A Posteriori Error Analysis for a Lagrange Multiplier Method for a Stokes/Biot Fluid-Poroelastic Structure Interaction Model

Wilfrid, Houédanou Koffi

doi:https://doi.org/10.1155/2021/8877012

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2021 | Article ID 8877012 | https://doi.org/10.1155/2021/8877012

A Posteriori Error Analysis for a Lagrange Multiplier Method for a Stokes/Biot Fluid-Poroelastic Structure Interaction Model

Houédanou Koffi Wilfrid¹

Academic Editor: Wei-Shih Du

Received31 Aug 2020

Accepted10 Feb 2021

Published04 Mar 2021

Abstract

In this work, we develop an a posteriori error analysis of a conforming mixed finite element method for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium on isotropic meshes in , . The approach utilizes a Lagrange multiplier method to impose weakly the interface conditions [Ilona Ambartsumyan et al., Numerische Mathematik, 140 (2): 513-553, 2018]. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. The proof of reliability makes use of suitable auxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved, Helmholtz decomposition, and local approximation properties of the Clément interpolant. On the other hand, inverse inequalities and the localization technique based on simplexe-bubble and face-bubble functions are the main tools for proving the efficiency of the estimator. Up to minor modifications, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

1. Introduction

In this paper, we develop an a posteriori error analysis for solving the interaction of a free incompressible viscous Newtonian fluid with a fluid within a poroelastic medium. This is a challenging multiphysics problem with applications to predicting and controlling processes arising in groundwater flow in fractured aquifers, oil and gas extraction, arterial flows, and industrial filters. In these applications, it is important to model properly the interaction between the free fluid with the fluid within the porous medium and to take into account the effect of the deformation of the medium. For example, geomechanical effects play an important role in hydraulic fracturing, as well as in modeling phenomena such as subsidence and compaction.

We adopt the Stokes equations to model the free fluid and the Biot system [1] for the fluid in the poroelastic media. In the latter, the volumetric deformation of the elastic porous matrix is complemented with the Darcy equation that describes the average velocity of the fluid in the pores. The model features two different kinds of coupling across the interface: Stokes-Darcy coupling [2–10] and fluid-structure interaction (FSI) [11–15].

The well-posedness of the mathematical model based on the Stokes-Biot system for the coupling between a fluid and a poroelastic structure is studied in [16]. A numerical study of the problem, using a Navier-Stokes equations for the fluid, is presented in [11, 17], utilizing a variational multiscale approach to stabilize the finite element spaces. The problem is solved using both a monolithic and a partitioned approach, with the latter requiring subiterations between the two problems.

Finite element analysis of an arbitrary Lagrangian-Eulerian method for Stokes/parabolic moving interface problem with jump coefficients has been studied in [18]. The authors in [19] study a numerical solution of the coupled system of the time-dependent Stokes and fully dynamic Biot equations. They establish stability of the scheme and derive error estimates for the fully discrete coupled scheme. Numerical errors and convergence rates for smooth problems as well as tests on realistic material parameters have been presented. In [20], Wen and He consider a strongly conservative discretization for the rearranged Stokes-Biot model based on interior penalty discontinuous Galerkin method and mixed finite element method. The existence and uniqueness of solution of the numerical scheme have been presented. Then, the analysis of stability and priori error estimates have been derived. The numerical examples under uniform meshes, which well validate the analysis of convergence and the strong mass conservation, are presented. A staggered finite element procedure for the coupled Stokes-Biot system with fluid entry resistance has been studied by Bergkamp et al. in [21] while Ambartsumyan et al. study in [22] flow and transport in fractured poroelastic media using Stokes flow in the fractures and the Biot model in the porous media. In [23], semidiscrete continuous-in-time approximation has been proposed for the weak coupled mixed formulation. For the discretization of the fluid velocity and pressure, the authors have used the finite elements which include the MINI elements, the Taylor-Hood elements, and the conforming Crouzeix-Raviart elements. For the discretization of the porous medium problem, they choose the spaces that include Raviart-Thomas and Brezzi-Douglas-Marini elements. An a priori error analysis is performed with some numerical tests confirming the convergence rates.

A posteriori error estimators are computable quantities, expressed in terms of the discrete solution and of the data that measure the actual discrete errors without the knowledge of the exact solution. They are essential to design adaptive mesh refinement algorithms which aqui-distribute the computational effort and optimize the approximation efficiency. Since the pioneering work of Babuška and Rheinboldt [24–27], adaptive finite element methods based on a posteriori error estimates have been extensively investigated. To our best knowledge, there is no a posteriori error estimation for the Stokes/Biot fluid-poroelastic structure interaction model for finite element methods. Here, we develop such a posteriori error analysis for the semidiscrete conforming finite element methods. We have got a new family of a local indicator error (see Definition 5, Equation (48)) and global (Equation (52)).

