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<p>In this work, starting from the lowest-order velocity-pressure pairs, we propose a stabilized virtual element scheme for solving the evolutionary Navier-Stokes equations. Its main idea is to rewrite a modified continuity equation (the modification refers to the addition of pressure jump/projection stabilization induced by the instability of a space pair) in a standard way involving an enriched divergence-free velocity field. This different velocity field is then fed back to the momentum equation, which allows fewer extra stabilized terms to control the dominating convection. Meanwhile, we provide the error estimates for the velocity and pressure with constants independent of any negative powers of viscosity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\nu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>, and present the matrix implementation of the enriched velocity field, including the velocity correction term and a local Brezzi-Douglas-Marini-like interpolation operator. Finally, numerical experiments are presented to show the performance of the proposed scheme on polygonal meshes and under small viscosity conditions.</p>

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The Lowest-Order Stabilized Virtual Element Method for the Evolutionary Navier-Stokes Equations with Small Viscosity

  • 期刊论文

  • 发表日期：

    2026年1月17日

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  • Xin Liu,
  • Qixuan Song,
  • Rui Zhang,
  • Zhangxin Chen

摘要

In this work, starting from the lowest-order velocity-pressure pairs, we propose a stabilized virtual element scheme for solving the evolutionary Navier-Stokes equations. Its main idea is to rewrite a modified continuity equation (the modification refers to the addition of pressure jump/projection stabilization induced by the instability of a space pair) in a standard way involving an enriched divergence-free velocity field. This different velocity field is then fed back to the momentum equation, which allows fewer extra stabilized terms to control the dominating convection. Meanwhile, we provide the error estimates for the velocity and pressure with constants independent of any negative powers of viscosity \(\nu \) ν , and present the matrix implementation of the enriched velocity field, including the velocity correction term and a local Brezzi-Douglas-Marini-like interpolation operator. Finally, numerical experiments are presented to show the performance of the proposed scheme on polygonal meshes and under small viscosity conditions.