TY - JOUR
T1 - Finite Element Methods for the Navier-Stokes Equations by $H(div)$ Elements
JO - Journal of Computational Mathematics
VL - 3
SP - 410
EP - 436
PY - 2008
DA - 2008/06
SN - 26
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/10360.html
KW - Finite element methods, Navier-Stokes equations, CFD.
AB - <p style="text-align: justify;">We derived and analyzed a new numerical scheme for the Navier-Stokes equations by using $H(div)$ conforming finite elements. A great deal of effort was given to an establishment of some Sobolev-type inequalities for piecewise smooth functions. In particular, the newly derived Sobolev inequalities were employed to provide a mathematical theory for the $H(div)$ finite element scheme. For example, it was proved that the new finite element scheme has solutions which admit a certain boundedness in terms of the input data. A solution uniqueness was also possible when the input data satisfies a certain smallness condition. Optimal-order error estimates for the corresponding finite element solutions were established in various Sobolev norms. The finite element solutions from the new scheme feature a full satisfaction of the continuity equation which is highly demanded in scientific computing.</p>