Piecewise divergence-free H(div )-nonconforming virtual elements are designed for Stokes problem in any dimensions. After introducing a local energy projector based on the Stokes problem and the stabilization, a divergence-free nonconforming virtual element method is proposed for Stokes problem. A detailed and rigorous error analysis is presented for the discrete method , including the norm equivalence of the stabilization on the kernel of the local energy projector, the interpolation error estimate, the discrete inf-sup condition, and the optimal error estimate of the discrete method . An important property in the analysis is that the local energy projector commutes with the divergence operator  . A reduced virtual element method is also discussed. Numerical results are provided to verify the theoretical convergence.
Piecewise divergence-free nonconforming virtual elements are designed for Stokes problem in any dimensions. After introducing a local energy projector based on the Stokes problem and the stabilization, a divergence-free nonconforming virtual element method is proposed for Stokes problem. A detailed and rigorous error analysis is presented for the discrete method . An important property in the analysis is that the local energy projector commutes with the divergence operator . With the help of a divergence-free interpolation operator onto a generalized Raviart-Thomas element space, a pressure-robust nonconforming virtual element method is developed by simply modifying the right hand side of the previous discretization . A reduced virtual element method is also discussed. Numerical results are provided to verify the theoretical convergence.
D
H(
D
div
R
)-nonconforming
nonconforming
D
, including the norm equivalence of the stabilization on the kernel of the local energy projector, the interpolation error estimate, the discrete inf-sup condition, and the optimal error estimate of the discrete method
A
. With the help of a divergence-free interpolation operator onto a generalized Raviart-Thomas element space, a pressure-robust nonconforming virtual element method is developed by simply modifying the right hand side of the previous discretization