Perfectly matched layers (PMLs) are formulated and applied to numerically solve nonlocal Helmholtz equations in one and two dimensions. In one dimension, we present the PML modifications for the nonlocal Helmholtz equation with general kernels and theoretically show its effectiveness in some sense. In two dimensions, we give the PML modifications in both Cartesian coordinates and polar coordinates. Based on the PML modifications, nonlocal Helmholtz equations are truncated in one and two dimensional spaces, and asymptotic compatibility schemes are introduced to discretize the resulting truncated problems. Finally, numerical examples are provided to study the "numerical reflections" by PMLs and demonstrate the effectiveness and validation of our nonlocal PML strategy.

翻译：完全匹配的层( PML) 用于一个和两个维的数值解非本地 Helmholtz 方程式。 在一个维度中, 我们用一般内核展示非本地 Helmholtz 方程式的 PML 修改, 并在理论上以某种意义上显示其有效性。 在两个维度中, 我们给Cartesian 坐标和极地坐标都提供 PML 修改 。 根据 PML 修改, 非本地 Helmholtz 方程式在一个和两个维的空格中被截断, 并引入了无症状兼容性计划, 将由此产生的脱轨问题分解 。 最后, 提供了数字示例, 用于研究 PMLs 的“ 数字反射”, 并展示我们非本地 PML 策略的有效性和有效性 。