How does the partition of unity influence SORAS preconditioner?

TL;DR

This work investigates how the partition of unity affects the convergence of the SORAS preconditioner for the reaction-convection-diffusion equation. By comparing two PU constructions—one with zero gradient at subdomain interfaces (PU1) and another with interior coverage across the overlap (PU2)—the study shows that PU2 yields faster GMRES convergence as the overlap width $\delta$ increases, while PU1 exhibits limited sensitivity to $\delta$. Eigenvalue considerations reveal that PU2 reduces the largest eigenvalue of the preconditioned operator without affecting the smallest, providing a theoretical explanation for the improved convergence observed in experiments. The results offer practical guidance for configuring PU in SORAS to enhance robustness and efficiency for non-self-adjoint domain decomposition problems, especially under increased overlap settings.

Abstract

We investigate the influence of the choice of the partition of unity on the convergence of the Symmetrized Optimized Restricted Additive Schwarz (SORAS) preconditioner for the reaction-convection-diffusion equation. We focus on two kinds of partitions of unity, and study the dependence on the overlap and on the number of subdomains. In particular, the second kind of partition of unity, which is non-zero in the interior of the whole overlapping region, gives more favorable convergence properties, especially when increasing the overlap width, in comparison with the first kind of partition of unity, whose gradient is zero on the subdomain interfaces and which would be the natural choice for ORAS solver instead.

Figures (2)

• Figure 1: Illustration in a one-dimensional two-subdomain case of the two kinds of partition of unity functions $\chi_j \colon \Omega \to [0,1]$ (PU1 on the left and PU2 on the right), with increasing width of the overlap $\delta$ from top to bottom.
• Figure 2: Numerical range of the preconditioned operator ($\mathbf{a}= [-x, -y]^T$).