Discrete Exterior Calculus

Tuesday, June 18, 2019 - 10:15am - 11:15am
Keller 3-180
Peter Schroeder (California Institute of Technology)
When switching from the smooth world to the discrete world in the computer we need fundamental building blocks to take smooth PDEs (for example) into the discrete setting. One way to do this is to ask, what is the discrete analog of the differential? And what properties should it satisfy? Is it enough if a divided difference converges in the limit, a limit we never actually reach in any given computation? Using ideas from exterior calculus, it turns out, there are simple and satisfying answers to how we can write differential operators on manifolds such that smooth properties are replicated exactly. For example we will find that Stokes' theorem can hold exactly in the discrete setting. This is fundamental consequences for the way we can perform numerical computations and be assured that, for example, invariants really are invariant.