A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem
Abstract
The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. Finally, we numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the $$\mathbb{P}_1$$ – $$\mathbb{P}_0$$ Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.
- Authors:
-
[1];
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Universita di Padova (Italy)
- Publication Date:
- Research Org.:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
- OSTI Identifier:
- 1844157
- Report Number(s):
- LA-UR-21-22961
Journal ID: ISSN 2158-2491
- Grant/Contract Number:
- 89233218CNA000001
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Computational Dynamics
- Additional Journal Information:
- Journal Volume: 9; Journal Issue: 2; Journal ID: ISSN 2158-2491
- Publisher:
- American Institute of Mathematical Sciences
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; Mathematics; Virtual Element Method; Stokes Equations; Scott-Vogelius finite element method
Citation Formats
Manzini, Gianmarco, and Mazzia, Annamaria. A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem. United States: N. p., 2021.
Web. doi:10.3934/jcd.2021020.
Manzini, Gianmarco, & Mazzia, Annamaria. A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem. United States. https://doi.org/10.3934/jcd.2021020
Manzini, Gianmarco, and Mazzia, Annamaria. Fri .
"A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem". United States. https://doi.org/10.3934/jcd.2021020. https://www.osti.gov/servlets/purl/1844157.
@article{osti_1844157,
title = {A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem},
author = {Manzini, Gianmarco and Mazzia, Annamaria},
abstractNote = {The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. Finally, we numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the $\mathbb{P}_1$ – $\mathbb{P}_0$ Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.},
doi = {10.3934/jcd.2021020},
journal = {Journal of Computational Dynamics},
number = 2,
volume = 9,
place = {United States},
year = {Fri Jan 01 00:00:00 EST 2021},
month = {Fri Jan 01 00:00:00 EST 2021}
}
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