Partial differential equations (PDE) are among the most
useful mathematical modeling tools, and numerical discretization of
PDE--approximating them by problems which can be solved on computers--is
one of the most important and widely used approaches to simulating
the physical world.  A vastly developed technology is built on such
discretizations.  Nonetheless, fundamental challenges remain in the
design and understanding of effective methods of discretization for
certain important classes of PDE problems.

The accuracy of a simulation depends on the consistency and stability of
the discretization method used.  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.