How does the partition of unity influence SORAS preconditioner?

TL;DR

This paper examines how different types of partitions of unity affect the convergence of the SORAS preconditioner for reaction-convection-diffusion equations, highlighting the advantages of a specific non-zero interior partition.

Contribution

It provides a comparative analysis of two partition of unity types and their impact on SORAS convergence, emphasizing the benefits of the second kind with interior support.

Findings

The second kind of partition improves convergence with increased overlap.

Partition choice significantly influences preconditioner performance.

Interior-support partitions outperform interface-zero partitions in this context.

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.