The NonConforming Virtual Element Method for the Stokes Equations
Abstract
In this paper, 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. Finally, numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.
- Authors:
-
[3]
- Univ. of Leicester (United Kingdom)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI) CNR (Italy)
- Publication Date:
- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES); USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program
- OSTI Identifier:
- 1337086
- Report Number(s):
- LA-UR-15-27293
Journal ID: ISSN 0036-1429
- Grant/Contract Number:
- AC52-06NA25396
- Resource Type:
- Accepted Manuscript
- Journal Name:
- SIAM Journal on Numerical Analysis
- Additional Journal Information:
- Journal Volume: 54; Journal Issue: 6; Journal ID: ISSN 0036-1429
- Publisher:
- Society for Industrial and Applied Mathematics
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mathematics; Virtual element method, finite element methods, polygonal and polyehdral mesh, high-order discretization, Stokes equations
Citation Formats
Cangiani, Andrea, Gyrya, Vitaliy, and Manzini, Gianmarco. The NonConforming Virtual Element Method for the Stokes Equations. United States: N. p., 2016.
Web. doi:10.1137/15M1049531.
Cangiani, Andrea, Gyrya, Vitaliy, & Manzini, Gianmarco. The NonConforming Virtual Element Method for the Stokes Equations. United States. https://doi.org/10.1137/15M1049531
Cangiani, Andrea, Gyrya, Vitaliy, and Manzini, Gianmarco. Fri .
"The NonConforming Virtual Element Method for the Stokes Equations". United States. https://doi.org/10.1137/15M1049531. https://www.osti.gov/servlets/purl/1337086.
@article{osti_1337086,
title = {The NonConforming Virtual Element Method for the Stokes Equations},
author = {Cangiani, Andrea and Gyrya, Vitaliy and Manzini, Gianmarco},
abstractNote = {In this paper, 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. Finally, numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.},
doi = {10.1137/15M1049531},
journal = {SIAM Journal on Numerical Analysis},
number = 6,
volume = 54,
place = {United States},
year = {Fri Jan 01 00:00:00 EST 2016},
month = {Fri Jan 01 00:00:00 EST 2016}
}
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