A Dimension-Oblivious Domain Decomposition Method Based on Space-Filling Curves

In this paper we present an algebraic dimension-oblivious two-level domain decomposition solver for discretizations of elliptic partial differential equations. The proposed parallel domain decomposition solver is based on a space-filling curve partitioning approach that is applicable to any discretization, i.e., it directly operates on the assembled matrix equations. Moreover, it allows for the effective use of arbitrary processor numbers independent of the dimension of the underlying partial differential equation while maintaining its optimal convergence behavior. This is the core property required to attain a combination technique solver with extreme scalability which can utilize exascale parallel systems efficiently. Moreover, this approach provides a basis for the development of a fault-tolerant solver for the numerical treatment of high-dimensional Problems. To achieve the required data redundancy we are therefore concerned with large overlaps of our domain decomposition, which we construct in any dimension via space-filling curves. In this paper, we propose a space-filling curve based domain decompositoin solver and present its convergence properties and scaling behavior. The results of our numerical experiments clearly show that our approach provides optimal convergence and scaling behavior in an arbitrary dimension utilizing arbitrary processor numbers.