# On condition numbers of symmetric and nonsymmetric domain decomposition methods

## TL;DR

This paper derives condition number estimates for nonsymmetric domain decomposition methods, including a specific Poisson equation case, providing insights into their convergence properties and operator characteristics.

## Contribution

It introduces a framework using oblique projections to estimate condition numbers and analyzes a restricted additive method for the Poisson equation.

## Key findings

- Bounded the condition number of the preconditioned operator
- Established non-negativity of the preconditioned operator
- Provided insights into convergence behavior of nonsymmetric methods

## Abstract

Using oblique projections and angles between subspaces we write condition number estimates for abstract nonsymmetric domain decomposition methods. In particular, we consider a restricted additive method for the Poisson equation and write a bound for the condition number of the preconditioned operator. We also obtain the non-negativity of the preconditioned operator. Condition number estimates are not enough for the convergence of iterative methods such as GMRES but these bounds may lead to further understanding of nonsymmetric domain decomposition methods.