The Geometrical Basis of Numerical Stability. The accuracy of a numerical solution to a partial differential equation depends on the consistency and stability of the discretization method. While consistency is usually elementary to establish, stability of numerical methods can be subtle, and for some key PDE problems the development of stable methods is extremely challenging. After illustrating the situation through simple but surprising examples, we will describe a powerful new approach--the finite element exterior calculus--to the design and understanding of discretizations for a variety of elliptic PDE problems. This approach achieves stability by developing discretizations which are compatible with the geometrical and topological structures, such as de Rham cohomology and Hodge decompositions, which underlie well-posedness of the PDE problem being solved.
"I will introduce the computer math system known as SAGE (www.sagemath.org). I will show science, math, and engineering students how to use SAGE in their area of study. SAGE is an open source program that aims to provide a viable free and open source alternative to Magma, Maple, Mathematica and MATLAB. Started in 2005 by UW mathematics professor William Stein, SAGE has grown into a highly functional general purpose mathematics computing package. Best of all, there are several UW undergraduates involved in SAGE development. You could be one too!"