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

TL;DR

This paper analyzes the Arrow-Hurwicz iteration applied to divergence-free virtual element discretizations of the incompressible Navier-Stokes equations, demonstrating geometric convergence independent of mesh size through theoretical and numerical validation.

Contribution

It provides a rigorous convergence analysis of the Arrow-Hurwicz iteration for virtual element discretizations of Navier-Stokes, showing mesh-independent geometric convergence.

Findings

Proves geometric convergence with mesh-independent contraction factor

Validates theoretical results with numerical experiments

Demonstrates computational efficiency of the method

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.