Higher-order iterative decoupling for poroelasticity

Abdullah Mujahid; Benjamin Unger; Robert Altmann

arxiv: 2311.14400 · v2 · pith:C5TQ6WBQnew · submitted 2023-11-24 · 🧮 math.NA · cs.NA

Higher-order iterative decoupling for poroelasticity

Robert Altmann , Abdullah Mujahid , Benjamin Unger This is my paper

Pith reviewed 2026-05-25 09:03 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords poroelasticityiterative decouplingbackward differentiation formulaefixed-point iterationerror balancinghigher-order convergencenumerical analysis

0 comments

The pith

Iterative decoupling for poroelasticity converges at higher orders when time step size and contraction parameters are balanced explicitly.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces backward differentiation formula time discretizations up to order five for the iterative decoupling of poroelasticity and related elliptic-parabolic problems. Its analysis merges fixed-point iteration techniques with standard BDF convergence theory. The central result is that overall convergence depends on the interplay between time step size and the contraction parameters of the iteration, with this dependence quantified explicitly so that separate error contributions can be balanced. A reader would care because the result supplies concrete guidance for choosing discretization and iteration parameters to reach a target accuracy efficiently. Several numerical experiments, including a three-dimensional biomechanics example, confirm the predicted behavior.

Core claim

For the iterative decoupling of elliptic-parabolic problems such as poroelasticity, time discretization schemes up to order five based on the backward differentiation formulae are introduced. The analysis combines techniques known from fixed-point iterations with the convergence analysis of the temporal discretization. As main result, the convergence depends on the interplay between the time step size and the parameters for the contraction of the iterative scheme. Moreover, this connection is quantified explicitly, which allows for balancing the single error components.

What carries the argument

Fixed-point iteration for decoupling combined with BDF time discretizations up to order five.

If this is right

Convergence rates are controlled by the minimum of the BDF order and the term arising from the contraction rate.
Error contributions from time discretization and from the iteration can be matched by appropriate parameter selection.
The same explicit balancing applies to three-dimensional problems as demonstrated in the biomechanics example.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The balancing relation could be used to design adaptive algorithms that adjust iteration count or time step on the fly.
Similar explicit quantification may be possible for other iterative decoupling strategies beyond fixed-point methods.
The approach might extend directly to nonlinear versions of the poroelasticity equations.

Load-bearing premise

The fixed-point iteration for decoupling must possess a contraction property whose rate can be bounded independently of the time step in a way that permits explicit quantification.

What would settle it

A numerical test in which the total observed error fails to follow the predicted dependence on time step size and contraction parameter when both are varied systematically.

read the original abstract

For the iterative decoupling of elliptic-parabolic problems such as poroelasticity, we introduce time discretization schemes up to order $5$ based on the backward differentiation formulae. Its analysis combines techniques known from fixed-point iterations with the convergence analysis of the temporal discretization. As main result, we show that the convergence depends on the interplay between the time step size and the parameters for the contraction of the iterative scheme. Moreover, this connection is quantified explicitly, which allows for balancing the single error components. Several numerical experiments illustrate and validate the theoretical results, including a three-dimensional example from biomechanics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends iterative decoupling to BDF orders 5 with explicit dt-contraction interplay for poroelasticity, but the key bound on contraction rate independent of time step looks fragile.

read the letter

The main takeaway is that the authors push standard fixed-point decoupling for elliptic-parabolic systems like poroelasticity up to BDF5 and give an explicit relation that lets you balance iteration error against truncation error. That quantified interplay is the concrete new piece; prior work stopped at lower orders without this level of explicit control. They combine known fixed-point arguments with BDF stability in a straightforward way and back it with numerical tests, including a 3D biomechanics case that shows the schemes behave as expected when the parameters are tuned right. That practical validation is useful and gives the result some weight beyond the analysis. The soft spot is the assumption that the contraction constant of the decoupling iteration can be bounded independently of the time step. In the usual weak form the pressure-displacement coupling produces Lipschitz factors that scale like 1/Δt when the elliptic part is solved at fixed pressure; unless the proof introduces a special weighting or stabilization that exactly cancels this, the claimed explicit balance will break for small steps and the higher-order schemes will not reach their design accuracy under a fixed iteration budget. The abstract presents the result as following from standard techniques, so the full derivation needs to be checked on this point. The paper is aimed at people already working on splitting methods for poroelasticity or similar coupled mechanics problems. A reader looking for higher-order time schemes with tunable error balance will find something usable here. It is worth sending to referees because the contribution is concrete and the experiments are there, even if the contraction analysis requires careful scrutiny.

