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.
翻译：我们考虑使用完全匹配的图层来计算非本地 Helmholtz 方程式。 我们首先通过将本地操作员的PML修改扩展至非本地整体形式的操作员, 得出非本地 PML 方程式。 我们然后对非本地 Helmholtz 方程式的某些加权平均值进行稳定估计, 并证明 (一) 非本地 PML 方程式的加权平均值在 PML 层中指数衰减;(二) 在另一例中, 非本地 Helmholtz 方程式的加权平均值本身在某些域之外加速衰变。 特别是对于典型的内核函数 $\ gamma_ 1 (s)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ $ $ 美元, 我们 美元, 我们的外外外外内核计算法中, 我们的理论分析结果的逻辑分析提供了一个理论分析, 的理论问题, 的理论分析。 最后, 的理论分析, 的理论分析, 的理论分析, 的理论分析, 的理论分析,, 的, 的, 的, 的理论分析, 的, 的, 的, 的, 的, 的, 的, 的, 的, 理论的, 的, 的, 的, 的, 的,提供了的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的,, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的, 的,,,,,,,,, 的, 的,,,, 的,,,,,,,,,,,
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