Variable Order, Directional H2-Matrices for Helmholtz Problems with Complex Frequency

The sparse approximation of high-frequency Helmholtz-type integral operators has many important physical applications such as problems in wave propagation and wave scattering. The discrete system matrices are huge and densely populated; hence their sparse approximation is of outstanding importance. In our paper we will generalize the directional $\mathcal{H}^{2}$-matrix techniques from the \textquotedblleft pure\textquotedblright\ Helmholtz operator $\mathcal{L}u=-\Delta u+\zeta^{2}u$ with $\zeta=-\operatorname*{i}k$, $k\in\mathbb{R}$, to general complex frequencies $\zeta\in\mathbb{C}$ with $\operatorname{Re}\zeta>0$. In this case, the fundamental solution decreases exponentially for large arguments. We will develop a new admissibility condition which contains $\operatorname{Re}\zeta$ in an explicit way and introduce the approximation of the integral kernel function on admissible blocks in terms of frequency-dependent \textit{directional expansion functions}. We develop an error analysis which is explicit with respect to the expansion order and with respect to $\operatorname{Re}\zeta$ and $\operatorname{Im}\zeta$. This allows to choose the \textit{variable }expansion order in a quasi-optimal way depending on $\operatorname{Re}\zeta$ but independent of, possibly large, $\operatorname{Im}\zeta$. The complexity analysis is explicit with respect to $\operatorname{Re}\zeta$ and $\operatorname{Im}\zeta$ and shows how higher values of $\operatorname{Re}$ reduce the complexity. In certain cases, it even turns out that the discrete matrix can be replaced by its nearfield part. Numerical experiments illustrate the sharpness of the derived estimates and the efficiency of our sparse approximation.