Parameter-robust discretization and preconditioning of Biot's consolidation model
For researchers in poroelasticity, this work provides discretization and preconditioning techniques that maintain robustness across parameter variations, addressing a key challenge in practical applications.
The paper develops finite element discretizations and block diagonal preconditioners for Biot's consolidation model that are robust with respect to variations in parameters such as time-step size, bulk/shear moduli, and hydraulic conductivity, ensuring stability across a wide range of parameter values.
Biot's consolidation model in poroelasticity has a number of applications in science, medicine, and engineering. The model depends on various parameters, and in practical applications these parameters ranges over several orders of magnitude. A current challenge is to design discretization techniques and solution algorithms that are well behaved with respect to these variations. The purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete Biot systems. The approach taken here is to consider the stability of the problem in non-standard or weighted Hilbert spaces and employ the operator preconditioning approach. We derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement. The parameters of interest are small time-step sizes, large bulk and shear moduli, and small hydraulic conductivity.