On the Well-Posedness of UPML Method for Wave Scattering in Layered Media (CSIAM Transactions on Applied Mathematics, 2024)
This paper proposes a novel method to establish the well-posedness of uniaxial perfectly matched layer (UPML) method for a two-dimensional acoustic scattering from a compactly supported source in a two-layered medium. We solve a longstanding problem by showing that the truncated layered medium scattering problem isalways resonance free regardless of the thickness and absorbing strength of UPML. Themain idea is based on analyzing an auxiliary waveguide problem obtained by truncating the layered medium scattering problem through PML in the vertical direction only.The Green function for this waveguide problem can be constructed explicitly based onthe separation of variables and Fourier transform. We prove that such a constructionis always well-defined regardless of the absorbing strength. The well-posedness of thefully UPML truncated scattering problem follows by assembling the waveguide Greenfunction through periodic extension.
Keywords: Helmholtz equation; perfectly matched layer; layered medium scattering; source scattering problem.