The incompressible constraint in fluid dynamics and
the nearly incompressible condition in solid mechanics post a major
difficulty in the numerical computation, especially in the finite
element method. In 1983, Scott and Vogelius showed that the
$P_k$-$P_{k-1}$ element (approximating the velocity by continuous
piecewise-polynomials of degree $k$ and approximating the pressure by
discontinuous piecewise-polynomials of degree $k-1$) is stable and
consequently of the optimal order on 2D triangular grids for any
$k\ge 4$, provided the grids have no nearly-singular vertex.
For such a combination of mixed elements, the finite element
velocity is divergence free point wise, truly incompressible.
The 3D version of $P_k$-$P_{k-1}$ problem is still open.
We give some partial answers and present some newly discovered
divergence-free elements in this talk.