For more information about this meeting, contact Jinchao Xu, Xiantao Li, Yuxi Zheng, Kris Jenssen, Hope Shaffer.

Title: | Finite element methods for singular perturbed PDEs |

Seminar: | Computational and Applied Mathematics Colloquium |

Speaker: | Michael Neilan, University of Pittsburgh |

Abstract: |

In this talk, we discuss various theoretical and practical issues of solving two different singular perturbation problems. The first problem is a fourth order biharmonic problem that degeneratesÂ to Poisson's equation as the perturbation parameter tends to zero. Therefore convergent finite element methods must be convergent for both the biharmonic problem as well as Poisson's equation. We construct such methods by enriching (locally) Lagrange elements with volume and edge bubbles. The second problem that we discuss is the Brinkman problem, a linear StokesÂ equation when the perturbation parameter is large, but degenerates to a mixed formulation of Poisson's equation when the parameter is small. As such, numerical methods that work well for Stokes behave poorly when the perturbation parameter is small, where as methods that are designed for the mixed formulation of Poisson's equation do not converge when the parameter is large. We discuss how to augment H(div) elements withÂ divergence free bubble functions to construct a family of stable
and convergent methods for the Brinkman problem. Furthermore, using exact sequences of function spaces (discrete de Rham complexes) we show how these two seemingly unrelated problems are closely connected. Finally, we show how to extend this methodology to obtainÂ conforming and divergence free elements
for the Stokes/Brinkman problem on general triangular meshes. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 11 / 11 / 2011 |

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