  The relaxed physical factorization (RPF) preconditioner is a recent algorithm
allowing for the efficient and robust solution to the block linear systems
arising from the three-field displacement-velocity-pressure formulation of
coupled poromechanics. For its application, however, it is necessary to invert
blocks with the algebraic form $\hat{C} = ( C + \beta F F^T)$, where $C$ is a
symmetric positive definite matrix, $FF^T$ a rank-deficient term, and $\beta$ a
real non-negative coefficient. The inversion of $\hat{C}$, performed in an
inexact way, can become unstable for large values of $\beta$, as it usually
occurs at some stages of a full poromechanical simulation. In this work, we
propose a family of algebraic techniques to stabilize the inexact solve with
$\hat{C}$. This strategy can prove useful in other problems as well where such
an issue might arise, such as augmented Lagrangian preconditioning techniques
for Navier-Stokes or incompressible elasticity. First, we introduce an
iterative scheme obtained by a natural splitting of matrix $\hat{C}$. Second,
we develop a technique based on the use of a proper projection operator
annihilating the near-kernel modes of $\hat{C}$. Both approaches give rise to a
novel class of preconditioners denoted as Enhanced RPF (ERPF). Effectiveness
and robustness of the proposed algorithms are demonstrated in both theoretical
benchmarks and real-world large-size applications, outperforming the native RPF
preconditioner.
