no. 1
A mixed elasticity formulation for fluid–poroelastic structure interaction
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 1, pp. 1-40

We develop a mixed finite element method for the coupled problem arising in the interaction between a free fluid governed by the Stokes equations and flow in deformable porous medium modeled by the Biot system of poroelasticity. Mass conservation, balance of stress, and the Beavers–Joseph–Saffman condition are imposed on the interface. We consider a fully mixed Biot formulation based on a weakly symmetric stress-displacement-rotation elasticity system and Darcy velocity-pressure flow formulation. A velocity-pressure formulation is used for the Stokes equations. The interface conditions are incorporated through the introduction of the traces of the structure velocity and the Darcy pressure as Lagrange multipliers. Existence and uniqueness of a solution are established for the continuous weak formulation. Stability and error estimates are derived for the semi-discrete continuous-in-time mixed finite element approximation. Numerical experiments are presented to verify the theoretical results and illustrate the robustness of the method with respect to the physical parameters.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1051/m2an/2021083
Keywords: Fluid-poroelastic structure interaction, Stokes–Biot model, mixed elasticity, mixed finite element method 
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     title = {A mixed elasticity formulation for fluid{\textendash}poroelastic structure interaction},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1--40},
     year = {2022},
     publisher = {EDP-Sciences},
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     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2021083/}
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Li, Tongtong; Yotov, Ivan. A mixed elasticity formulation for fluid–poroelastic structure interaction. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 1, pp. 1-40. doi: 10.1051/m2an/2021083

Ambartsumyan, I., Khattatov, E., Yotov, I., & Zunino, P. 2015, in Simulation of flow in fractured poroelastic media: a comparison of different discretization approaches. Finite Difference Methods, Theory and Applications, (SciCham: Springer), Lecture Notes in Comput., 9045, 3–14 | MR

I. Ambartsumyan, E. Khattatov, I. Yotov and P. Zunino, A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model. Numer. Math., 140, (2018) 513–553 | MR | DOI

I. Ambartsumyan, V. J. Ervin, T. Nguyen and I. Yotov, A nonlinear Stokes-Biot model for the interaction of a non-Newtonian fluid with poroelastic media. ESAIM: M2AN, 53, (2019) 1915–1955 | MR | Zbl | Numdam | DOI

I. Ambartsumyan, E. Khattatov, T. Nguyen and I. Yotov, Flow and transport in fractured poroelastic media. GEM Int. J. Geomath., 10, (2019) 1–34 | MR

I. Ambartsumyan, E. Khattatov, J. M. Nordbotten and I. Yotov, A multipoint stress mixed finite element method for elasticity on simplicial grids. SIAM J. Numer. Anal., 58, (2020) 630–656 | MR | DOI

I. Ambartsumyan, E. Khattatov and I. Yotov, A coupled multipoint stress-multipoint flux mixed finite element method for the Biot system of poroelasticity. Comput. Methods Appl. Mech. Eng., 372, (2020) 113407 | MR | DOI

I. Ambartsumyan, E. Khattatov, J. M. Nordbotten and I. Yotov, A multipoint stress mixed finite element method for elasticity on quadrilateral grids. Numer. Methods Part. Differ. Equ., 37, (2021) 1886–1915 | MR | DOI

D. Arnold and J. Lee, Mixed methods for elastodynamics with weak symmetry. SIAM J. Numer. Anal., 52, (2014) 2743–2769 | MR | DOI

D. N. Arnold, R. S. Falk and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comp., 76, (2007) 1699–1723 | MR | Zbl | DOI

D. N. Arnold, G. Awanou and W. Qiu, Mixed finite elements for elasticity on quadrilateral meshes. Adv. Comput. Math., 41, (2015) 553–572 | MR | DOI

S. Badia, A. Quaini and A. Quarteroni, Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interaction. J. Comput. Phys., 228, (2009) 7986–8014 | MR | Zbl | DOI

