A pressure-robust virtual element method for the Navier-Stokes problem on polygonal mesh

We propose and analyze a new pressure-robust virtual element method for the incompressible Navier-Stokes equation. Our approach is based on the introduction of the reconstruction operator that is defined from the virtual function space to the piecewise RT finite element space with extra conditions enforced. By employing this operator in the discretizations of the nonlinear term and the body force, we introduce a pressure-robust virtual element method on polygonal meshes. The well-posedness and convergence analyses are carried out, which imply that the velocity error in energy norm is not affected by the continuous pressure. A series of numerical examples are shown to support the theoretical findings, including the optimal convergence rates for the velocity and pressure, the pressure-robust property and an application in a coupled flow-transport problem. Finally, the implementation of the reconstruction operator is shown in detail.