Perfectly Matched Layers for nonlocal Helmholtz equations II: multi-dimensional cases

Abstract: 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.