Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains (Q2882321)

A perfectly matched layer (PML) is an artificial absorbing layer used for the numerical solution of wave equations by reducing the problem to a bounded domain whose interface does not reflect outgoing waves; PMLs correspond to complex scalings in one or more directions. The main result of this paper is that, under some not too restrictive assumptions on the right-hand side, the PML method for the Helmholtz equation NEWLINE\[NEWLINE (\Delta - \mu)u = f NEWLINE\]NEWLINE in quasi-cylindrical domains in \(\mathbb R^{n+1}\) is stable and of exponential convergence. This result is obtained by first proving a limiting-absorption principle for the Dirichlet Laplacian in quasi-cylindrical domains, and then showing that the problem in a truncated domain, which corresponds to a finite PML, describes in- and outgoing solutions with an error that vanishes exponentially fast as the length of the domain tends to infinity.NEWLINENEWLINERemarks: PMLs were originally introduced by \textit{J.-P. Berenger} in [J. Comput. Phys. 114, No. 2, 185--200 (1994; Zbl 0814.65129)]. For a short description of the principle of limiting absorption, see, e.g., the introduction in [\textit{Y. Saito}, Publ. Res. Inst. Math. Sci., Kyoto Univ. 7, 581--619 (1972; Zbl 0251.34043)]. A quasi-cylindrical domain is a domain which outside some bounded set is diffeomorphic to a semi-cylinder.