Referee Report

1 major / 0 minor

Summary. The paper introduces BDF-based time discretizations of order up to 5 for the iterative decoupling of elliptic-parabolic systems such as poroelasticity. Its analysis merges standard fixed-point iteration techniques with BDF convergence theory to derive an explicit relation between the time-step size, the contraction parameters of the decoupling iteration, and the overall error; this relation is claimed to permit balancing of the iterative and discretization error components. Numerical experiments, including a three-dimensional biomechanics example, are presented to illustrate and validate the theoretical statements.

Significance. If the claimed Δt-independent contraction bound holds and the explicit quantification is correct, the result would supply a practical tool for choosing iteration tolerances in high-order decoupled poroelastic simulations, which is useful in applications where balancing computational cost and accuracy is important. The combination of existing techniques is not novel in itself, but the explicit error-component balancing constitutes a modest incremental contribution provided the technical premise is verified.

major comments (1)

[Abstract / main result paragraph] Abstract and main-result paragraph: the central claim requires that the contraction constant of the fixed-point decoupling map admits a bound independent of Δt so that the fixed-point error can be balanced against the BDF truncation error of order up to 5. In the standard weak formulation of the Biot system the pressure-to-displacement coupling produces terms whose Lipschitz constants scale as 1/Δt when the elliptic equation is solved for fixed pressure; the manuscript must identify the precise location (section, equation, or weighting) where this 1/Δt factor is cancelled or neutralized, otherwise the explicit quantification cannot hold for arbitrarily small Δt.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the central claim. We address the point below.

read point-by-point responses

Referee: [Abstract / main result paragraph] Abstract and main-result paragraph: the central claim requires that the contraction constant of the fixed-point decoupling map admits a bound independent of Δt so that the fixed-point error can be balanced against the BDF truncation error of order up to 5. In the standard weak formulation of the Biot system the pressure-to-displacement coupling produces terms whose Lipschitz constants scale as 1/Δt when the elliptic equation is solved for fixed pressure; the manuscript must identify the precise location (section, equation, or weighting) where this 1/Δt factor is cancelled or neutralized, otherwise the explicit quantification cannot hold for arbitrarily small Δt.

Authors: The central claim does not require a contraction constant that is independent of Δt. As stated in the abstract and proved in Section 3, the explicit error bound (Theorem 3.1) quantifies the interplay: the fixed-point error term is controlled by the contraction parameter γ of the iteration, while the discretization error is O(Δt^k) for k ≤ 5; balancing is achieved by relating γ to Δt. The 1/Δt factor from the pressure-displacement coupling in the standard weak form is neutralized by the weighted norm used to establish the contraction (see the inner product defined in (3.4) and the estimate (3.9) that absorbs the factor into the iteration parameter). This is the precise location requested. We will add a short clarifying sentence in the revised version pointing to these equations. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation combines independent standard techniques

full rationale

The abstract states that the analysis 'combines techniques known from fixed-point iterations with the convergence analysis of the temporal discretization' to derive an explicit relation between time-step size and contraction parameters. No quoted step reduces a prediction to a fitted input by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames an input as an output. The central result is presented as a quantified balance of independent error components rather than a tautology. This is the normal case of a self-contained numerical-analysis argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit list of free parameters or invented entities; the central claim rests on the domain assumption that the iterative decoupling admits a contraction whose rate interacts with the time step in a quantifiable way.

axioms (1)

domain assumption The elliptic-parabolic poroelasticity problem admits an iterative decoupling whose fixed-point map is contractive with parameters that can be related to the time step size.
Invoked in the main result paragraph when stating that convergence depends on the interplay between time step size and contraction parameters.

pith-pipeline@v0.9.0 · 5619 in / 1324 out tokens · 34363 ms · 2026-05-25T09:03:16.937046+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

[1]

Altmann and M

R. Altmann and M. Deiml . A novel iterative time integration scheme for linear poroelasticity. Electron. Trans. Numer. Anal., 60:256–275, 2024

work page 2024
[2]

Altmann and M

R. Altmann and M. Deiml . A second-order iterative time integration scheme for linear poroelasticity. ArXiv preprint 2403.12699, 2024

work page arXiv 2024
[3]

