ArXiv (open access)
Key ideas
- We decompose a discrete exact sequence $(P\Lambda^\bullet, d)$ as
\(P \Lambda^k = \bar{P} \Lambda^k \oplus \mathring{P} \Lambda^k\)
for each $k$ such that each
\(\mathring{\pi} \bar{d} : \bar{P} \Lambda^k \to \mathring{P} \Lambda^{k+1}\)
is invertible.
- This can be done for the Whitney forms on simplicial grids by employing spanning trees.
Main findings
- The operator $p = (\mathring{\pi} \bar{d})^{-1} \mathring{\pi}$ is a Poincaré operator that satisfies the homotopy formula
\(dp + pd = I\)
- Each space decomposes as
\(P \Lambda^k = \bar{P} \Lambda^k \oplus d \bar{P} \Lambda^{k-1}\)
- This describes a basis for the function spaces in which the Hodge-Laplace problem translates to (at most) four 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.
