10

Reading today about the theory of differential forms, I was left impressed how much it reminded me of second order Finite Volume Method (FVM).

I'm struggling to figure out is thinking this way just trivial or is there some deeper connection.

Well, differential forms serve to generalize some concepts deeply rooted in second order FVM, like flux of fluid trough a surface, and we are all about fluxes in FVM. Then integral theorem (of Stokes) is one of the central objects in theory of differential forms. It's proving involves an integration of differential forms on a manifold-where simplexes (triangles, tetrahedrons,etc.) appear. Manifold is actually tessellated in a same manner we represent a smooth shape over which fluid passes using straight edged cells.

These are just some of the similar things. The fact is that reading about differential forms made me not being able to stop thinking about FVM.

Does second order Finite Volume method actually represent Computational manifestation of Differential Forms theory?

Johntra Volta
  • 1,403
  • 9
  • 14
  • 5
    Your thoughts are in line with some work of E. Tonti, see his page on "Discrete Physics" and also try a search on "mimetic discretization". – Stefano M Feb 05 '13 at 23:22
  • 1
    I recall seeing something called "discrete differential forms" that maybe is related to this. I think its primary use is in computational geometry, but have seen some uses in simulation. A google will give you some ideas. – Reid.Atcheson Feb 06 '13 at 19:30
  • 1
    @Reid - This leads me, among others, to Desbrun's papers - the author I learned about earlier during the day - very interesting! – Johntra Volta Feb 06 '13 at 21:10

2 Answers2

1

One way to think about a differential $k$-form is "something integrable over a $k$-dimensional manifold". The most familiar example would be the volume form $dx$ in $\int_0^1 x^2 dx$, but also the $x^2$, which a $0$-form.

Stokes' Theorem generalizes many of the identities you're familiar with from vector calculus, such as the divergence theorem. These identities are applied to integral conservation laws to compute fluxes across boundaries in Finite Volume Methods so one should, as you suspect, be able to write everything in terms of differential forms.

Patrick Sanan
  • 849
  • 5
  • 16
1

Differential geometric techniques are used in formulations / understanding of finite-element (-volume) methods.

See here and here

Vijay Murthy
  • 176
  • 7