Altmann and R

R. Altmann and R. Maier . A decoupling and linearizing discretization for poroelasticity with nonlinear permeability.SIAM J. Sci. Comput., 44(3):B457–B478, 2022

work page 2022
[4]

Altmann, R

R. Altmann, R. Maier, and B. Unger . Semi-explicit discretization schemes for weakly-coupled elliptic-parabolic problems.Math. Comp., 90:1089–1118, 2021

work page 2021
[5]

Altmann, R

R. Altmann, R. Maier, and B. Unger . A semi-explicit integration scheme for weakly-coupled poroelasticity with nonlinear permeability. InProc. Appl. Math. Mech., volume 20, page e202000061, 2021

work page 2021
[6]

Altmann, R

R. Altmann, R. Maier, and B. Unger . Semi-explicit integration of second order for weakly coupled poroelasticity. BIT Numer. Math., 64(20), 2024

work page 2024
[7]

Altmann, V

R. Altmann, V. Mehrmann, and B. Unger . Port-Hamiltonian formulations of poroelastic network models. Math. Comput. Model. Dyn. Sys., 27(1):429–452, 2021

work page 2021
[8]

I. A. Baratta, J. P. Dean, J. S. Dokken, M. Habera, J. S. Hale, C. N. Richardson, M. E. Rognes, M. W. Scroggs, N. Sime, and G. N. Wells . DOLFINx: The next generation FEniCS problem solving environment, 2023

work page 2023
[9]

N. A. Barnafi Wittwer, S. D. Gregorio, L. Dede’, P. Zunino, C. Vergara, and A. Quar- teroni. A multiscale poromechanics model integrating myocardial perfusion and the epicardial coronary vessels. SIAM J. Appl. Math., 82(4):1167–1193, 2022

work page 2022
[10]

M. A. Biot . General theory of three-dimensional consolidation.J. Appl. Phys, 12(2):155–164, 1941

work page 1941
[11]

Corti, P.F

M. Corti, P.F. Antonietti, L. Dede’, and A.M. Quarteroni . Numerical modelling of the brain poromechanics by high-order discontinuous Galerkin methods.ArXiv e-print 2210.02272, 2022. 18 ROBERT ALTMANN †, ABDULLAH MUJAHID ⋆, AND BENJAMIN UNGER ⋆ 0.0e+00 4.7e-02 (a) Displacement u 1.40e-04 1.50e-04 (b) Arteriole pressure p1 -2.5e-09 4.8e-09 (c) Venous press...

work page arXiv 2022
[12]

Dahlquist

G. Dahlquist. G-stability is equivalent to A-stability.BIT Numer. Math., 18(4):384–401, 1978

work page 1978
[13]

Eliseussen, M

E. Eliseussen, M. E. Rognes, and T. B. Thompson . A posteriorierror estimation and adaptivity for multiple-network poroelasticity.ESAIM Math. Model. Numer. Anal., 57(4):1921–1952, 2023

work page 1921
[14]

E. Emmrich. Discrete versions of Gronwall’s lemma and their application to the numerical analysis of parabolic problems.Preprint 637, Technische Universität Berlin, Germany, 1999. HIGHER-ORDER ITERATIVE DECOUPLING FOR POROELASTICITY 19

work page 1999
[15]

Ern and S

A. Ern and S. Meunier . A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems.ESAIM: Math. Model. Numer. Anal., 43(2):353–375, 2009

work page 2009
[16]

Zheltov, and I.N

G.I.Barenblatt, I.P. Zheltov, and I.N. Kochina . Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata].J. Appl. Math. Mech., 24(5):1286–1303, 1960

work page 1960
[17]

Hairer, S

E. Hairer, S. P. Nørsett, and G. W anner . Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Heidelberg, London, 2nd rev. edition, 2009

work page 2009
[18]

Hairer and G

E. Hairer and G. W anner.Solving Ordinary Differential Equations II: Stiff and Differential–Algebraic Problems. Springer Berlin Heidelberg, Berlin, Heidelberg, 1996

work page 1996
[19]

T. Kato. Perturbation Theory for Linear Operators. Springer Berlin Heidelberg, Berlin, Heidelberg, 1995

work page 1995
[20]

Kim, H.A

J. Kim, H.A. Tchelepi, and R. Juanes . Stability and convergence of sequential methods for coupled flow and geomechanics: Drained and undrained splits.Comput. Meth. Appl. Mech. Eng., 200(23):2094–2116, 2011

work page 2094
[21]

