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Ray-based Finite Element Method for High-frequency Helmholtz Equations


In this dissertation we propose a ray-based finite element method (ray-FEM) for the high-frequency Helmholtz equation in smooth media, whose basis are learned adaptively from the medium and source. The method requires a fixed number of grid points per wavelength to represent the wave field; moreover, it achieves an asymptotic convergence rate of $\mathcal{O}(\omega^{-\frac{1}{2}})$, where $\omega$ is the frequency parameter in the Helmholtz equation.

The local basis are motivated by the geometric optics ansatz and are composed of polynomials modulated by plane waves propagating in a few dominant ray directions. The ray directions are learned by processing a low-frequency wave field that probes the medium with the same source. Once the local ray directions are extracted, they are incorporated into the local basis to solve the high-frequency Helmholtz equation. This process can be continued to further improve the approximations for both local ray directions and high-frequency wave fields iteratively.

In addition, a fast sweeping-type preconditioner is used to solve the resulting linear system. We present numerical examples in 2D to show both efficiency and convergence of our method as the frequency becomes larger and larger. In particular, we show empirically that the overall complexity is $\mathcal{O}(\omega^2)$ up to a poly-logarithmic factor.

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