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

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.