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Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations. (English) Zbl 1425.35016

Summary: We present a ray-based finite element method for the high-frequency Helmholtz equation in smooth media, whose basis is 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 is motivated by the geometric optics ansatz and is 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. Finally, a fast solver is developed for solving 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.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Software:

UMFPACK; Matlab
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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