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Accelerating the Improved Arrow--Hurwicz Iteration via the Anderson Algorithm for Steady-State Navier--Stokes Equations
Sinan Ergen, Mustafa Ağgül, Mustafa Türkyılmazoğlu
TL;DR
Steady incompressible Navier–Stokes solvers need to reduce the cost of nonlinear, coupled velocity–pressure iterations, especially when IAH still requires many iterations at high Reynolds numbers. The paper applies Anderson acceleration to IAH, formulates the scheme as a smooth nonlinear fixed-point operator, and reports lower iteration counts and CPU time while retaining tested accuracy. The evidence is limited by the centerline-velocity diagnostic and by theory whose sufficient parameter restrictions are not enforced in experiments.
Problem
Finite-element Navier–Stokes computation is expensive because nonlinear convection and velocity–pressure coupling require costly coupled solves, while IAH can still need many iterations at high Reynolds numbers.
Method
The paper applies Anderson acceleration to IAH and reformulates the iteration as a grad-div augmented nonlinear fixed-point operator whose well-posedness, Lipschitz continuity, and Fréchet differentiability are established.
Results
Anderson acceleration significantly reduces IAH iteration counts and CPU time while preserving manufactured-solution convergence rates, centerline-velocity agreement, and stability through Re = 15,000 and full-step channel flow.
Takeaways & Limitations
AA-IAH is an efficient and robust alternative for the tested steady incompressible-flow problems.
Takeaways & Limitations
The reported centerline-velocity comparison is a diagnostic rather than a norm-based full-field error estimate, and experiments do not enforce the sufficient local-theory parameter bounds.
Abstract
from arXiv · showhide
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.
1 Introduction
Steady incompressible Navier–Stokes discretization is computationally difficult because nonlinear convection and velocity–pressure saddle-point coupling make standard iterations expensive. The paper applies Anderson acceleration to the improved Arrow–Hurwicz method and establishes the resulting fixed-point operator's analytical regularity.
- Motivation: Finite element Navier–Stokes solves are costly because convection creates nonlinearity and velocity–pressure coupling requires large coupled linear systems.These costs increase for fine meshes or complex three-dimensional problems.
- Existing approaches: Decoupling methods reduce computational costs by separating velocity and pressure computations into smaller subproblems.The Arrow–Hurwicz method is a notable example of this strategy.
- Existing approaches: The improved Arrow–Hurwicz method is more stable and efficient than classical Arrow–Hurwicz, but its iteration counts remain high at large Reynolds numbers.The paper identifies these iteration counts as an open target for improvement.
- Proposed acceleration: Anderson acceleration uses previous iterates in a least-squares optimization to speed fixed-point iterations.Prior applications include Picard, fixed-point, and grad-div-stabilized methods for Navier–Stokes and other nonlinear systems.
- Contribution: The paper reformulates IAH as a nonlinear fixed-point operator and proves its well-posedness, Lipschitz continuity, and Fréchet differentiability.The operator is defined on the product finite-element space for the grad-div augmented formulation.
2 Preliminaries
The preliminaries define the velocity–pressure spaces, weak Navier–Stokes formulation, finite-element discretization, and skew-symmetric convection form used by the improved Arrow–Hurwicz analysis. Discrete inf-sup compatibility and a small-data assumption guarantee a unique discrete solution with an a priori velocity bound.
- Function spaces: The natural spaces are X = [H1_0(Ω)]^d for velocity and Q = L2_0(Ω) for pressure, with X′ denoting the dual of X.The zero-mean pressure space and homogeneous velocity space provide the functional setting for the weak formulation.
- Weak formulation: The skew-symmetric trilinear form represents convection and satisfies a continuity bound proportional to ||∇u|| ||∇v|| ||∇w||.The bound uses a constant M independent of the particular velocity fields.
- Weak formulation: The continuous weak formulation seeks a velocity–pressure pair satisfying viscous, convective, pressure, and forcing terms against arbitrary test functions.The finite-element formulation replaces the continuous spaces and test functions with conforming discrete subspaces.
- Discrete stability: The discrete spaces must satisfy a mesh-independent LBB inf-sup condition to support the coupled finite-element formulation.This compatibility condition is imposed on Xh and Qh.
- Discrete stability: A discrete inf-sup condition together with the small-data assumption guarantees a unique nonlinear discrete solution and an a priori velocity stability bound.The bound is stated as ||∇uh|| ≤ ν^-1||f||_-1.
