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 $\omega$, but is especially efficient for high-frequency problems. It is based on a time-domain approach and consists of three steps: \emph{i)} computation of a suitable incoming plane wavelet with compact support in the propagation direction; \emph{ii)} solving a scattering problem in the time domain for the incoming plane wavelet; \emph{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 \emph{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 $\omega$, as shown in the numerical experiments. We also present a new algorithm for computing the Fourier transform in \emph{iii)} that exploits the reduced number of degrees of freedom corresponding to the adapted meshes.