen [3] for Maxwell scattering problems.…”

Domain decomposition method for Maxwell’s equations: Scattering off periodic structures

“…The PML method goes back to Bérenger, [2]. Convergence of the method was proven in [15,16] and [14] for non-periodic problems. As is shown in Section 3 the PML method fails for periodic domains in the presence of Wood anomalies.…”

Additive Schwarz Method for Scattering Problems Using the PML Method at Interfaces

“…Our derivation of the PML method, described in detail in [8] is based on an analytic continuation, as in [9,2] Details of the implementation in 2D can by found in [13]. The basic idea is an analytic continuation of the solution in the exterior along a distance variable.…”

“…The error introduced by cutting the PML is analyzed for example in [9,8], where it is shown that the PML system is well-posed and the error decays exponentially with ρ. Remark: Toselli [11] coupled the incoming field at the external boundary of the PML, this way the incoming field is damped in the PML and this might explain, why he concluded that it is best to use a very thin layer.…”