The Arrow-Hurwicz iteration for virtual element discretizations of the incompressible Navier-Stokes equations

2025/07/16 by Binbin Du, Du, Binbin, Cheng, Shenxiang +4 · 1 citation
Engineering · #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2507.12036

openalex publication_date 2025/07/16 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

This article presents a detailed analysis of the Arrow-Hurwicz iteration applied to the solution of the incompressible Navier-Stokes equations, discretized by a divergence-free mixed virtual element method. Under a set of appropriate assumptions, it is rigorously demonstrated that the method exhibits geometric convergence, with a contraction factor that remains independent of the mesh sizes. A series of numerical experiments are conducted to validate the theoretical findings and to assess the computational performance of the proposed method.

Citations

• Stability and interpolation properties for Stokes-like virtual element spaces
• Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem
• The Stokes complex for Virtual Elements in three dimensions
• The Stokes complex for Virtual Elements with application to Navier--Stokes flows
• Inf-sup stable finite elements on barycentric refinements producing divergence--free approximations in arbitrary dimensions
• Some Error Analysis on Virtual Element Methods
• Virtual Elements for the Navier-Stokes problem on polygonal meshes
• The non-conforming virtual element method for the Stokes equations
• Stability Analysis for the Virtual Element Method
• Divergence free Virtual Elements for the Stokes problem on polygonal meshes
• Conforming and nonconforming virtual element methods for elliptic\n problems

Cited by

• Optimal error analysis of an interior penalty virtual element method for fourth-order singular perturbation problems

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• A Virtual Element method for non-Newtonian fluid flows
• A posteriori error analysis and adaptivity for a VEM discretization of the Navier-Stokes equations