A dimension-oblivious domain decomposition method based on space-filling curves

TL;DR

This paper introduces a dimension-oblivious, space-filling curve-based domain decomposition solver for elliptic PDEs that scales efficiently across arbitrary dimensions and processor counts.

Contribution

It presents a novel algebraic, dimension-oblivious solver using space-filling curves, enabling optimal scalability and convergence for high-dimensional problems.

Findings

Achieves optimal convergence in arbitrary dimensions.

Demonstrates excellent scalability on parallel systems.

Provides a basis for fault-tolerant high-dimensional solvers.

Abstract

In this paper we present an algebraic dimension-oblivious two-level domain decomposition solver for discretizations of elliptic partial differential equations. The proposed parallel 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 optimal convergence behavior. This is the core property required to attain a sparse grid based combination method 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 via space-filling curves. In this paper, we propose our space-filling curve based domain decomposition solver and present its convergence properties and scaling behavior. The results of numerical experiments clearly show that our approach provides optimal convergence and scaling behavior in arbitrary dimension utilizing arbitrary processor numbers.