A modified Anderson acceleration with sharp linear convergence rate predictions and application to incompressible flows

TL;DR

This paper introduces a modified Anderson acceleration method, AAg, that accelerates Picard iterations for Navier-Stokes equations, providing sharp convergence rate predictions and an adaptive depth selection strategy, validated through numerical experiments.

Contribution

The work extends Anderson acceleration with a new residual-based variant, offering sharp convergence rate predictions and an adaptive depth strategy for improved efficiency.

Findings

AAg accelerates convergence of Picard iterations for Navier-Stokes.

The method provides sharp linear convergence rate predictions.

Numerical experiments demonstrate the effectiveness of AAg and the adaptive strategy.

Abstract

In this work, we extend a modified Anderson acceleration proposed in [Y. He, arXiv:2603.25983, 2026] to accelerate the Picard iteration for the Navier-Stokes equations. In this variant of Anderson acceleration, named AAg, the nonlinear residual--rather than the standard fixed-point iteration residual--is used to define the associated least-squares problem. We establish a convergence analysis for this method with any depth that shows how AAg accelerates convergence through the gain of the optimization problem, and obtain a sharp prediction of its linear convergence rate (a feature that is not part of the known theory for classical Anderson acceleration). Additionally, motivated by this sharp convergence prediction, we introduce an adaptive strategy that automatically selects the depth parameter. Results of several numerical experiments are given that illustrate the new theory and also demonstrate the effectiveness of the proposed adaptive approach. Comparisons of AAg to usual AA and nonlinear GMRES are also provided.