Piecewise Divergence-Free Nonconforming Virtual Elements for Stokes Problem in Any Dimensions

TL;DR

This paper introduces a novel divergence-free nonconforming virtual element method for solving the Stokes problem in any dimension, featuring rigorous error analysis and pressure-robust modifications, validated by numerical experiments.

Contribution

It develops a divergence-free virtual element method with a local energy projector that commutes with divergence, enhancing accuracy and pressure robustness for the Stokes problem.

Findings

The method achieves optimal convergence rates.

Pressure-robustness improves solution accuracy.

Numerical results confirm theoretical error estimates.

Abstract

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.