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# Discrete exterior calculus

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It's been known for a long time that smooth differential forms form a cochain complex (De Rham, 1931). However the opposite direction of this correspondence has been neglected until recently: co-chains are differential forms. Neglected in what sense?

If a manifold is a cell complex (cubical complex in the simplest case), co-chains are thought of as discrete differential forms while the co-boundary operator is the exterior derivative. Turns out this is a good way to discretize things, such as PDEs. Better yet, the result is a theory that parallels the smooth calculus. It's discrete calculus rather than discretized calculus.

This is an illustration of how calculus can be discretized:

Here, $d$ stands for the exterior derivative and $R$ is the de Rham map. Course:

Project:

It is educational to see the heat PDE next to the heat transfer model written with differential forms: $$\frac{\partial u}{\partial t}=-\frac{\partial ^2u}{\partial x^2}$$ $$d_t T=-d_x(d_xT)^*,$$ where $d$ is the exterior derivative and * stands for the Hodge duality.

Sources:

Inexplicably, cubical complexes are mostly ignored in the discrete exterior calculus literature. The reason must be the inertia from the decades of having to deal with meshes. Of course, discretization on a square grid is easy...