Open Access Publications from the University of California

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

• Author(s): Fang, Jun
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.
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.