"The divergence-free finite element is a type of mixed finite element for solving computational fluid problems.
A divergence-free element is a combination of continuous piecewise polynomial vectors of degree k, approximating the velocity, and discontinuous piecewise polynomials of degree k minus one for the pressure, where the divergence of the discrete velocity space is exactly the discrete pressure space.
Nevertheless, most such spaces are not well-matched and result in unstable methods.
Scott and Vegelius discovered first such a finite element in 1984 that for all polynomial degree 4 or higher,  the element is stable on 2D triangular grids.
They posted explicitly the 3D version of this problem, which still remains open.
In this talk,  we present some divergence-free elements on composite grids, rectangular grids,  and tetrahedral grids.
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