An adaptive finite element method for high-frequency scattering problems with variable coefficients

We introduce a new method for the numerical approximation of time-harmonic acoustic scattering problems stemming from material inhomogeneities. The method works for any frequency ω, but is especially efficient for high-frequency problems. It is based on a time-domain approach and consists of three steps: i) computation of a suitable incoming plane wavelet with compact support in the propagation direction; ii) solving a scattering problem in the time domain for the incoming plane wavelet; iii) reconstruction of the time-harmonic solution from the time-domain solution via a Fourier transform in time. An essential ingredient of the new method is a front-tracking mesh adaptation algorithm for solving the problem in ii). By exploiting the limited support of the wave front, this allows us to make the number of the required degrees of freedom to reach a given accuracy significantly less dependent on the frequency ω, as shown in the numerical experiments. We also present a new algorithm for computing the Fourier transform in iii) that exploits the reduced number of degrees of freedom corresponding to the adapted meshes.