# Preconditioned nonlinear iterations for overlapping Chebyshev discretizations with independent grids

## TL;DR

This paper introduces a nonlinear preconditioning approach using overlapping Chebyshev discretizations with independent grids, reducing communication and enabling parallel solutions for PDEs.

## Contribution

It presents a novel nonlinear Schwarz preconditioning method that operates on independent subdomain discretizations, avoiding restrictive updates and minimizing communication.

## Key findings

- Reduces interprocess communication in nonlinear PDE solvers
- Enables parallel independent subdomain problem solving
- Offers a flexible preconditioning framework for nonlinear discretizations

## Abstract

The additive Schwarz method is usually presented as a preconditioner for a PDE linearization based on overlapping subsets of nodes from a global discretization. It has previously been shown how to apply Schwarz preconditioning to a nonlinear problem. By first replacing the original global PDE with the Schwarz overlapping problem, the global discretization becomes a simple union of subdomain discretizations, and unknowns do not need to be shared. In this way restrictive-type updates can be avoided, and subdomains need to communicate only via interface interpolations. The resulting preconditioner can be applied linearly or nonlinearly. In the latter case nonlinear subdomain problems are solved independently in parallel, and the frequency and amount of interprocess communication can be greatly reduced compared to linearized preconditioning.