Divergence free virtual elements for the stokes problem on polygonal meshes

Beirão da Veiga, Lourenco; Lovadina, Carlo; Vacca, Giuseppe

doi:10.1051/m2an/2016032

Beirão da Veiga, Lourenco ¹ ; Lovadina, Carlo ^2,³ ; Vacca, Giuseppe ⁴

¹ Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via Roberto Cozzi 55, 20125 Milano, Italy.
² Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy.
³ IMATI-CNR, Via Ferrata 1, 27100 Pavia, Italy.
⁴ Dipartimento di Matematica, Università degli Studi di Bari, Via Edoardo Orabona 4, 70125 Bari, Italy.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 509-535.

Résumé

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.

MR Zbl | 4 citations dans Numdam

DOI : 10.1051/m2an/2016032

Classification : 65N30, 65N12, 65N15, 76D07
Mots-clés : Virtual element method, polygonal meshes, Stokes problem, divergence free approximation

Affiliations des auteurs :

Beirão da Veiga, Lourenco ¹ ; Lovadina, Carlo ^{2, 3} ; Vacca, Giuseppe ⁴

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Beirão da Veiga, Lourenco; Lovadina, Carlo; Vacca, Giuseppe. Divergence free virtual elements for the stokes problem on polygonal meshes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 509-535. doi : 10.1051/m2an/2016032. http://archive.numdam.org/articles/10.1051/m2an/2016032/

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