The nonconforming virtual element method for the stokes equations

Cangiani, Andrea; Gyrya, Vitaliy; Manzini, Gianmarco

Abstract

© 2016 Society for Industrial and Applied Mathematics. We present the nonconforming 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 nonpolynomial 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 nonconforming 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.

Citation

Cangiani, A., Gyrya, V., & Manzini, G. (2016). The nonconforming virtual element method for the stokes equations. SIAM Journal on Numerical Analysis, 54(6), 3411-3435. https://doi.org/10.1137/15M1049531

• Journal Article Type: Article
• Acceptance Date: Aug 26, 2016
• Online Publication Date: Nov 29, 2016
• Publication Date: Nov 29, 2016
• Deposit Date: Aug 9, 2019
• Journal: SIAM Journal on Numerical Analysis
• Print ISSN: 0036-1429
• Electronic ISSN: 1095-7170
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 54
• Issue: 6
• Pages: 3411-3435
• DOI: https://doi.org/10.1137/15M1049531
• Public URL: https://nottingham-repository.worktribe.com/output/2411676
• Publisher URL: https://epubs.siam.org/doi/10.1137/15M1049531