3 Improved Arrow–Hurwicz Scheme and Anderson Acceleration
The AA-IAH method reformulates the improved Arrow–Hurwicz iteration as a fixed-point problem and establishes the local analytical conditions required for Anderson acceleration. The analysis also clarifies the scope of the local theory and its relationship to the computational implementation.
- Method: The IAH iteration avoids redundancy with earlier theory by using the established boundedness, convergence, and error results while focusing this work on the AA-IAH formulation and analysis.The method is stated with the initialization and update steps used in the present work.
- Method: The IAH scheme is presented as a fixed-point operator G, with Anderson acceleration applied to the problem G(x) = x.The accelerated method uses residuals and a finite history of iterates, with depth m = 0 recovering standard fixed-point iteration.
- Fixed-point operator: The fixed-point equations correspond to a grad-div augmented discrete Navier–Stokes formulation, coinciding with the standard discrete system when the grad-div term vanishes on discretely divergence-free velocities.Otherwise, the fixed-point problem is interpreted as the grad-div stabilized discrete formulation.
- Fixed-point operator: The operator G is well-defined for every input pair in Xh × Qh because the associated finite-dimensional system has a unique solution.The uniqueness argument uses the triviality of the homogeneous system and finite-dimensional linear algebra.
- Local verification: Locally near a fixed point, G satisfies the boundedness and Lipschitz-continuity conditions required by Assumption 1.A bounded neighborhood U provides uniform constants through suprema of the relevant estimates.
- Residual analysis: The local theory verifies the assumptions needed for an Anderson residual bound when AA-IAH iterates remain in the neighborhood of the fixed point.The residual estimate includes the Anderson gain factor and constants depending on local derivative bounds, depth, and geometric conditions.
4 Numerical Experiments
The experiments evaluate accuracy, convergence, acceleration, and flow-pattern fidelity for AA-IAH against IAH across manufactured-solution, cavity-flow, and full-step problems. AA-IAH preserves spatial convergence and velocity agreement while reducing computational effort across tested Reynolds numbers and memory depths.
- Experimental design: The experiments assess accuracy, cavity-flow performance through Re = 15,000, and channel flow over a full step.The implementation uses Q2 velocity and Q1 pressure finite elements, with manufactured solutions for convergence analysis.
- Convergence test: IAH and AA-IAH with m = 2 produce nearly identical errors and convergence rates, exhibiting expected spatial convergence.Anderson acceleration changes the nonlinear iteration path but not the underlying Q2–Q1 spatial discretization.
- Lid-driven cavity flow: AA-IAH is more efficient than IAH for every tested memory depth and Reynolds number from 1,000 to 15,000.The comparison reports total iteration counts, CPU times, and relative iterate-change histories; m = 4 is shown for the history plots.
- Lid-driven cavity flow: The cavity convergence histories show slower, smoother IAH decreases and oscillatory AA-IAH behavior caused by combining previous mapped iterates.The oscillations are examined alongside the influence of memory depth on converged centerline profiles.
- Lid-driven cavity flow: AA-IAH and IAH produce closely matching cavity velocity diagnostics despite the measured reduction in computation time.Centerline profiles and streamline structures agree closely with reference data, including tests at Re ≥ 10,000.
- Channel flow over a full step: The full-step channel simulation captures the recirculation zone behind the obstacle and agrees with reported literature flow behavior.The computation uses a 480 × 160 mesh with 692,179 degrees of freedom.
5 Conclusion
The study shows that Anderson acceleration substantially improves the IAH method for steady incompressible-flow problems while preserving accuracy, stability, and its mathematical foundation.
- The proposed AA-IAH method achieves expected optimal error convergence rates with Q2–Q1 finite elements and manufactured-solution tests.
- The underlying operator is well-posed and Lipschitz continuously differentiable, providing a theoretical foundation for the method.
- Anderson acceleration significantly reduces iteration counts and CPU time compared with the IAH method despite additional least-squares calculations.The reported reduction in total solution time supports the method’s practical applicability.
- AA-IAH remains stable up to Re = 15,000 and for channel flow over a full step, while retaining centerline-velocity and manufactured-solution accuracy.
- The accelerated IAH algorithm is an efficient and robust alternative for the tested steady incompressible-flow problems.
Data Availability and Acknowledgements
The authors provide the source code and numerical scripts, and report that the numerical data are generated by the included experiment drivers without external datasets.
- Source code and scripts for the numerical results, tables, and figures are available at the stated GitHub repository.
- The numerical data are generated by the included experiment drivers, with no external experimental dataset used.
- The article is derived from the first author’s master’s thesis.
- The first author acknowledges TÜBİTAK financial support through the 2210-A National Scholarship Programme for MSc Students.