Block Triangular Preconditioners for Double Saddle-Point Linear Systems Arising in the Mixed Form of Poroelasticity Equations

In this paper, we study a class of inexact block triangular preconditioners for double saddle-point symmetric linear systems arising from the mixed finite element and mixed hybrid finite element discretization of Biot's poroelasticity equations. We extend the theoretical results presented in [Balani et al., NLAA 2024] for general double saddle-point problems with nonzero diagonal blocks, by developing a novel spectral analysis of the preconditioned matrix. This shows that the complex eigenvalues lie in a circle of center (1,0) and radius smaller than 1. In contrast, the real eigenvalues are described in terms of the roots of a third-degree polynomial with real coefficients. Numerical results are reported to show the quality of the theoretical bounds, which generally turn out to be quite tight, and illustrate the efficiency of the proposed preconditioners in the acceleration of GMRES, especially in comparison with similar block diagonal preconditioning strategies along with the MINRES iteration.