# A posteriori error analysis for Schwarz overlapping domain decomposition methods

## TL;DR

This paper introduces an adjoint-based a posteriori error analysis for Schwarz overlapping domain decomposition methods, enabling targeted error reduction in quantities of interest through a two-stage strategy.

## Contribution

It develops a novel error decomposition framework for overlapping Schwarz methods, linking iteration and discretization errors to improve solution accuracy.

## Key findings

- Error decomposition into iteration and discretization contributions
- Efficient error reduction strategy for quantities of interest
- Applicability to high-performance computing scenarios

## Abstract

Domain decomposition methods are widely used for the numerical solution of partial differential equations on high performance computers. We develop an adjoint-based a posteriori error analysis for both multiplicative and additive overlapping Schwarz domain decomposition methods. The numerical error in a user-specified functional of the solution (quantity of interest) is decomposed into contributions that arise as a result of the finite iteration between the subdomains and from the spatial discretization. The spatial discretization contribution is further decomposed into contributions arising from each subdomain. This decomposition of the numerical error is used to construct a two stage solution strategy that efficiently reduces the error in the quantity of interest by adjusting the relative contributions to the error.