Natural damping of time-harmonic waves and its influence on Schwarz methods

TL;DR

The paper addresses solving time-harmonic wave problems with Schwarz methods and analyzes how natural damping from the underlying time-domain dynamics influences convergence. It introduces damped Helmholtz models arising from first-order damping ($r$) and viscoelastic damping ($\gamma$), and uses Fourier analysis to compare waveguide and cavity configurations under impedance interfaces. The main finding is that physical damping can dramatically improve Schwarz convergence, sometimes transforming otherwise intractable closed-cavity problems into easily solvable ones, with distinct scaling behaviors depending on the damping type and frequency $\omega$. These insights guide the use of natural damping as a practical tool to enhance Schwarz methods for high-frequency wave problems in engineered geometries.

Abstract

The influence of various damping on the performance of Schwarz methods for time-harmonic waves is visualized by Fourier analysis.

Figures (10)

• Figure 1: Greens function for 3 Helmholtz problems. Left: closed cavity; middle: wave guide; right: free space, where optimized Schwarz solvers are as effective as for Laplace problems Gander_Zhang_2022.
• Figure 2: Greens functions with first order damping $r=1$ (top) and viscoelastic damping $\gamma=0.003$ (bottom).
• Figure 3: Convergence factor dependence on $r$ for the waveguide with the operator $\Delta + \omega^2 - \I\omega r$
• Figure 4: Convergence factor dependence on $\omega$ (top), number of subdomains $N$ (middle) and overlap $L$ (bottom) for the waveguide with the operator $\Delta + \omega^2 - \I\omega r$
• Figure 5: Convergence factor dependence on $\gamma$ for the waveguide with the operator $(1+ \I\omega \gamma)\Delta + \omega^2$