Kim, H.A

J. Kim, H.A. Tchelepi, and R. Juanes . Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits.Comput. Meth. Appl. Mech. Eng., 200(13):1591–1606, 2011

work page 2011
[22]

Kosloff, R

D. Kosloff, R. F. Scott, and J. Scranton . Finite element simulation of Wilmington oil field subsidence: I. Linear modelling.Tectonophysics, 65(3-4):339–368, 1980

work page 1980
[23]

Kunkel and V

P. Kunkel and V. Mehrmann . Differential-Algebraic Equations. Analysis and Numerical Solution. European Mathematical Society, Zürich, 2006

work page 2006
[24]

J. J. Lee, K.-A. Mardal, and R. Winther . Parameter-robust discretization and preconditioning of Biot’s consolidation model.SIAM J. Sci. Comput., 39(1):A1–A24, 2017

work page 2017
[25]

J. J. Lee, E. Piersanti, K.-A. Mardal, and M. E. Rognes . A mixed finite element method for nearly incompressible multiple-network poroelasticity.SIAM J. Sci. Comput., 41(2):A722–A747, 2019

work page 2019
[26]

Lubich and A

C. Lubich and A. Ostermann . Runge–Kutta approximation of quasi-linear parabolic equations. Math. Comp., 64(210):601–627, 1995

work page 1995
[27]

Mikelić and M

A. Mikelić and M. F. Wheeler . Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci., 17(3):455–461, 2013

work page 2013
[28]

A. Mujahid. Monolithic, non-iterative and iterative time discretization methods for linear coupled elliptic-parabolic systems.GAMM Archive for Students, 4(1), 2022

work page 2022
[29]

Nevanlinna and F

O. Nevanlinna and F. Odeh . Multiplier techniques for linear multistep methods.Numer. Funct. Anal. Optim., 3(4):377–423, 1981

work page 1981
[30]

Piersanti and M

E. Piersanti and M. E. Rognes . Supplementary material (code) for ‘a mixed finite element method for nearly incompressible multiple-network poroelasticity’ by j. j. lee, e. piersanti, k.-a. mardal and m. e. rognes., 2018

work page 2018
[31]

R. E. Showalter. Diffusion in poro-elastic media.J. Math. Anal. Appl., 251(1):310–340, 2000

work page 2000
[32]

Storvik, J

E. Storvik, J. W. Both, K. Kumar, J. M. Nordbotten, and F. A. Radu . On the optimization of the fixed-stress splitting for Biot’s equations.Internat. J. Numer. Methods Engrg., 120(2):179–194, 2019

work page 2019
[33]

J. C. V ardakis, D. Chou, B. J. Tully, C. C. Hung, T. H. Lee, P. H. Tsui, and Y. Ventikos . Investigating cerebral oedema using poroelasticity.Med. Eng. Phys., 38(1):48–57, 2016

work page 2016
[34]

M. F. Wheeler and X. Gai . Iteratively coupled mixed and Galerkin finite element methods for poro-elasticity. Numer. Meth. Part. D. E., 23(4):785–797, 2007

work page 2007
[35]

J. A. White and R. I. Borja . Stabilized low-order finite elements for coupled solid-deformation/fluid- diffusionandtheirapplicationtofaultzonetransients. Comput. Method. Appl. M., 197(49-50):4353–4366, 2008

work page 2008
[36]

E. Zeidler . Nonlinear functional analysis and its applications. 2A: Linear monotone operators. Springer, New York, Berlin, Heidelberg, 1990

work page 1990
[37]

M. D. Zoback . Reservoir geomechanics. Cambridge University Press, Cambridge, 2010. †Institute of Analysis and Numerics, Otto von Guericke University Magdeburg, Univer- sitätsplatz 2, 39106 Magdeburg, Germany Email address: robert.altmann@ovgu.de 20 ROBERT ALTMANN †, ABDULLAH MUJAHID ⋆, AND BENJAMIN UNGER ⋆ ⋆ Stuttgart Center for Simulation Science (SC Si...

work page 2010

[1] [1]

Altmann and M

R. Altmann and M. Deiml . A novel iterative time integration scheme for linear poroelasticity. Electron. Trans. Numer. Anal., 60:256–275, 2024

work page 2024

[2] [2]

Altmann and M

R. Altmann and M. Deiml . A second-order iterative time integration scheme for linear poroelasticity. ArXiv preprint 2403.12699, 2024

work page arXiv 2024

[3] [3]

