# Meeting Details

Title: Divergence-free finite elements on tetrahedral grids * Computational and Applied Mathematics Colloquium Shangyou Scott Zhang (U. Del) In 1983, Scott and Vogelius showed that the $P_k$-$P_{k-1}$ element (approximating the velocity by continuous $P_k$ piecewise-polynomials and approximating the pressure by discontinuous $P_{k-1}$ piecewise-polynomials) is stable and consequently of the optimal order on 2D triangular grids for any $k\ge 4$, provided the grids have no singular or nearly-singular vertex. For such a combination of mixed elements, the discrete velocity is divergence free pointwise. We call all such finite elements divergence-free elements. For $k\le 3$, Scott and Vogelius showed that the divergence free element would not be stable, and would not produce approximating solutions on general triangular grids. What is this magic number $k$ in 3D? Scott and Vogelius posted this question explicitly in their paper after discovering that $k=4$ in 2D, which lacks an answer for more than two decades. We will answer partially this question in the talk. We will also review some low-order 2D and 3D divergence-free elements, on special grids.