  In this work, we propose two-level space-time domain decomposition
preconditioners for parabolic problems discretized using finite elements. They
are motivated as an extension to space-time of balancing domain decomposition
by constraints preconditioners. The key ingredients to be defined are the
sub-assembled space and operator, the coarse degrees of freedom (DOFs) in which
we want to enforce continuity among subdomains at the preconditioner level, and
the transfer operator from the sub-assembled to the original finite element
space. With regard to the sub-assembled operator, a perturbation of the time
derivative is needed to end up with a well-posed preconditioner. The set of
coarse DOFs includes the time average (at the space-time subdomain) of
classical space constraints plus new constraints between consecutive subdomains
in time. Numerical experiments show that the proposed schemes are weakly
scalable in time, i.e., we can efficiently exploit increasing computational
resources to solve more time steps in the same {total elapsed} time. Further,
the scheme is also weakly space-time scalable, since it leads to asymptotically
constant iterations when solving larger problems both in space and time.
Excellent {wall clock} time weak scalability is achieved for space-time
parallel solvers on some thousands of cores.