Altmann and R

R. Altmann and R. Maier . A decoupling and linearizing discretization for poroelasticity with nonlinear permeability.SIAM J. Sci. Comput., 44(3):B457–B478, 2022

work page 2022

[4] [4]

Altmann, R

R. Altmann, R. Maier, and B. Unger . Semi-explicit discretization schemes for weakly-coupled elliptic-parabolic problems.Math. Comp., 90:1089–1118, 2021

work page 2021

[5] [5]

Altmann, R

R. Altmann, R. Maier, and B. Unger . A semi-explicit integration scheme for weakly-coupled poroelasticity with nonlinear permeability. InProc. Appl. Math. Mech., volume 20, page e202000061, 2021

work page 2021

[6] [6]

Altmann, R

R. Altmann, R. Maier, and B. Unger . Semi-explicit integration of second order for weakly coupled poroelasticity. BIT Numer. Math., 64(20), 2024

work page 2024

[7] [7]

Altmann, V

R. Altmann, V. Mehrmann, and B. Unger . Port-Hamiltonian formulations of poroelastic network models. Math. Comput. Model. Dyn. Sys., 27(1):429–452, 2021

work page 2021

[8] [8]

I. A. Baratta, J. P. Dean, J. S. Dokken, M. Habera, J. S. Hale, C. N. Richardson, M. E. Rognes, M. W. Scroggs, N. Sime, and G. N. Wells . DOLFINx: The next generation FEniCS problem solving environment, 2023

work page 2023

[9] [9]

N. A. Barnafi Wittwer, S. D. Gregorio, L. Dede’, P. Zunino, C. Vergara, and A. Quar- teroni. A multiscale poromechanics model integrating myocardial perfusion and the epicardial coronary vessels. SIAM J. Appl. Math., 82(4):1167–1193, 2022

work page 2022

[10] [10]

M. A. Biot . General theory of three-dimensional consolidation.J. Appl. Phys, 12(2):155–164, 1941

work page 1941

[11] [11]

Corti, P.F

M. Corti, P.F. Antonietti, L. Dede’, and A.M. Quarteroni . Numerical modelling of the brain poromechanics by high-order discontinuous Galerkin methods.ArXiv e-print 2210.02272, 2022. 18 ROBERT ALTMANN †, ABDULLAH MUJAHID ⋆, AND BENJAMIN UNGER ⋆ 0.0e+00 4.7e-02 (a) Displacement u 1.40e-04 1.50e-04 (b) Arteriole pressure p1 -2.5e-09 4.8e-09 (c) Venous press...

work page arXiv 2022

[12] [12]

Dahlquist

G. Dahlquist. G-stability is equivalent to A-stability.BIT Numer. Math., 18(4):384–401, 1978

work page 1978

[13] [13]

Eliseussen, M

E. Eliseussen, M. E. Rognes, and T. B. Thompson . A posteriorierror estimation and adaptivity for multiple-network poroelasticity.ESAIM Math. Model. Numer. Anal., 57(4):1921–1952, 2023

work page 1921

[14] [14]

E. Emmrich. Discrete versions of Gronwall’s lemma and their application to the numerical analysis of parabolic problems.Preprint 637, Technische Universität Berlin, Germany, 1999. HIGHER-ORDER ITERATIVE DECOUPLING FOR POROELASTICITY 19

work page 1999

[15] [15]

Ern and S

A. Ern and S. Meunier . A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems.ESAIM: Math. Model. Numer. Anal., 43(2):353–375, 2009

work page 2009

[16] [16]

Zheltov, and I.N

G.I.Barenblatt, I.P. Zheltov, and I.N. Kochina . Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata].J. Appl. Math. Mech., 24(5):1286–1303, 1960

work page 1960

[17] [17]

Hairer, S

E. Hairer, S. P. Nørsett, and G. W anner . Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Heidelberg, London, 2nd rev. edition, 2009

work page 2009

[18] [18]

Hairer and G

E. Hairer and G. W anner.Solving Ordinary Differential Equations II: Stiff and Differential–Algebraic Problems. Springer Berlin Heidelberg, Berlin, Heidelberg, 1996

work page 1996

[19] [19]

T. Kato. Perturbation Theory for Linear Operators. Springer Berlin Heidelberg, Berlin, Heidelberg, 1995

work page 1995

[20] [20]

