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 ω → ∞ without the pollution effect. A fast solver is developed for the resulting linear system with an empirical complexity O(ω d ) up to a poly-logarithmic factor. Numerical examples in 2D are presented to corroborate the claims.
Recently, a ray-based finite element method (ray-FEM) is proposed to learn and incorporate dominant directions into the basis in a stable, efficient and systematic way REF .
Learning Dominant Wave Directions For Plane Wave Methods For High-Frequency Helmholtz Equations
arXiv: Numerical Analysis
Mathematics