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.
Accepté le :
DOI : 10.1051/m2an/2016032
Mots clés : Virtual element method, polygonal meshes, Stokes problem, divergence free approximation
@article{M2AN_2017__51_2_509_0, author = {da Veiga, Lourenco Beir\~ao and Lovadina, Carlo and Vacca, Giuseppe}, title = {Divergence free virtual elements for the stokes problem on polygonal meshes}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {509--535}, publisher = {EDP-Sciences}, volume = {51}, number = {2}, year = {2017}, doi = {10.1051/m2an/2016032}, mrnumber = {3626409}, zbl = {1398.76094}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016032/} }
TY - JOUR AU - da Veiga, Lourenco Beirão AU - Lovadina, Carlo AU - Vacca, Giuseppe TI - Divergence free virtual elements for the stokes problem on polygonal meshes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 509 EP - 535 VL - 51 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016032/ DO - 10.1051/m2an/2016032 LA - en ID - M2AN_2017__51_2_509_0 ER -
%0 Journal Article %A da Veiga, Lourenco Beirão %A Lovadina, Carlo %A Vacca, Giuseppe %T Divergence free virtual elements for the stokes problem on polygonal meshes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 509-535 %V 51 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016032/ %R 10.1051/m2an/2016032 %G en %F M2AN_2017__51_2_509_0
da Veiga, Lourenco Beirão; 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://www.numdam.org/articles/10.1051/m2an/2016032/
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