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 talk we will generalize the directional H2-matrix techniques from the "original" Helmholtz operator (purely imaginary wave number) to general complex frequencies z with Re(z) >0.

In this case, the fundamental solution decreases exponentially for large arguments. We will develop a new admissibility condition which contain Re(z) and Im(z) in an explicit way and introduce the approximation of the integral kernel function on admissible blocks in terms of frequency-dependent directional expansion functions. We present an error analysis which is explicit with respect to the expansion order and with respect to the real and imaginary part of z. This allows us to choose the variable expansion order in a quasi-optimal way depending on Re(z) but independent of, possibly large, Im(z). The complexity analysis is explicit with respect to Re(z) and Im(z) and shows how higher values of Re(z) 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.

This talk comprises joint work with S. Börm, Christian-Albrechts-Universität Kiel, Germany and M. Lopez-Fernandez, Sapienza Universita di Roma, Italy