Y. Bazilevs, K. Takizawa and T. E. Tezduyar, Computational Fluid-Structure Interaction: Methods and Applications. John Wiley& Sons (2013) | Zbl | DOI

G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally impermeable wall. J. Fluid. Mech, 30, (1967) 197–207 | DOI

E. A. Bergkamp, C. V. Verhoosel, J. J. C. Remmers and D. M. J. Smeulders, A staggered finite element procedure for the coupled Stokes-Biot system with fluid entry resistance. Comput. Geosci., 24, (2020) 1497–1522 | MR | DOI

M. Biot, General theory of three-dimensional consolidation. J. Appl. Phys., 12, (1941) 155–164 | JFM | DOI

Boffi, D., Brezzi, F., Demkowicz, L. F., Durán, R. G., & Falk, R. S. 2008, in Mixed Finite Elements, Compatibility Conditions, and Applications, (Berlin; Fondazione C.I.M.E., Florence: Springer-Verlag), Lecture Notes in Mathematics, 1939 | MR | DOI

Brezzi, F. 1991, in Mixed and Hybrid Finite Element Methods, (New York: Springer-Verlag), Springer Series in Computational Mathematics, 15 | MR | Zbl | DOI

Bukač, M., Yotov, I., Zakerzadeh, R., & Zunino, P. in Effects of poroelasticity on fluid-structure interaction in arteries: a computational sensitivity study. In: Modeling the Heart and the Circulatory System, (Cham: Springer), MS&A. Model. Simul. Appl., 14, 197–220 | MR

M. Bukač, I. Yotov, R. Zakerzadeh and P. Zunino, Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche’s coupling approach. Comput. Methods Appl. Mech. Eng., 292, (2015) 138–170 | MR | DOI

M. Bukač, I. Yotov and P. Zunino, An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure. Numer. Methods Part. Differ. Equ., 31, (2015) 1054–1100 | MR | DOI

M. Bukač, I. Yotov and P. Zunino, Dimensional model reduction for flow through fractures in poroelastic media. ESAIM: M2AN, 51, (2017) 1429–1471 | MR

H. -J. Bungartz, & M. Schäfer, Fluid-structure Interaction: Modelling, Simulation, Optimisation. Vol. 53. Springer Science& Business Media (2006). | MR | DOI

S. Cesmelioglu, Analysis of the coupled Navier-Stokes/Biot problem. J. Math. Anal. Appl., 456, (2017) 970–991 | MR | DOI

A. Cesmelioglu and P. Chidyagwai, Numerical analysis of the coupling of free fluid with a poroelastic material. Numer. Methods Part. Differ. Equ., 36, (2020) 463–494 | MR | DOI

Cesmelioglu, A., Lee, H., Quaini, A., Wang, K., & Yi, S. Y. 2016, in Optimization-based decoupling algorithms for a fluid-poroelastic system. In: Topics in Numerical Partial Differential Equations and Scientific Computing., (New York: Springer), IMA Vol. Math. Appl., 160, 79–98 | MR | DOI

P. Ciarlet, The Finite Element Method for Elliptic Problems, 4. Amsterdam-New York-Oxford: Studies in Mathematics and its Applications. North-Holland Publishing Co. (1978) | MR | Zbl

T. Davis, Algorithm 832: UMFPACK V4.3 - an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw., 30, (2004) 196–199 | MR | Zbl | DOI

M. Discacciati, E. Miglio and A. Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math., 43, (2002) 57–74 | MR | Zbl | DOI

H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press (2014) | MR | Zbl | DOI

V. J. Ervin, E. W. Jenkins and S. Sun, Coupled generalized nonlinear Stokes flow with flow through a porous medium. SIAM J. Numer. Anal., 47, (2009) 929–952 | MR | Zbl | DOI

M. Fernández, Incremental displacement-correction schemes for the explicit coupling of a thin structure with an incompressible fluid. C.R. Math., 349, (2011) 473–477 | MR | Zbl | DOI

Galdi, G. P. Fundamental Trends in Fluid-structure Interaction. Vol. 1 of Contemporary Challenges in Mathematical Fluid Dynamics and Its Applications (Hackensack, NJ: World Scientific Publishing Co., Pte. Ltd.)

