Guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the Biot problem

Kumar, Kundan; Kyas, Svetlana; Nordbotten, Jan Martin; Repin, Sergey

doi:10.1016/j.camwa.2020.05.005

Cited by 8 publications

(4 citation statements)

References 55 publications

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“…Moreover, we refer the reader to [31] for iterative algorithms applied to an extension of Biot model that includes multiple pressure fields with biological applications. Furthermore, a priori error estimates and a posteriori error estimates of these iterative algorithms have been studied and developed in [1,29,37].…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of single rate and multirate fixed stress split iterative coupling schemes in heterogeneous poroelastic media

Almani

Kumar

Wheeler

2023

Numerical Methods Partial

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Recently, the accurate modeling of flow-structure interactions has gained more attention and importance for both petroleum and environmental engineering applications. Of particular interest is the coupling between subsurface flow and reservoir geomechanics. Different single rate and multirate iterative and explicit coupling schemes have been proposed and analyzed in the past. In addition, Banach fixed point contraction results were obtained for iterative coupling schemes, and conditionally stable results were obtained for explicit coupling schemes. In this work, we will consider the mathematical analysis of the single rate and multirate fixed stress split iterative coupling schemes for spatially heterogeneous poroelastic media. We will re-establish the contractivity for both schemes in the localized case, and we will show that heterogeneities come at the expense of imposing more restricted conditions on the number of fine flow time steps that can be taken within one coarse mechanics time step in the multirate case. Our mathematical analysis is supplemented by numerical simulations validating our derived upper bounds. To the best of our knowledge, this is the first rigorous mathematical analysis of the multirate fixed-stress split iterative coupling scheme in heterogeneous poroelastic media.

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Section: Introductionmentioning

confidence: 99%

Convergence analysis of single rate and multirate fixed stress split iterative coupling schemes in heterogeneous poroelastic media

Almani

Kumar

Wheeler

2023

Numerical Methods Partial

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“…In Riedlbeck et al [2017], the authors consider the standard two-field formulation of Biot's equation in two spatial dimensions, develop an a-posteriori error analysis based on H(div) reconstructions of the flux and effective stress and apply the resulting estimators to construct a time-space adaptive algorithm. Kuman et al [2021] used a-posteriori estimators have to provide error estimates for the popular fixed-stress iterative solution scheme applied to the two-field formulation. Formulations with additional fields have also been considered for Biot's equations.…”

Section: Introductionmentioning

confidence: 99%

A-posteriori error estimation and adaptivity for multiple-network poroelasticity

Emilie¹,

Rognes²,

Thompson³

2021

Preprint

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The multiple-network poroelasticity (MPET) equations describe deformation and pressures in an elastic medium permeated by interacting fluid networks. In this paper, we (i) place these equations in the theoretical context of coupled elliptic-parabolic problems, (ii) use this context to derive residual-based a-posteriori error estimates and indicators for fully discrete MPET solutions and (iii) evaluate the performance of these error estimators in adaptive algorithms for a set of test cases: ranging from synthetic scenarios to physiologically realistic simulations of brain mechanics.

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“…This special issue contains several contributions related to the Biot's model. In [3], the system is decoupled by employing the fixed-stress splitting scheme, which leads to a semi-discrete system that is solved iteratively. The error bounds are derived by combining a posteriori estimates for contractive mappings with functional-type error control for elliptic partial differential equations.…”

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confidence: 99%

Robust and reliable finite element methods in poromechanics

Bertrand¹,

Ern²,

Radu³

2021

Computers & Mathematics with Applications

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Guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the Biot problem

Cited by 8 publications

References 55 publications

Convergence analysis of single rate and multirate fixed stress split iterative coupling schemes in heterogeneous poroelastic media

Convergence analysis of single rate and multirate fixed stress split iterative coupling schemes in heterogeneous poroelastic media

A-posteriori error estimation and adaptivity for multiple-network poroelasticity

Robust and reliable finite element methods in poromechanics

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