Series: Mathematics Colloquium

Date: Thursday, October 24, 2002

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Host: Jinchao Xu

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: Douglas Arnold, IMA, University of Minnesota

Title: 

From Exact Sequences to Colliding Black Holes: Differential Complexes
in Numerical Analysis

Abstract: 

Differential complexes such as the de Rham complex have recently come
to play an important role in the design and analysis of numerical
methods for partial differential equations.  The design of stable
discretizations of systems of partial differential equations often
hinges on capturing subtle aspects of the structure of the system in
the discretization.  In many cases the differential geometric
structure captured by a differential complex has been found to be an
essential element, and a discrete differential complex which is
appropriately related to the original complex is essential.  This new
geometric viewpoint provides a unifying understanding of a variety of
innovative numerical methods developed over recent decades, in
particular for the stable approximation of electromagnetic problems.
Very recently it has enabled the development of new algorithms for
elasticity problems with properties previously unattainable.  And it
seems likely to provide an important element for the solution of
numerical problems beyond current capabilities, such as the simulation
of gravitational wave emission from colliding black holes.