Convergence analysis and proof of acceleration for NGMRES applied to the Picard iteration for Navier-Stokes equations

TL;DR

This paper analyzes the convergence and acceleration mechanisms of NGMRES when applied to the Picard iteration for Navier-Stokes equations, providing the first proof of its convergence properties.

Contribution

It introduces a convergence analysis for NGMRES applied to Navier-Stokes Picard iteration, identifying the mechanism behind its acceleration.

Findings

NGMRES scales the Picard Lipschitz constant by the optimization gain.

Convergence estimates are sharp and predictive.

NGMRES improves performance even when unaccelerated iteration diverges.

Abstract

We consider nonlinear GMRES (NGMRES) as an acceleration technique for the Navier-Stokes Picard iteration, a direction that has not previously been explored. We identify the optimal norm for the least squares optimization problem arising in the NGMRES algorithm, and establish a convergence analysis for NGMRES with general depth that proves NGMRES scales the Picard Lipschitz constant by the gain of the optimization problem. To our knowledge, this is the first convergence proof for NGMRES that identifies the mechanism responsible for convergence acceleration. Numerical experiments demonstrate that the convergence estimates are remarkably sharp. In addition, NGMRES greatly improves the performance of the Picard iteration, even in cases where the unaccelerated iteration diverges.