For more information about this meeting, contact Svetlana Katok, Anatole Katok.

Title: | First-Order System LL*(FOSLL*) for Maxwell's equations in 3D with edge singularities |

Seminar: | Department of Mathematics Colloquium |

Speaker: | Thomas Manteuffel, University of Colorado |

Abstract: |

Many important applications involve the solution of the eddy current
approximation to Maxwell's equations. For example, mathematical models
of magnetically confined plasma in tokomak geometry lead to
magnetohydrodynamic equations (MHD) in which Maxwell's equations play
a major role. Sandia's Z-pinch reactor provides another important
application.
In this talk we focus on a first-order system least-squares (FOSLS)
formulation of the eddy current approximation to Maxwell's equations.
The $L2$--norm version of FOSLS yields a least-squares functional
whose bilinear part is $H1$--elliptic. This means
that the minimization process amounts to solving a loosely coupled
system of scalar elliptic equations.
An unfortunate limitation of the $L2$--norm FOSLS approach is
that this product $H1$ equivalence generally requires sufficient
smoothness of the original problem. Inverse--norm FOSLS overcomes
this limitation, but at a substantial loss of real efficiency. The
FOSLL* approach described in this talk is a promising alternative
that is based on recasting the original problem as a minimization
principle involving the adjoint equations.
This talk provides a theoretical foundation for the FOSLL*
methodology and application to the eddy current form of Maxwell's equations.
It is shown that singularities due to discontinuous coefficients
are easily treated. However, singularities due to reentrant edges
require a further modification. A partially weighted norm is used
only on the slack equations. The solution retains optimal order accuracy
and the resulting linear systems are easily solved by multigrid methods.
Comparison is made to the curl/curl formulation, which requires Nedelec
finite elements, and the weighted regularization
approach, which requires a finite element space with a $C1$
subspace. The FOSLL$^*$ approach uses standard $H1$ conforming
finite element spaces, is shown to have equal or better
accuracy, obtained at a smaller cost. Numerical examples are presented
that support the theory. |

### Room Reservation Information

Room Number: | MB114 |

Date: | 11 / 08 / 2007 |

Time: | 04:00pm - 05:00pm |