  Fault tolerant algorithms for the numerical approximation of elliptic partial
differential equations on modern supercomputers play a more and more important
role in the future design of exa-scale enabled iterative solvers. Here, we
combine domain partitioning with highly scalable geometric multigrid schemes to
obtain fast and fault-robust solvers in three dimensions. The recovery strategy
is based on a hierarchical hybrid concept where the values on lower dimensional
primitives such as faces are stored redundantly and thus can be recovered
easily in case of a failure. The lost volume unknowns in the faulty region are
re-computed approximately with multigrid cycles by solving a local Dirichlet
problem on the faulty subdomain. Different strategies are compared and
evaluated with respect to performance, computational cost, and speed up.
Especially effective are strategies in which the local recovery in the faulty
region is executed in parallel with global solves and when the local recovery
is additionally accelerated. This results in an asynchronous multigrid
iteration that can fully compensate faults. Excellent parallel performance on a
current peta-scale system is demonstrated.
