What is the current state of research on the existence of incomplete cholesky factorizations (in the context of preconditioning) for symmetric positive definite matrices?
I wonder in particular whether the question of existence is settled for finite element mass matrices.
Meijerink and van der Vorst showed that incomplete Cholesky dose not break down for $M$-matrices. As for finite element mass matrices, you will have to specify a basis to have any hope of making that claim.
Given any basis $\Phi = [\phi_0 | \phi_1 | \phi_2 | \phi_3]$ such that the mass matrix $A = \Phi^T \Phi$ is SPD, we can construct a new basis $\hat \Phi = \Phi B$ such that the new mass matrix
$$\hat\Phi^T \hat \Phi = \begin{pmatrix} 3 & -2 & 0 & 2 \\ -2 & 3 & -2 & 0 \\ 0 & -2 & 3 & -2 \\ 2 & 0 & -2 & 3 \end{pmatrix} =: K$$
is the Kershaw matrix, an SPD matrix for which incomplete Cholesky produces a negative pivot. The transformation $B$ satisfies $B^T A B = K$, and given the Cholesky factorizations $A = L_AL_A^T$ and $K = L_K L_K^T$, is readily computable as $B = L_A^{-T} L_K^T$.
I would suggest you the reading of Wang's thesis on the subject. http://repository.tamu.edu/bitstream/handle/1969.1/ETD-TAMU-3135/WANG-DISSERTATION.pdf?sequence=1
Using local element stiffness matrices modification it allows to produce a M-matrix that approximates correctly the original global stiffness matrix (after assembly). Hope this helps. Tom
