Accelerating the Improved Arrow--Hurwicz Iteration via the Anderson Algorithm for Steady-State Navier--Stokes Equations
Abstract
We apply Anderson acceleration to the improved Arrow--Hurwicz (IAH) method for the finite element solution of the steady-state incompressible Navier--Stokes equations. The IAH scheme avoids saddle-point solves by decoupling the velocity and pressure updates, but can require prohibitively many iterations, particularly at high Reynolds numbers. To place the acceleration on a rigorous footing, we reformulate the IAH iteration as a nonlinear fixed-point operator G for the grad-div augmented discrete formulation induced by the scheme and establish its well-definedness, Lipschitz continuity, and Fréchet differentiability, thereby verifying the required smoothness conditions locally near the fixed point. Numerical experiments on problems with known analytical solutions, lid-driven cavity flow up to Re = 15,000, and channel flow over a full step demonstrate that the resulting Anderson-accelerated improved Arrow--Hurwicz algorithm substantially reduces iteration counts and CPU time while retaining the reported manufactured-solution convergence rates and centerline-velocity agreement.
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BibTeX
@misc{ergen2026accelerating,
title = {{Accelerating the Improved Arrow--Hurwicz Iteration via the Anderson Algorithm for Steady-State Navier--Stokes Equations}},
author = {Sinan Ergen and Mustafa Ağgül and Mustafa Türkyılmazoğlu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01288},
eprint = {2609.01288},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01288},
note = {CC0},
}
RIS
TY - UNPB
TI - Accelerating the Improved Arrow--Hurwicz Iteration via the Anderson Algorithm for Steady-State Navier--Stokes Equations
AU - Sinan Ergen
AU - Mustafa Ağgül
AU - Mustafa Türkyılmazoğlu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01288
UR - https://arxiv.org/abs/2609.01288
AB - We apply Anderson acceleration to the improved Arrow--Hurwicz (IAH) method for the finite element solution of the steady-state incompressible Navier--Stokes equations. The IAH scheme avoids saddle-point solves by decoupling the velocity and pressure updates, but can require prohibitively many iterations, particularly at high Reynolds numbers. To place the acceleration on a rigorous footing, we reformulate the IAH iteration as a nonlinear fixed-point operator G for the grad-div augmented discrete formulation induced by the scheme and establish its well-definedness, Lipschitz continuity, and Fréchet differentiability, thereby verifying the required smoothness conditions locally near the fixed point. Numerical experiments on problems with known analytical solutions, lid-driven cavity flow up to Re = 15,000, and channel flow over a full step demonstrate that the resulting Anderson-accelerated improved Arrow--Hurwicz algorithm substantially reduces iteration counts and CPU time while retaining the reported manufactured-solution convergence rates and centerline-velocity agreement.
ER -