Divergence free Virtual Elements for the Stokes problem on polygonal meshes

By L. Beirão da Veiga, Carlo Lovadina, Giuseppe Vacca. 2015

331 citations recorded by the source corpus.

In the present paper we develop a new family of Virtual Elements for the Stokes problem on polygonal meshes. By a proper choice of the Virtual space of velocities and the associated degrees of freedom, we can guarantee that the final discrete velocity is pointwise divergence-free, and not only in a relaxed (projected) sense, as it happens for more standard elements. Moreover, we show that the discrete problem is immediately equivalent to a reduced problem with less degrees of freedom, thus yielding a very efficient scheme. We provide a rigorous error analysis of the method and several numerical tests, including a comparison with a different Virtual Element choice.

Subfields: Computational Mechanics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/4c1181185d1ec47815ce399e4fee760109ca679f

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In Computational Mechanics

In Mechanics of Materials

Citations in the corpus

Virtual element for the buckling problem of Kirchhoff–Love plates (2019, background): “in a large range of problems in fluid and solid mechanics; see for instance [2, 3, 4, 7, 8, 9, 11, 12, 15, 16, 21, 22, 23, 24, 33, 40, 43, 44].”
A Hybrid High-Order method for the incompressible Navier-Stokes equations based on Temam's device (2018, neutral): “We cite here, in particular, the two-dimensional Virtual Element Method (VEM) of [8]; see also the related work [7].”
Serendipity Virtual Elements for General Elliptic Equations in Three Dimensions (2018, background): “The use of such an approach introduces other potential advantages, such as exact satisfaction of linear constraints as in [12] or [3], and the possibility to build easily discrete spaces of high global regularity [2–3, 16].”