AI 中文总结
该研究将安德森加速应用于改进的Arrow–Hurwicz方法，通过将其迭代重表述为非线性不动点算子并验证光滑性，大幅减少了稳态Navier–Stokes方程求解的迭代次数与CPU时间，且保持了收敛特性。
AI 中文摘要
我们将安德森加速（Anderson acceleration）应用于改进的Arrow–Hurwicz（IAH）方法，以求解稳态不可压Navier–Stokes方程的有限元解。IAH格式通过解耦速度与压力更新避免了鞍点求解，但可能需要极多的迭代次数，尤其在高雷诺数（Re）下。为使加速具备严谨基础，我们将IAH迭代重新表述为该格式诱导的梯度-散度增强离散格式的非线性不动点算子G，并证明其适定性、利普希茨连续性及弗雷歇可微性，从而验证不动点附近局部所需的光滑性条件。对具有已知解析解的问题、雷诺数高达15000的顶盖驱动空腔流及全阶梯通道流的数值实验表明，所得安德森加速的改进Arrow–Hurwicz算法大幅减少了迭代次数与CPU时间，同时保持了已报道的 manufactured解收敛率与中心线速度一致性。
英文摘要
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.
