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 ‘pure’ Helmholtz operator $\mathcal{L}u=-\varDelta u+\zeta ^{2}u$ with $\zeta =-\operatorname *{i}k$, $k\in \mathbb{R}$ to general complex frequencies $\zeta \in \mathbb{C}$ with $\operatorname{Re}\zeta\geq0$. In this case, the fundamental solution decreases exponentially for large arguments. We will develop a new admissibility condition that contains $\operatorname{Re}\zeta $ in an explicit way, and introduces the approximation of the integral kernel function on admissible blocks in terms of frequency-dependent directional expansion functions. We develop an error analysis that is explicit with respect to the expansion order and with respect to $\operatorname{Re}\zeta $ and $\operatorname{Im}\zeta $. This allows for choosing the 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} \zeta $ 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.