A Lagrange multiplier method for a Stokes–Biot fluid–poroelastic structure interaction model

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.

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

$$\begin{aligned} a_n + \tau \sum _{l=1}^n b_l \le \tau \sum _{l=1}^{n-1} d_la_l + \tau \sum _{l=1}^n c_l +B, \quad n\ge 1. \end{aligned}$$

Then,

$$\begin{aligned} a_n + \tau \sum _{l=1}^n b_l \le \exp \left( \tau \sum _{l=1}^{n-1}d_l\right) \left( \tau \sum _{l=1}^n c_l + B \right) , \quad n \ge 1. \end{aligned}$$

Proof of Theorem 6.1

We choose

$$\begin{aligned} (\mathbf{v}_{f,h},w_{f,h},\mathbf{v}_{p,h},w_{p,h},\varvec{\xi }_{p,h},\mu _h) = \left( \mathbf{u}^{n}_{f,h},p^{n}_{f,h},\mathbf{u}^{n}_{p,h},p^{n}_{p,h},d_{\tau } \varvec{\eta }^{n}_{p,h},\lambda _h\right) \end{aligned}$$

in (6.1)–(6.3) and use the discrete analog of (3.17):

$$\begin{aligned} \int _{S}u^n d_{\tau }\phi ^n = \frac{1}{2} d_{\tau }\Vert \phi ^n\Vert ^2_{L^2(S)} + \frac{1}{2} \tau \Vert d_{\tau }\phi ^n\Vert ^2_{L^2(S)} \end{aligned}$$

(9.1)

to obtain the energy equality

$$\begin{aligned}&\frac{1}{2} d_{\tau }\left( s_0 \left\| p^n_{p,h}\right\| ^2_{L^2\left( \Omega _p\right) } + a^e_{p}\left( \varvec{\eta }^n_{p,h},\varvec{\eta }^n_{p,h}\right) \right) \nonumber \\&\quad + \frac{\tau }{2} \left( s_0 \left\| d_{\tau }p^n_{p,h}\right\| ^2_{L^2\left( \Omega _p\right) } +a^e_{p}\left( d_{\tau }\varvec{\eta }^n_{p,h},d_{\tau }\varvec{\eta }^n_{p,h}\right) \right) \nonumber \\&\quad + a_{f}\left( \mathbf{u}^n_{f,h},\mathbf{u}^n_{f,h}\right) + a^d_{p}\left( \mathbf{u}^n_{p,h},\mathbf{u}^n_{p,h}\right) + \left| \mathbf{u}^n_{f,h} - d_{\tau }\varvec{\eta }^n_{p,h} \right| ^2_{a_{BJS}} = \mathcal {F} \left( t_n\right) . \end{aligned}$$

(9.2)

The right-hand side can be bounded as follows, using inequalities (4.1) and (4.3),

$$\begin{aligned} \mathcal {F}\left( t_n\right)&= \left( \mathbf{f}_{f}\left( t_n\right) ,\mathbf{u}^n_{f,h}\right) + \left( \mathbf{f}_{p}\left( t_n\right) ,d_\tau \varvec{\eta }^n_{p,h}\right) +\left( q_{f}\left( t_n\right) ,p^n_{f,h}\right) + \left( q_{p}\left( t_n\right) ,p^n_{p,h}\right) \nonumber \\&\le \left( \mathbf{f}_{p}\left( t_n\right) ,d_\tau \varvec{\eta }^n_{p,h}\right) + \frac{\epsilon _1}{2} \left( \left\| \mathbf{u}^n_{f,h}\right\| ^2_{L^2\left( \Omega _f\right) } + \left\| p^n_{f,h}\right\| ^2_{L^2\left( \Omega _f\right) } + \left\| p^n_{p,h}\right\| ^2_{L^2\left( \Omega _p\right) }\right) \nonumber \\&\quad + C \frac{1}{2\epsilon _1}\left( \left\| \mathbf{f}_{f}\left( t_n\right) \right\| ^2_{L^2\left( \Omega _f\right) } + \left\| q_{f}\left( t_n\right) \right\| ^2_{L^2\left( \Omega _f\right) } + \left\| q_{p}\left( t_n\right) \right\| ^2_{L^2\left( \Omega _p\right) } \right) . \qquad \end{aligned}$$

(9.3)

