  We consider the computation of a nonlocal Helmholtz equation by using
Perfectly Matched Layer (PML). We first derive the nonlocal PML equation by
extending PML modifications from the local operator to the nonlocal operator of
integral form. We then give stability estimates of some weighted average value
of the nonlocal Helmholtz solution and prove that (i) the weighted average
value of the nonlocal PML solution decays exponentially in PML layers in one
case; (ii) in the other case, the weighted average value of the nonlocal
Helmholtz solution itself decays exponentially outside some domain.
Particularly for a typical kernel function $\gamma_1(s)=\frac12 e^{-| s|}$, we
obtain the Green's function of the nonlocal Helmholtz equation, and use the
Green's function to further prove that (i) the nonlocal PML solution decays
exponentially in PML layers in one case; (ii) in the other case, the nonlocal
Helmholtz solution itself decays exponentially outside some domain. Based on
our theoretical analysis, the truncated nonlocal problems are discussed and an
asymptotic compatibility scheme is also introduced to solve the resulting
truncated problems. Finally, numerical examples are provided to verify the
effectiveness and validation of our nonlocal PML strategy and theoretical
findings.
