FEEC was developed in the early 2000s by Douglas N. Arnold, Richard S. Falk and Ragnar Winther, [1] [2] [3] among others.
One starts with the recognition that the used differential operators are often part of complexes: successive application results in zero.
A related variational saddle-point formulation for mixed quantities is then generated.
One can then prove uniqueness and optimal convergence as function of mesh density.
FEEC is of immediate relevancy for diffusion, elasticity, electromagnetism, Stokes flow.