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

$$\begin{aligned}&s_0 \left\| p^N_{p,h}\right\| ^2_{L^2\left( \Omega _p\right) } + \left\| \varvec{\eta }^N_{p,h}\right\| ^2_{H^1\left( \Omega _p\right) }\nonumber \\&\quad \quad + \tau \sum _{n=1}^{N}\left( \left\| \mathbf{u}^n_{f,h}\right\| ^2_{H^1\left( \Omega _{f}\right) } + \left\| \mathbf{u}^n_{p,h}\right\| ^2_{L^2\left( \Omega _{p}\right) }+ \left| \mathbf{u}^n_{f,h} - d_\tau \varvec{\eta }^n_{p,h}\right| ^2_{a_{BJS}} \right) \nonumber \\&\quad \quad + \tau ^2 \sum _{n=1}^{N} \left( s_0\left\| d_\tau p^n_{p,h}\right\| ^2_{L^2\left( \Omega _p\right) } + \left\| d_\tau \varvec{\eta }^n_{p,h}\right\| ^2_{H^1\left( \Omega _p\right) }\right) \nonumber \\&\quad \le C\left( s_0 \left\| p^0_{p,h}\right\| ^2_{L^2\left( \Omega _p\right) } + \left\| \varvec{\eta }^0_{p,h}\right\| ^2_{H^1\left( \Omega _p\right) }\right. \nonumber \\&\qquad + \epsilon _1 \tau \sum _{n=1}^{N} \left( \left\| \mathbf{u}^n_{f,h}\right\| ^2_{L^2\left( \Omega _f\right) } + \left\| p^n_{f,h}\right\| ^2_{L^2\left( \Omega _f\right) } +\left\| p^n_{p,h}\right\| ^2_{L^2\left( \Omega _p\right) }\right) \nonumber \\&\qquad + \epsilon _1^{-1}\tau \sum _{n=1}^{N}\left( \left\| \mathbf{f}_{f}\left( t_n\right) \right\| ^2_{L^2\left( \Omega _f\right) } + \left\| q_{f}\left( t_n\right) \right\| ^2_{L^2\left( \Omega _f\right) } + \left\| q_{p}\left( t_n\right) \right\| ^2_{L^2\left( \Omega _p\right) } \right) \nonumber \\&\qquad \left. +\,\tau \sum _{n=1}^{N}\left( \mathbf{f}_{p}\left( t_n\right) ,d_\tau \varvec{\eta }^n_{p,h}\right) \right) . \end{aligned}$$

(9.4)

To bound the last term on the right we use summation by parts:

$$\begin{aligned}&\tau \sum _{n=1}^{N}\left( \mathbf{f}_{p}\left( t_n\right) ,d_\tau \varvec{\eta }^n_{p,h}\right) = \left( \mathbf{f}_p\left( t_N\right) ,\varvec{\eta }_{p,h}^N\right) - \left( \mathbf{f}_p\left( 0\right) ,\varvec{\eta }_{p,h}^0\right) - \tau \sum _{n=1}^{N-1}\left( d_\tau \mathbf{f}_{p}^n,\varvec{\eta }^n_{p,h}\right) \nonumber \\&\qquad \le \frac{\epsilon _1}{2} \left\| \varvec{\eta }_{p,h}^N\right\| ^2_{L^2\left( \Omega _p\right) } + \frac{1}{2\epsilon _1}\left\| \mathbf{f}_p\left( t_N\right) \right\| ^2_{L^2\left( \Omega _p\right) } + \frac{\tau }{2}\sum _{n=1}^{N-1}\left\| \varvec{\eta }^n_{p,h}\right\| ^2_{L^2\left( \Omega _p\right) } \nonumber \\&\qquad \qquad + \frac{1}{2} \left( \left\| \varvec{\eta }_{p,h}^0\right\| ^2_{L^2\left( \Omega _p\right) } + \left\| \mathbf{f}_{p}\left( 0\right) \right\| ^2_{L^2\left( \Omega _p\right) } + \tau \sum _{n=1}^{N-1}\left\| d_\tau \mathbf{f}_{p}^n\right\| ^2_{L^2\left( \Omega _p\right) }\right) . \end{aligned}$$

(9.5)

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),

$$\begin{aligned}&\epsilon _2\tau \sum _{n=1}^N \Big (\left\| p^n_{f,h}\right\| ^2_{L^2(\Omega _f)} + \left\| p^n_{p,h}\right\| ^2_{L^2(\Omega _p)} + \left\| \lambda ^n_h\right\| ^2_{ \Lambda _h} \Big ) \nonumber \\&\quad \le C \epsilon _2\tau \sum _{n=1}^N \Big ( \left\| \mathbf{f}_{f}(t_n)\right\| ^2_{L^2(\Omega _f)} + \left\| \mathbf{f}_{p}(t_n)\right\| ^2_{L^2(\Omega _p)} + \left\| \mathbf{u}^n_{f,h}\right\| ^2_{H^1(\Omega _f)} + \left\| \mathbf{u}^n_{p,h}\right\| ^2_{L^2(\Omega _p)} \nonumber \\&\quad \quad + \left\| \varvec{\eta }^n_{p,h}\right\| ^2_{H^1(\Omega _p)} + \left| \mathbf{u}^n_{f,h}-d_\tau \varvec{\eta }^n_{p,h}\right| ^2_{a_{BJS}}\Big ). \end{aligned}$$

