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<p>In this paper, we investigate a stabilization technique for the Navier–Stokes equations for incompressible fluid flow using equal-order virtual element pairs on general polygonal meshes. We propose a residual-based SUPG-like stabilization term to address the violation of the discrete inf-sup condition, which leads to pressure instability, and to mitigate the effects of the convection-dominated regime. Additionally, we employ a grad-div stabilization term to address the violation of divergence-free constraints. We extend the concept of nonlinear stability derived in (López-Marcos and Sanz-Serna, IMA J. Numer. Anal. <b>8</b>(1), 71–84, <CitationRef CitationID="CR32">1998</CitationRef>) to a stabilized virtual element framework. Following the results of Lopez-Marcos &amp; Sanz-Serna, we establish the well-posedness and optimal convergence estimates in the energy norm using the branches of non-singular solutions. We perform several numerical experiments to validate the theoretical findings.</p>

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A SUPG-stabilized virtual element method for the Navier–Stokes equation: approximations of branches of non-singular solutions

  • 期刊论文

  • 发表日期：

    2025年7月7日

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  • Sudheer Mishra,
  • E. Natarajan,
  • Sundararajan Natarajan

摘要

In this paper, we investigate a stabilization technique for the Navier–Stokes equations for incompressible fluid flow using equal-order virtual element pairs on general polygonal meshes. We propose a residual-based SUPG-like stabilization term to address the violation of the discrete inf-sup condition, which leads to pressure instability, and to mitigate the effects of the convection-dominated regime. Additionally, we employ a grad-div stabilization term to address the violation of divergence-free constraints. We extend the concept of nonlinear stability derived in (López-Marcos and Sanz-Serna, IMA J. Numer. Anal. 8(1), 71–84, 1998) to a stabilized virtual element framework. Following the results of Lopez-Marcos & Sanz-Serna, we establish the well-posedness and optimal convergence estimates in the energy norm using the branches of non-singular solutions. We perform several numerical experiments to validate the theoretical findings.