J. Galvis and M. Sarkis, Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. Electron. Trans. Numer. Anal., 26, (2007) 350–384 | MR | Zbl

G. Gatica, S. Meddahi and R. Oyarzúa, A conforming mixed finite-element method for the coupling of fluid flow with porous media flow. IMA J. Numer. Anal., 29, (2009) 86–108 | MR | Zbl | DOI

G. Gatica, A. Márquez, R. Oyarzúa and R. Rebolledo, Analysis of an augmented fully-mixed approach for the coupling of quasi-Newtonian fluids and porous media. Comput. Methods Appl. Mech. Eng., 270, (2014) 76–112 | MR | Zbl | DOI

F. Hecht, New development in FreeFem++. J. Numer. Math., 20, (2012) 251–265 | MR | Zbl | DOI

M. Jayadharan, E. Khattatov and I. Yotov, Domain decomposition methods for mixed finite element discretizations of the Biot system of poroelasticity. Comput. Geosci., 25, (2021) 1919–1938 | MR | DOI

E. Khattatov and I. Yotov, Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry. ESAIM: M2AN, 53, (2019) 2081–2108 | MR | Zbl | Numdam | DOI

H. Kunwar, H. Lee and K. Seelman, Second-order time discretization for a coupled quasi-Newtonian fluid-poroelastic system. Int. J. Numer. Methods Fluids, 92, (2020) 687–702 | MR | DOI

W. J. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow. SIAM J. Numer. Anal., 40, (2002) 2195–2218 | MR | Zbl | DOI

J. Lee, Robust error analysis of coupled mixed methods for Biot’s consolidation model. J. Sci. Comput., 69, (2016) 610–632 | MR | DOI

V. Martin, J. Jaffré and J. E. Roberts, Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput., 26, (2005) 1667–1691 | MR | Zbl | DOI

T. Richter, Fluid-structure Interactions: Models, Analysis and Finite Elements, 118. Springer (2017) | MR | DOI

B. Riviere and I. Yotov, Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal., 42, (2005) 1959–1977 | MR | Zbl | DOI

P. G. Saffman, On the boundary condition at the surface of a porous media. Stud. Appl. Math., 2, (1971) 93–101 | Zbl | DOI

Showalter, R. E., Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Vol. 49 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (1997) | MR | Zbl

Showalter, R. E. in Poroelastic filtration coupled to Stokes flow. In: Control Theory of Partial Differential Equations, (Boca Raton, FL: Chapman& Hall/CRC), Lect. Notes Pure Appl. Math., 242, 229–241 | MR | Zbl

R. E. Showalter, Nonlinear degenerate evolution equations in mixed formulation. SIAM J. Math. Anal., 42, (2010) 2114–2131 | MR | Zbl | DOI

D. Vassilev, C. Wang and I. Yotov, Domain decomposition for coupled Stokes and Darcy flows. Comput. Methods Appl. Mech. Eng., 268, (2014) 264–283 | MR | Zbl | DOI

J. Wen and Y. He, A strongly conservative finite element method for the coupled Stokes-Biot model. Comput. Math. Appl., 80, (2020) 1421–1442 | MR | Zbl | DOI

H. K. Wilfrid, Nonconforming finite element methods for a Stokes/Biot fluid-poroelastic structure interaction model. Results Appl. Math., 7, (2020) | MR | Zbl | DOI

S.-Y. Yi, Convergence analysis of a new mixed finite element method for Biot’s consolidation model. Numer. Meth. Partial. Differ. Equ., 30, (2014) 1189–1210 | MR | Zbl | DOI

S.-Y. Yi, A study of two modes of locking in poroelasticity. SIAM J. Numer. Anal., 55, (2017) 1915–1936 | MR | Zbl | DOI

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