Parameter Robust Preconditioning for Multi-Compartmental Darcy Equations

Piersanti, Eleonora; Rognes, Marie E.; Mardal, Kent‐André

doi:10.1007/978-3-030-55874-1_69

Cited by 2 publications

(11 citation statements)

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“…A naive block diagonal preconditioner B MPT can be constructed by taking the inverse of the diagonal blocks of A MPT . However, as we demonstrated in [20], the resulting preconditioner is not robust with respect to variations in the conductivity and exchange parameters. In fact, the condition numbers increased linearly with the ratio between the exchange and conductivity coefficients.…”

Section: Notationmentioning

confidence: 84%

“…We begin by summarizing our novel approach to variational formulations and associated preconditioning of the MPT equations. This approach was introduced in [20]. The core idea is to reformulate the MPT (and MPET) equations using a change of pressure variables p. In particular, we aim to find a transformation of the variables p → p such that the transformed system of pressure equations decouple.…”

Section: Notationmentioning

confidence: 99%

“…The core idea is to reformulate the MPT (and MPET) equations using a change of pressure variables p. In particular, we aim to find a transformation of the variables p → p such that the transformed system of pressure equations decouple. Here, we briefly illustrate the core idea, formulation of the MPT equations and resulting preconditioner, and refer to [20] for more details. This approach is then extended to the MPET equations in Section 4.…”

Section: Notationmentioning

confidence: 99%

“…As in [20], we can immediately identify the following preconditioner for the transformed system (3.5):…”

Section: Preconditioning the Transformed Mpt Systemmentioning

confidence: 99%

“…In this paper, we present a new approach for preconditioning linear systems of equations resulting from a conforming finite element discretization of the total pressure variational formulation of the MPET equations. The key idea, as introduced for the MPT equations in [20], to design a parameter-dependent transformation of the pressure variables p = (p 1 , . .…”

mentioning

confidence: 99%

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Parameter robust preconditioning by congruence for multiple-network poroelasticity

Piersanti¹,

Lee²,

Thompson³

et al. 2020

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The mechanical behaviour 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 multi-porosity/multi-permeability 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 introduce a new approach to formulating finite element methods and associated preconditioners for the MPET equations. 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. Our methodology is supported by theoretical considerations as well as by numerical results.

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Section: Notationmentioning

confidence: 84%

Section: Notationmentioning

confidence: 99%

Section: Notationmentioning

confidence: 99%

“…As in [20], we can immediately identify the following preconditioner for the transformed system (3.5):…”

Section: Preconditioning the Transformed Mpt Systemmentioning

confidence: 99%

mentioning

confidence: 99%