For more information about this meeting, contact Jinchao Xu.

Title: | Divergence-free finite elements on tetrahedral grids * |

Seminar: | Computational and Applied Mathematics Colloquium |

Speaker: | Shangyou Scott Zhang (U. Del) |

Abstract: |

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. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 10 / 05 / 2007 |

Time: | 03:35pm - 04:25pm |