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This work presents an overview of the most relevant results obtained by the authors regarding the numerical solution of the Biot’s consolidation problem by preconditioning techniques. The emphasis here is on the design of parameter-robust preconditioners for the efficient solution of the algebraic system of equations resulting after proper discretization of such poroelastic problems. The classical two- and three-field formulations of the problem are considered, and block preconditioners are presented for some of the discretization schemes that have been proposed by the authors for these formulations. These discretizations have been proved to be well-posed with respect to the physical and discretization parameters, what provides a framework to develop preconditioners that are robust with respect to such parameters as well. In particular, we construct both norm-equivalent (block diagonal) and field-of-value-equivalent (block triangular) preconditioners, which are proved to be parameter-robust. The theoretical results on this parameter-robustness are demonstrated by considering typical benchmark problems in the literature for Biot’s model.

Discretization of flow in fractured porous media commonly lead to large systems of linear equations that require dedicated solvers. In this work, we develop an efficient linear solver and its practical implementation for mixed‐dimensional scalar elliptic problems. We design an effective preconditioner based on approximate block factorization and algebraic multigrid techniques. Numerical results on benchmarks with complex fracture structures demonstrate the effectiveness of the proposed linear solver and its robustness with respect to different physical and discretization parameters.