Learning Dominant Wave Directions For Plane Wave Methods For High-Frequency Helmholtz Equations

TL;DR

This paper introduces a ray-based finite element method that learns local dominant wave directions to efficiently solve high-frequency Helmholtz equations, achieving asymptotic convergence without pollution effects.

Contribution

The method adaptively learns local ray directions from low-frequency probes and incorporates them into finite element bases, improving high-frequency Helmholtz solutions iteratively.

Findings

Achieves asymptotic convergence as frequency increases

Requires fixed grid points per wavelength

Develops a fast solver with empirical complexity O(ω^d)

Abstract

We present a ray-based finite element method (ray-FEM) by learning basis adaptive to the underlying high-frequency Helmholtz equation in smooth media. Based on the geometric optics ansatz of the wave field, we learn local dominant ray directions by probing the medium using low-frequency waves with the same source. Once local ray directions are extracted, they are incorporated into the finite element basis to solve the high-frequency Helmholtz equation. This process can be continued to further improve approximations for both local ray directions and the high frequency wave field iteratively. The method requires a fixed number of grid points per wavelength to represent the wave field and achieves an asymptotic convergence as the frequency $\omega\rightarrow \infty$ without the pollution effect. A fast solver is developed for the resulting linear system with an empirical complexity $\mathcal{O}(\omega^d)$ up to a poly-logarithmic factor. Numerical examples in 2D are presented to corroborate the claims.