A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media

TL;DR

This paper introduces a hybrid numerical method combining ray-FEM and asymptotic expansion to accurately and efficiently solve high-frequency Helmholtz equations with point source singularities in smooth heterogeneous media, demonstrating favorable convergence and complexity.

Contribution

The paper presents a novel hybrid approach that effectively handles source singularities in high-frequency Helmholtz problems, improving accuracy and computational efficiency over existing methods.

Findings

Achieves asymptotic convergence rate of O(ω^{-1/2})

Demonstrates empirical overall complexity of O(ω^2) up to poly-logarithmic factors

Provides numerical evidence of accuracy and efficiency in 2D examples

Abstract

We propose a hybrid approach to solve the high-frequency Helmholtz equation with point source terms in smooth heterogeneous media. The method is based on the ray-based finite element method (ray-FEM), whose original version can not handle the singularity close to point sources accurately. This pitfall is addressed by combining the ray-FEM, which is used to compute the smooth far-field of the solution accurately, with a high-order asymptotic expansion close to the point source, which is used to properly capture the singularity of the solution in the near-field. The method requires a fixed number of grid points per wavelength to accurately represent the wave field with an asymptotic convergence rate of $\mathcal{O}(\omega^{-1/2})$, where $\omega$ is the frequency parameter in the Helmholtz equation. In addition, a fast sweeping-type preconditioner is used to solve the resulting linear system. We present numerical examples in 2D to show both accuracy and efficiency of our method as the frequency increases. In particular, we provide numerical evidence of the convergence rate, and we show empirically that the overall complexity is $\mathcal{O}(\omega^2)$ up to a poly-logarithmic factor.