Can classical Schwarz methods for time-harmonic elastic waves converge?
This work identifies a fundamental limitation of classical Schwarz methods for time-harmonic elastic wave problems, which is important for researchers developing domain decomposition methods for linear elasticity.
The paper demonstrates that the classical Schwarz method applied to time-harmonic Navier equations diverges for low to intermediate frequencies, which is worse than for Helmholtz and Maxwell equations where low frequencies only stagnate. Numerical examples illustrate divergent modes, and using the method as a preconditioner for Krylov methods still leads to convergence difficulties.
We show that applying a classical Schwarz method to the time harmonic Navier equations, which are an important model for linear elasticity, leads in general to a divergent method for low to intermediate frequencies. This is even worse than for Helmholtz and time harmonic Maxwell's equations, where the classical Schwarz method is also not convergent, but low frequencies only stagnate, they do not diverge. We illustrate the divergent modes by numerical examples, and also show that when using the classical Schwarz method as a preconditioner for a Krylov method, convergence difficulties remain.