A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator

The Helmholtz equation governing wave propagation and scattering phenomena
is difficult to solve numerically. Its discretization with piecewise linear finite
elements results in typically large linear systems of equations. The inherently
parallel domain decomposition methods constitute hence a promising class of
preconditioners. An essential element of these methods is a good coarse space.
Here, the Helmholtz equation presents a particular challenge, as even slight
deviations from the optimal choice can be devastating.

In this paper, we present a coarse space that is based on local eigenproblems
involving the Dirichlet-to-Neumann operator. Our construction is completely
automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. The resulting method is parallel by design and its efficiency is demonstrated on 2D homogeneous and heterogeneous numerical examples.