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.