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

TL;DR

This paper develops and validates perfectly matched layers (PMLs) for nonlocal Helmholtz equations in multi-dimensional spaces, enabling effective numerical solutions with minimal reflections.

Contribution

It introduces PML modifications for nonlocal Helmholtz equations in 1D and 2D, including Cartesian and polar coordinates, and demonstrates their effectiveness through numerical validation.

Findings

PMLs effectively reduce numerical reflections in nonlocal Helmholtz problems.

The proposed PML strategies are validated through numerical experiments.

Asymptotic compatibility schemes successfully discretize the truncated problems.

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.