  Motivated by recent work on coarse spaces for Helmholtz problems, we provide
in this paper a comparative study on the use of spectral coarse spaces of GenEO
type for heterogeneous indefinite elliptic problems within an additive
overlapping Schwarz method. In particular, we focus here on two different but
related formulations of local generalised eigenvalue problems and compare their
performance numerically. Even though their behaviour seems to be very similar
for several well-known heterogeneous test cases that are mildly indefinite,
only one of the coarse spaces has so far been analysed theoretically, while the
other one leads to a significantly more robust domain decomposition method when
the indefiniteness is increased. We present a summary of recent results
developing such a theory and describe how the numerical experiments illustrate
it.
