Abstract
Perfectly matched layers (PMLs) are formulated and applied to numerically solve the nonlocal Helmholtz equations in one and two dimensions. In one dimension, we present a PML modification for the nonlocal Helmholtz equation with a general kernel and demonstrate theoretically how effective it is in some sense. In two dimensions, we present PML modifications in both Cartesian and polar coordinates. Based on the PML modifications, the nonlocal Helmholtz equations are truncated in one and two dimensional spaces, and an asymptotic compatibility scheme is 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.
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