Enhanced Relaxed Physical Factorization preconditioner for coupled poromechanics
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In this work, we focus on the relaxed physical factorization (RPF) preconditioner for the block linear systems arising from the three-field formulation (displacement/velocity/pressure) of coupled poromechanics. Inspired by the relaxed dimensional factorization developed for the Navier-Stokes equations by Benzi et al. [J. Comput. Phys., 230 (2011), pp. 6185–6202; Comput. Methods Appl. Mech. Engrg., 300 (2016), pp. 129–145], the RPF preconditioner was recently advanced by Frigo et al. [SIAM J. Sci. Comp., 41 (2019), pp. B694–B720] and relies on: (i) combining proper physics-based splittings of the block matrix by field type, and (ii) introducing an optimal relaxation parameter α. However, a possible drawback arises from the need of inverting blocks in the form Ĉ = ( C + β F F^T) for large values of the real coefficient β, where C is a regular square matrix and FF^T is a rank-deficient term. In this work, we propose a family of algebraic techniques to stabilize the inexact solve with Ĉ, which can also prove useful in other problems where such an issue might arise, such as augmented Lagrangian preconditioning strategies for Navier-Stokes or incompressible elasticity. First, we introduce an iterative scheme obtained by a natural splitting of matrix Ĉ. Second, we develop a technique based on the use of a proper projection operator onto the range of F. 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.
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