Divergence free virtual elements for the stokes problem on polygonal meshes (original) (raw)

ESAIM: M2AN 51 (2017) 509-535

1 Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via Roberto Cozzi 55, 20125 Milano, Italy.
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2 Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy.
3 IMATI-CNR, Via Ferrata 1, 27100 Pavia, Italy.
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4 Dipartimento di Matematica, Università degli Studi di Bari, Via Edoardo Orabona 4, 70125 Bari, Italy.
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Received: 6 October 2015
Revised: 3 March 2016
Accepted: 21 April 2016

Abstract

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 fewer 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.

Mathematics Subject Classification: 65N30 / 65N12 / 65N15 / 76D07

Key words: Virtual element method / polygonal meshes / Stokes problem / divergence free approximation