I am doing research on the structure in the Schur complements and find an interesting phenomenon:
Suppose that A is from 5--pt laplacian. If I use nested dissection ordering and multifrontal method to compute the LU factorization and then check the last schur complement block, it has low-rank for the off-diagonal blocks.
But, when I use the same method to factorize $A - \lambda I$, where $\lambda$ is some positive value near the eigenvalues of A, then the last schur complement does not have the low-rank property.
I do not know whether the indefinite will change the structure in the schur complement or not. Can anyone provide some reference for this topic?