Symmetric saddle-point system, Schur complement, and block preconditioners

Let $n,m\ge 1$, $V=M_{n\times 1}(\mathbb{R})$, and $Q=M_{m\times 1}(\mathbb{R})$. Let $A\in M_n(\mathbb{R})$ be symmetric positive definite, let $B\in M_{m\times n}(\mathbb{R})$ have full row rank (meaning $B^{T}q=0$ implies $q=0$), and let $M_Q\in M_m(\mathbb{R})$ be symmetric positive definite. The saddle system and its matrix are $K=\left(\begin{array}{cc}A& B^T\\ B& 0\end{array}\right),Au+B^{T}p=f,Bu=g\mathrm{.}$ Define the positive pressure Schur matrix $S:=B{A}^{-1}{B}^T\mathrm{.}$ In the block-LU convention of Block LU factorisation and the associated Schur complement, the trailing Schur complement of $A$ in $K$ is $-S$. The exact block solve first solves $A{u}_0=f$, then $Sp=B{u}_0-g$, and finally $Au=f-B^{T}p$; the theorem Exact Schur solve for a full-row-rank symmetric saddle system ↗ verifies this procedure and well-definedness.

For a scaling $\gamma >0$, the pressure-mass block-diagonal preconditioner is $P_{\gamma }=\mathrm{diag}(A,\gamma M_Q)\mathrm{.}$ It is symmetric positive definite. If $P_{\gamma }=C{C}^T$ is its Cholesky factorization with $C$ lower triangular and positive on the diagonal, the symmetric preconditioned system is $C^{-1}K{C}^{-T}y=C^{-1}r$, with the usual change of unknown. This is the symmetric-preconditioning convention of Left preconditioning, right preconditioning, and symmetric preconditioning of a linear system. No spectral bound, Krylov iteration count, complexity estimate, or generic speed improvement is part of this definition.