A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

Manzini, Gianmarco; Mazzia, Annamaria

doi:10.3934/jcd.2021020

Title: A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

Journal Article · Fri Jan 01 00:00:00 EST 2021 · Journal of Computational Dynamics

DOI:https://doi.org/10.3934/jcd.2021020· OSTI ID:1844157

^[1]; Mazzia, Annamaria ^[2]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Universita di Padova (Italy)

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.

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Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 1844157

Report Number(s):: LA-UR-21-22961

Journal Information:: Journal of Computational Dynamics, Vol. 9, Issue 2; ISSN 2158-2491

Publisher:: American Institute of Mathematical SciencesCopyright Statement

Country of Publication:: United States

Language:: English

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Title: A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

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