The non-conforming virtual element method for the Stokes equations

TL;DR

This paper introduces a non-conforming Virtual Element Method for the steady Stokes problem, enabling high-order accurate velocity and pressure approximations on complex meshes without explicit non-polynomial function evaluation.

Contribution

The paper develops a novel non-conforming VEM for the Stokes equations that works on general polygonal/polyhedral meshes and arbitrary polynomial degrees, ensuring stability and optimal error estimates.

Findings

Method achieves high-order accuracy in velocity and pressure.

Numerical results confirm theoretical convergence rates.

Approach is effective on complex mesh geometries.

Abstract

We present the non-conforming Virtual Element Method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable non-polynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non-polynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two-and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the non-conforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.