(9.6)

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

$$\begin{aligned}&s_0 \left\| p^N_{p,h}\right\| ^2_{L^2(\Omega _p)} + \left\| \varvec{\eta }^N_{p,h}\right\| ^2_{H^1(\Omega _p)}\\&\quad \quad +\tau \sum _{n=1}^{N} \left[ \left\| \mathbf{u}^n_{f,h}\right\| ^2_{H^1(\Omega _{f})} +\left\| \mathbf{u}^n_{p,h}\right\| ^2_{L^2(\Omega _{p})}+\,|\mathbf{u}^n_{f,h} - d_\tau \varvec{\eta }^n_{p,h}|^2_{a_{BJS}} \right] \\&\quad \quad + \tau ^2\sum _{n=1}^{N} \left[ s_0\left\| d_\tau p^n_{p,h}\right\| ^2_{L^2(\Omega _p)} + \left\| d_\tau \varvec{\eta }^n_{p,h}\right\| ^2_{H^1(\Omega _p)} \right] \\&\quad \quad + \tau \sum _{n=1}^{N} \left[ \left\| p^n_{p,h}\right\| ^2_{L^2(\Omega _p)} +\left\| p^n_{f,h}\right\| ^2_{L^2(\Omega _f)}+\left\| \lambda ^n_{h}\right\| ^2_{\Lambda _h} \right] \\&\quad \le C\exp (T)\Big (s_0 \left\| p^0_{p,h}\right\| ^2_{L^2(\Omega _p)} + \left\| \varvec{\eta }^0_{p,h}\right\| ^2_{H^1(\Omega _p)}+ \left\| \mathbf{f}_{p}(0)\right\| ^2_{L^2(\Omega _p)} \\&\qquad +\tau \sum _{n=1}^{N} \Big [ \left\| \mathbf{f}_{f}(t_n)\right\| ^2_{L^2(\Omega _f)}+\left\| \mathbf{f}_{p}(t_n)\right\| ^2_{L^2(\Omega _p)} + \left\| q_{f}(t_n)\right\| ^2_{L^2(\Omega _f)}\\&\qquad + \left\| q_{p}(t_n)\right\| ^2_{L^2(\Omega _p)}+\left\| d_\tau \mathbf{f}_{p}\right\| ^2_{L^2(\Omega _p)}\Big ] \Big ), \end{aligned}$$

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):

$$\begin{aligned}&a_{f}\left( \mathbf{e}^n_f,\mathbf{v}_{f,h}\right) + a^d_{p}\left( \mathbf{e}^n_p,\mathbf{v}_{p,h}\right) + a^e_{p}\left( \mathbf{e}^n_s,\varvec{\xi }_{p,h}\right) \nonumber \\&\qquad +a_{BJS}\left( \mathbf{e}^n_f,d_\tau \mathbf{e}^n_{s};\mathbf{v}_{f,h},\varvec{\xi }_{p,h}\right) + b_f\left( \mathbf{v}_{f,h},e^n_{fp}\right) + b_p\left( \mathbf{v}_{p,h},e^n_{pp}\right) \nonumber \\&\qquad +\alpha b_p\left( \varvec{\xi }_{p,h},e^n_{pp}\right) + b_{\Gamma }\left( \mathbf{v}_{f,h},\mathbf{v}_{p,h},\varvec{\xi }_{p,h};e^n_\lambda \right) +\left( s_0\,d_\tau e^n_{pp},w_{p,h}\right) \nonumber \\&\qquad - \alpha b_p\left( d_\tau e^n_{s},w_{p,h}\right) - b_p\left( \mathbf{e}^n_{p},w_{p,h}\right) - b_f\left( \mathbf{e}^n_{f},w_{f,h}\right) \nonumber \\&\qquad = \left( s_0 r_n\left( p_p\right) ,w_{p,h}\right) +a_{BJS}\left( 0,r_n\left( \varvec{\eta }_{p}\right) ;\mathbf{v}_{f,h},\varvec{\xi }_{p,h}\right) - \alpha b_p\left( r_n\left( \varvec{\eta }_p\right) \!,w_{p,h}\right) , \end{aligned}$$

(9.7)

where $r_n$ denotes the difference between the time derivative and its discrete analog:

$$\begin{aligned} r_n(\theta )&= \partial _t \theta (t_n) -d_\tau \theta ^n. \end{aligned}$$

It is easy to see that [11, Lemma 4] for sufficiently smooth $\theta$,

$$\begin{aligned} \tau \sum _{n=1}^N\Vert r_n(\theta )\Vert ^2_{H^k(S)} \le C\tau ^2\Vert \partial _{tt}\theta \Vert ^2_{L^2(0,T;H^k(S))}. \end{aligned}$$

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.