A Quick and Dirty Introduction to Discrete Exterior Calculus – Geometry Processing and Applications WS19

So far we’ve been exploring exterior calculus purely in the smooth setting. Unfortunately this theory was developed by some old-timers who did not know anything about computers, hence it cannot be used directly by machines that store only a finite amount of information. For instance, if we have a smooth vector field or a smooth 1-form we can’t possibly store the direction of every little arrow” at each point — there are far too many of them! Instead, we need to keep track of a discrete (or really, finite) number of pieces of information that capture the essential behavior of the objects we’re working with; we call this scheme discrete exterior calculus (or DEC for short). The big secret about DEC is that it’s literally nothing more than the good-old fashioned (continuous) exterior calculus we’ve been learning about, except that we integrate differential forms over elements of our mesh.

So far we’ve been exploring exterior calculus purely in the smooth setting. Unfortunately this theory was developed by some old-timers who did not know anything about computers, hence it cannot be used directly by machines that store only a finite amount of information. For instance, if we have a smooth vector field or a smooth 1-form we can’t possibly store the direction of every little arrow” at each point — there are far too many of them! Instead, we need to keep track of a discrete (or really, finite) number of pieces of information that capture the essential behavior of the objects we’re