Abstract
We study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium. The free fluid is governed by the Stokes equations, while the flow in the poroelastic medium is modeled using the Biot poroelasticity system. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting in continuity of flux condition of essential type. A Lagrange multiplier method is employed to impose weakly this condition. A stability and error analysis is performed for the semi-discrete continuous-in-time and the fully discrete formulations. A series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the robustness of the model with respect to its parameters.
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Ilona Ambartsumyan, Eldar Khattatov and Ivan Yotov are partially supported by DOE Grant DE-FG02-04ER25618 and NSF Grant DMS 1418947. Paolo Zunino is partially supported by DOE Grant DE-FG02-04ER25618 and by the INdAM Research group GNCS.
Appendix: Fully discrete analysis
Appendix: Fully discrete analysis
In this section we provide a detailed analysis of the stability and convergence of the fully discrete method (6.1)–(6.3). We will utilize the following discrete Gronwall inequality [43].
Lemma 9.1
(Discrete Gronwall lemma) Let \(\tau > 0\), \(B \ge 0\), and let \(a_n,b_n,c_n,d_n\), \(n \ge 0\), be non-negative sequences such that \(a_0 \le B\) and
Then,
Proof of Theorem 6.1
We choose
in (6.1)–(6.3) and use the discrete analog of (3.17):
to obtain the energy equality
The right-hand side can be bounded as follows, using inequalities (4.1) and (4.3),
Combining (9.2) and (9.3), summing up over the time index \(n=1,...,N\), multiplying by \(\tau \) and using the coercivity of the bilinear forms (3.4)–(3.6), we obtain
To bound the last term on the right we use summation by parts:
Next using the inf–sup condition (3.9) for \((p^n_{f,h},p^n_{p,h}, \lambda ^n_h)\) we obtain, in a similar way to (4.8),
Combining (9.4)–(9.6), and taking \(\epsilon _2\) small enough, and then \(\epsilon _1\) small enough, and using Lemma 9.1 with \(a_n = \Vert \varvec{\eta }^n_{p,h}\Vert ^2_{H^1(\Omega _p)}\), gives
which implies the statement of the theorem using the appropriate space-time norms. \(\square \)
For the sake of space, we do not present the proof of Theorem 6.2. The error equations are obtained by subtracting the first two equations of the fully discrete formulation (6.1)–(6.2) from the their continuous counterparts (2.12)–(2.13):
where \(r_n\) denotes the difference between the time derivative and its discrete analog:
It is easy to see that [11, Lemma 4] for sufficiently smooth \(\theta \),
The proof of Theorem 6.2 follows the structure of the proof of Theorem 5.1, using discrete-in-time arguments as in the proof of Theorem 6.1.
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Ambartsumyan, I., Khattatov, E., Yotov, I. et al. A Lagrange multiplier method for a Stokes–Biot fluid–poroelastic structure interaction model. Numer. Math. 140, 513–553 (2018). https://doi.org/10.1007/s00211-018-0967-1
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DOI: https://doi.org/10.1007/s00211-018-0967-1