Finite Element Methods for the Navier-Stokes Equations by $H(div)$ Elements (Journal of Computational Mathematics, 2018)
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.
Keywords: Finite element methods; Navier-Stokes equations; CFD.