A relaxed physical factorization preconditioner for mixed finite element coupled poromechanics

In this paper, we introduce a relaxed physical factorization (RPF) preconditioner for the efficient iterative solution of the linearized algebraic system arising from the mixed finite element discretization of coupled poromechanics equations. The preconditioner is obtained by using a proper factorization of the 3 x 3 block matrix and setting a relaxation parameter alpha. The preconditioner is inspired by the relaxed dimensional factorization introduced by Benzi et al. [J. Comput. Phys., 230 (2011), pp. 6185-6202; Comput. Methods Appl. Mech. Engrg., 300 (2016), pp. 129-145]. A stable algorithm is advanced to compute the optimal value of alpha, along with a lower bound to control the possible ill-conditioning of the alpha-dependent inner blocks. Numerical experiments in both theoretical benchmarks and real-world applications are presented and discussed to investigate the RPF properties, performance, and robustness.