Geodesic
Divergence free virtual elements for the stokes problem on polygonal meshes
Beirão da Veiga, Lourenco1  ; Lovadina, Carlo2, 3 ; Vacca, Giuseppe4
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

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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.

DOI : 10.1051/m2an/2016032
Classification : 65N30, 65N12, 65N15, 76D07
Keywords: Virtual element method, polygonal meshes, Stokes problem, divergence free approximation

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. 
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     title = {Divergence free virtual elements for the stokes problem on polygonal meshes},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {509--535},
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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

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