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:10.48550/arxiv.1808.08036

Cited by 4 publications

(4 citation statements)

References 83 publications

(129 reference statements)

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“…As for the undrained split, we establish convergence based on an abstract convergence result for alternating minimization. After all, the following convergence result is consistent with results based on previous problem-specific a priori error analyses [37,41,72] and an a posteriori error analysis based on Ostrowski-type estimates for contraction mappings [80].…”

Section: Derivation and Analysis Of The Fixed-stress Split As Alterna...supporting

confidence: 89%

The gradient flow structures of thermo-poro-visco-elastic processes in porous media

Both,

Kumar,

Nordbotten

et al. 2019

Preprint

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In this paper, the inherent gradient flow structures of thermo-poro-visco-elastic processes in porous media are examined for the first time. In the first part, a modelling framework is introduced aiming for describing such processes as generalized gradient flows requiring choices of physical states, corresponding energies, dissipation potentials and external work rates. It is demonstrated that various existing models can be in fact written within this framework. Ultimately, the particular structure allows for a unified well-posedness analysis performed for different classes of linear and non-linear models. In the second part, the gradient flow structures are utilized for constructing efficient discrete approximation schemes for thermo-poro-visco-elasticity -in particular robust, physical splitting schemes. Applying alternating minimization to naturally arising minimization formulations of (semi-)discrete models is proposed. For such, the energy decrease per iteration is quantified by applying abstract convergence theory only utilizing convexity and Lipschitz continuity properties of the problem -a fairly simple but flexible machinery. By this approach, e.g., the widely used undrained and fixed-stress splits for the linear Biot equations are derived and analyzed. By application of the framework to more advanced models, novel splitting schemes with guaranteed theoretical convergence rates are naturally derived. Moreover, based on the minimization character of the (semi-)discrete equations, relaxation of splitting schemes by line search is proposed; numerical results show a potentially great impact on the acceleration of splitting schemes for both linear and nonlinear problems.

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Section: Derivation and Analysis Of The Fixed-stress Split As Alterna...supporting

confidence: 89%

The gradient flow structures of thermo-poro-visco-elastic processes in porous media

Both,

Kumar,

Nordbotten

et al. 2019

Preprint

Self Cite

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show abstract

“…We constitute their adaptive counterpart upon the distinction of the different error components arising in the standard fixed stress algorithm, namely the spatial discretization, the temporal discretization, and the fixed stress iteration components. To arrive to this aim, we take ideas from [8,22,25,29], for general a posteriori error techniques taking into account inexact iterative solvers, but most closely form [1], where a domain decomposition problem is solved via space-time iterative methods. Particularly, we will rely on [3,Theorem 6.2] where an energy-type-norm differences between the exact and the approximate pressure and displacement is shown to be bounded by the dual norm of the residuals.…”

Section: Domentioning

confidence: 99%

Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems

Ahmed

Nordbotten

Radu

2020

Journal of Computational and Applied Mathematics

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In this paper we develop adaptive iterative coupling schemes for the Biot system modeling coupled poromechanics problems. We particularly consider the space-time formulation of the fixed-stress iterative scheme, in which we first solve the problem of flow over the whole space-time interval, then exploiting the space-time information for solving the mechanics. Two common discretizations of this algorithm are then introduced based on two coupled mixed finite element methods in-space and the backward Euler scheme in-time. Therefrom, adaptive fixed-stress algorithms are build on conforming reconstructions of the pressure and displacement together with equilibrated flux and stresses reconstructions. These ingredients are used to derive a posteriori error estimates for the fixed-stress algorithms, distinguishing the different error components, namely the spatial discretization, the temporal discretization, and the fixed-stress iteration components. Precisely, at the iteration k ≥ 1 of the adaptive algorithm, we prove that our estimate gives a guaranteed and fully computable upper bound on the energy-type error measuring the difference between the exact and approximate pressure and displacement. These error components are efficiently used to design adaptive asynchronous time-stepping and adaptive stopping criteria for the fixed-stress algorithms. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive iterative coupling algorithms.

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“…Regarding the mathematical theory of poroelasticity, in particular Biot's quasi-static, linear consolidation model has been well-studied. Well-posedness including the existence, uniqueness, and regularity of solutions, has been established [8][9][10]; recent advances in the numerical analysis include, e.g., stable finite discretizations [11][12][13][14][15][16][17][18], efficient numerical iterative solvers [19][20][21][22][23][24], and a posteriori error estimates [25][26][27]. Lately, linear and non-linear extensions have become of increased interest.…”

Section: Introductionmentioning

confidence: 99%

Global existence of a weak solution to unsaturated poroelasticity

Both¹,

Pop²,

Yotov³

2019

Preprint

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In this paper, we consider unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in unsaturated porous media, modelled by a non-linear extension of Biot's quasistatic consolidation model. The coupled, elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in deformable porous media obtained under similar assumptions as usually considered for Richards' equation. In this work, the existence of a weak solution is established using regularization techniques, the Galerkin method, and compactness arguments. The final result holds under non-degeneracy conditions and natural continuity properties for the non-linearities. The assumptions are demonstrated to be reasonable in view of geotechnical applications.

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

Cited by 4 publications

References 83 publications

The gradient flow structures of thermo-poro-visco-elastic processes in porous media

The gradient flow structures of thermo-poro-visco-elastic processes in porous media

Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems

Global existence of a weak solution to unsaturated poroelasticity

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