  We propose a simple domain decomposition method for $d$-dimensional elliptic
PDEs which involves an overlapping decomposition into local subdomain problems
and a global coarse problem. It relies on a space-filling curve to create
equally sized subproblems and to determine a certain overlap based on the
one-dimensional ordering of the space-filling curve. Furthermore we employ
agglomeration and a purely algebraic Galerkin discretization in the
construction of the coarse problem. This way, the use of $d$-dimensional
geometric information is avoided. The subproblems are dealt with in an
additive, parallel way, which gives rise to a subspace correction type linear
iteration and a preconditioner for the conjugate gradient method. To make the
algorithm fault-tolerant we store on each processor, besides the data of the
associated subproblem, a copy of the coarse problem and also the data of a
fixed amount of neighboring subproblems with respect to the one-dimensional
ordering of the subproblems induced by the space-filling curve. This redundancy
then allows to restore the necessary data if processors fail during the
computation. Theory supports that the convergence rate of such a linear
iteration method stays the same in expectation, and only its order constant
deteriorates slightly due to the faults. We observe this in numerical
experiments for the preconditioned conjugate gradient method in slightly weaker
form as well. Altogether, we obtain a fault-tolerant, parallel and efficient
domain decomposition method based on space-filling curves which is especially
suited for higher-dimensional elliptic problems.
