Sparse Aitken–Schwarz domain decomposition with application to Darcy flow

Abstract

This paper focuses on the acceleration of the Schwarz method by the Aitken’s acceleration of the convergence technique by considering the special structure of the Schwarz’s error operator between two consecutive iterates on the domain decomposition’s interfaces. The new proposed method, called Sparse Aitken–Schwarz method, builds a low-rank space for each subdomain to approximate the true solution on its interfaces instead of building a low-rank space associated to the gathered subdomains’ interfaces solution as the Aitken–Schwarz technique does. We show that the resulting method is better than the full GMRES method applied to the global interface problem as the low-rank space generated is a space whose dimension is larger than the size of Krylov space for the same number of local solving. The Sparse Aitken–Schwarz has better robustness to noise than the Aitken–Schwarz. Weak and strong scaling results on a two-level parallel implementation show the improvements of the Sparse Aitken–Schwarz over the Aitken–Schwarz on a 3D Darcy flow application with randomly distributed permeability fields with high contrast values (Berenguer & Al, Aitken’s acceleration of the Schwarz process using singular value decomposition for heterogeneous 3D groundwater flow problems, Computers & Fluids 80:320-326, 2013).