Parameter Robust Preconditioning by Congruence for Multiple-Network Poroelasticity

Abstract

The mechanical behavior of a poroelastic medium permeated by multiple interacting fluid networks can be described by a system of time-dependent partial differential equations known as the multiple-network poroelasticity (MPET) equations or multiporosity/multipermeability systems. These equations generalize Biot's equations, which describe the mechanics of the one network case. The efficient numerical solution of the MPET equations is challenging, in part due to the complexity of the system and in part due to the presence of interacting parameter regimes. In this paper, we present a new strategy for efficiently and robustly solving the MPET equations numerically. In particular, we discuss an approach to formulating finite element methods and associated preconditioners for the MPET equations based on simultaneous diagonalization of the element matrices. We demonstrate the technique for the multicompartment Darcy equations, with large exchange variability, and the MPET equations for a nearly incompressible medium with large exchange variability. The approach is based on designing transformations of variables that simultaneously diagonalize (by congruence) the equations' key operators and subsequently constructing parameter-robust block diagonal preconditioners for the transformed system. The proposed approach is supported by theoretical considerations as well as by numerical results.

Robust Preconditioners for Perturbed Saddle-Point Problems and Conservative Discretizations of Biot's Equations Utilizing Total Pressure

“…A main challenge for these equations is the construction of solvers that scale properly for nearly incompressible solids where the Lam\' e dilation modulus tends to infinity, as well as in the case of nearly incompressible fluids, for which the constrained specific storage coefficient approaches zero, or the nearly impermeable regime where the hydraulic conductivity is very small. These scenarios entail not only a complication at the practical and implementation level, but also a difficulty inherent to the functional setting of the abstract formulation (see, e.g., [31,18,43]). In more detail, for almost incompressible solids (\lambda \gg \mu ), the primal form of the elasticity equation, used in (1.1a), here scaled by \lambda , - \bfnabla \cdot \Bigl( 2 \mu \lambda \bfitvar (\bfitu ) + (\nabla \cdot \bfitu )I \Bigr) = \bfitb in \Omega , is known to suffer from locking when using standard elements such as Lagrange elements.…”

Poroelastic Benchmarking of a Finite Element Multiphysics Code

“…The iterative approach has traditionally been preferred due to the potentially high cost of solving the coupled nonlinear equations. However, scalable preconditioners for the fully-coupled poroelastic problem (Piersanti et al, 2021) and high performance implementations such as in the MOOSE Framework (Gaston et al, 2009;Andrs et al, n.d.), have demonstrated that a fully coupled approach can be both performant and flexible. Along with MOOSE itself, applications built off of MOOSE have taken advantage of the modular nature of the code to produce fully coupled finite element multiphysics.…”

Waterscales: Mathematical and computational foundations for modelling cerebral fluid flow

“…Preconditioning by congruence. When next turning to the efficient solution of the MPET equations via preconditioned iterative methods, an interesting puzzle appeared with a charming solution motif [27,28]. To illustrate, consider a coupled Darcy flow with exchange system like (9) written as…”

“…This observation gives a direct construction for P in terms of the eigenvectors of K −1 E. The transformed system is then block-diagonal and easily preconditioned with condition numbers that are uniform for a range of exchange and conductance parameters [28]. Interestingly, the construction extends to the MPET system [27].…”