I have several questions about Discrete Exterior Calculus (DEC) in numerical methods for solving partial differential equations in physics:

(Discrete Exterious Calculus is a newly developed subject mainly used in numerical computing, one reference is, for example, Hirani's PhD thesis: Discrete Exterior Calculus.)

Has any kind of convergence property been proved? I mean, under what conditions will the numerical solution of a DEC scheme converge to the actual solution of the PDE? I checked many papers in the literature and didn't see any material concerning the convergence property, because I am studying an engineering problem on a computer and if we can't guarantee it will converge then the precision will be a problem.

What is the current status of using DEC to numerically solve equations or create simulations in fluid mechnanics, elasticity and electromagnetism, respectively? Should anyone give me some relevant papers, I have found some but just don't know if I missed anything.