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arXiv · 2609.01288

Accelerating the Improved Arrow--Hurwicz Iteration via the Anderson Algorithm for Steady-State Navier--Stokes Equations

Sinan Ergen·Mustafa Ağgül·Mustafa Türkyılmazoğlu
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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 Fr\'{e}chet 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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Sinan Ergen, Mustafa Ağgül, Mustafa Türkyılmazoğlu. 2026-09-01. Accelerating the Improved Arrow--Hurwicz Iteration via the Anderson Algorithm for Steady-State Navier--Stokes Equations. https://arxiv.org/abs/2609.01288

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