Key ideas
- We decompose a Hilbert complex $(V^\bullet, d)$ as \(P^k = \hat{V}^k \oplus \breve{V}^k \oplus \mathring{V}^k\) for each $k$ such that each \(\breve{\pi} \hat{d} : \hat{V}^k \to \breve{V}^{k+1}\) is invertible and $\mathring{V}^k$ is isomorphic to the space of harmonic $k$-forms.
- This can be done for the Whitney forms on simplicial grids by employing spanning trees.
Main findings
-
The operator $p = (\breve{\pi} \hat{d})^{-1} \breve{\pi}$ is a Poincaré operator that satisfies the formula
\[dp + pd = I - \rho\]with $\rho=0$ on exact complexes.
- Each space decomposes as \(P^k = \hat{V}^k \oplus d \hat{V}^{k-1} \oplus \mathfrak{H}^k\)
- A basis is formed for the function spaces in which the Hodge-Laplace problem translates to (at most) seven symmetric positive definite systems. This leads to significant computational speed-up without loss of accuracy.
- Auxiliary space preconditioners for elliptic finite element problems are proposed based on the decomposition.