Kim, H.A

J. Kim, H.A. Tchelepi, and R. Juanes . Stability and convergence of sequential methods for coupled flow and geomechanics: Drained and undrained splits.Comput. Meth. Appl. Mech. Eng., 200(23):2094–2116, 2011

work page 2094

[21] [21]

Kim, H.A

J. Kim, H.A. Tchelepi, and R. Juanes . Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits.Comput. Meth. Appl. Mech. Eng., 200(13):1591–1606, 2011

work page 2011

[22] [22]

Kosloff, R

D. Kosloff, R. F. Scott, and J. Scranton . Finite element simulation of Wilmington oil field subsidence: I. Linear modelling.Tectonophysics, 65(3-4):339–368, 1980

work page 1980

[23] [23]

Kunkel and V

P. Kunkel and V. Mehrmann . Differential-Algebraic Equations. Analysis and Numerical Solution. European Mathematical Society, Zürich, 2006

work page 2006

[24] [24]

J. J. Lee, K.-A. Mardal, and R. Winther . Parameter-robust discretization and preconditioning of Biot’s consolidation model.SIAM J. Sci. Comput., 39(1):A1–A24, 2017

work page 2017

[25] [25]

J. J. Lee, E. Piersanti, K.-A. Mardal, and M. E. Rognes . A mixed finite element method for nearly incompressible multiple-network poroelasticity.SIAM J. Sci. Comput., 41(2):A722–A747, 2019

work page 2019

[26] [26]

Lubich and A

C. Lubich and A. Ostermann . Runge–Kutta approximation of quasi-linear parabolic equations. Math. Comp., 64(210):601–627, 1995

work page 1995

[27] [27]

Mikelić and M

A. Mikelić and M. F. Wheeler . Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci., 17(3):455–461, 2013

work page 2013

[28] [28]

A. Mujahid. Monolithic, non-iterative and iterative time discretization methods for linear coupled elliptic-parabolic systems.GAMM Archive for Students, 4(1), 2022

work page 2022

[29] [29]

Nevanlinna and F

O. Nevanlinna and F. Odeh . Multiplier techniques for linear multistep methods.Numer. Funct. Anal. Optim., 3(4):377–423, 1981

work page 1981

[30] [30]

Piersanti and M

E. Piersanti and M. E. Rognes . Supplementary material (code) for ‘a mixed finite element method for nearly incompressible multiple-network poroelasticity’ by j. j. lee, e. piersanti, k.-a. mardal and m. e. rognes., 2018

work page 2018

[31] [31]

R. E. Showalter. Diffusion in poro-elastic media.J. Math. Anal. Appl., 251(1):310–340, 2000

work page 2000

[32] [32]

Storvik, J

E. Storvik, J. W. Both, K. Kumar, J. M. Nordbotten, and F. A. Radu . On the optimization of the fixed-stress splitting for Biot’s equations.Internat. J. Numer. Methods Engrg., 120(2):179–194, 2019

work page 2019

[33] [33]

J. C. V ardakis, D. Chou, B. J. Tully, C. C. Hung, T. H. Lee, P. H. Tsui, and Y. Ventikos . Investigating cerebral oedema using poroelasticity.Med. Eng. Phys., 38(1):48–57, 2016

work page 2016

[34] [34]

M. F. Wheeler and X. Gai . Iteratively coupled mixed and Galerkin finite element methods for poro-elasticity. Numer. Meth. Part. D. E., 23(4):785–797, 2007

work page 2007

[35] [35]

J. A. White and R. I. Borja . Stabilized low-order finite elements for coupled solid-deformation/fluid- diffusionandtheirapplicationtofaultzonetransients. Comput. Method. Appl. M., 197(49-50):4353–4366, 2008

work page 2008

[36] [36]

E. Zeidler . Nonlinear functional analysis and its applications. 2A: Linear monotone operators. Springer, New York, Berlin, Heidelberg, 1990

work page 1990

[37] [37]

M. D. Zoback . Reservoir geomechanics. Cambridge University Press, Cambridge, 2010. †Institute of Analysis and Numerics, Otto von Guericke University Magdeburg, Univer- sitätsplatz 2, 39106 Magdeburg, Germany Email address: robert.altmann@ovgu.de 20 ROBERT ALTMANN †, ABDULLAH MUJAHID ⋆, AND BENJAMIN UNGER ⋆ ⋆ Stuttgart Center for Simulation Science (SC Si...

work page 2010