A fault-tolerant domain decomposition method based on space-filling curves

Abstract

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.