The schedule of the paper is as follows. Section 2 is devoted to notations and basic results that are used throughout the document. Our main results regarding a posteriori error analysis are stated in Section 3. We prove that our indicator errors are efficiency and reliability, and then, are optimal. The global inf-sup condition is the main tool yielding the reliability. In turn, the local efficiency result is derived using the technique of bubble function introduced by Verfürth [28] and used in similar context by Carstensen and Dolzmann [29]. Finally, this paper is summarized with further works in Section 4.

2. Preliminaries and Notations

2.1. Stokes-Biot Model Problem

We consider a multiphysics model problem for free fluid’s interaction with a flow in a deformable porous media, where the simulation domain , , is a union of nonoverlapping regions and . Here, is a free fluid region with flow governed by the Stokes equations, and is a poroelastic material governed by the Biot system. For simplicity of notation, we assume that each region is connected. The extension to nonconnected regions is straightforward. Let (see Figure 1).

Let be the velocity-pressure pair in , , and let be the displacement in . Let be the fluid viscosity, let be the body force terms, and let be external source or sink terms. Let and denote, respectively, the deformation rate tensor and the stress tensor:

In the free fluid region , satisfy the Stokes equations: where is the final time. Let and be the elastic and poroelastic stress tensors, respectively: where and are the Lamé parameters and is the Biot-Willis constant. The poroelasticity region is governed by the quasistatic Biot system [23]: where is a storage coefficient and the symmetric and uniformly positive definite rock permeability tensor, satisfying, for some constants ,

Following [1], the interface conditions on the fluid-poroelasticity interface are mass conservation, balance of stresses, and the Beavers-Joseph-Saffman (BJS) condition [30] modeling slip with friction: where and are the outward unit normal vectors to and , respectively, , , is an orthogonal system of unit tangent vectors on , , and is an experimentally determined friction coefficient. We note that continuity of flux constraints the normal velocity of the solid skeleton, while the condition accounts for its tangential velocity.

The above system of equations needs to be complemented by a set of boundary and initial conditions. Let and . Let . We assume for simplicity homogeneous boundary conditions:

To ovoid the issue with restricting the mean value of the pressure, we assume that . We also assume that is not adjacent to the interface , i.e., . Nonhomogeneous displacement and velocity conditions can be handled in a standard way by adding suitable extensions of the boundary data. The pressure boundary condition is natural in the mixed Darcy formulation, so nonhomogeneous pressure data would lead to an additional boundary term. We further sat the initial conditions:

Equations (2)–(10) consist of the model of the coupled Stokes and Biot flows problem that we will study below.

2.2. Weak Formulation

In this part, we first introduce some Sobolev spaces [31] and norms. If is a bounded domain of and is a nonnegative integer, the Sobolev space is defined in the usual way with the usual norm and seminorm . In particular, , and we write for . Similarly, we denote by the or inner product. For shortness if is equal to , we will drop the index , while for any , , and , for . The space denotes the closure of in . Let be the space of vector valued functions with components in . The norm and the seminorm on are given by

For a connected open subset of the boundary , we write for the inner product (or duality pairing), that is, for scalar valued functions , one defines:

In the following, we derive a Lagrange multiplier type weak formulation of the system, which will be the basis for our finite element approximation. Let where is the space of vectors with divergence in with a norm

We define the global velocity and pressure spaces as with norms

The weak formulation is obtained by mutiplying the equations in each region by suitable test functions, integrating by parts the second-order terms in space, and utilizing the interface and boundary conditions.

Let define be the bilinear forms related to Stokes, Darcy, and the elasticity operator, respectively. Let

Integration by parts in (2) and the two equations in (5) lead to the interface term

Using the first condition for balance of normal stress in (9), we set which will be used as a Lagrange multiplier to impose the mass conservation interface condition (8). Utilizing the BJS condition (10) and the second condition for balance of stresses in (9), we obtain where

For the well-posedness of , we require that . According to the normal trace theorem, since , then . Furthermore, since on and , then (see, e.g., [32]). Therefore, we take .

The Lagrange multiplier variational formulation is for , find , , , , , and such that , , and for all , , , , , and , where we used the notation .

The assumptions on the fluid viscosity and the material coefficients , , and imply that the bilinear forms , , and are coercive and continuous in the appropriate norms. In particular, there exist positive constants , , , , , such that: where (28)–(29) and (32)–(33) hold true thanks to Poincaré inequality and (32)–(33) also relies on Korn’s inequality.

In summary, from ([23], Corollary 3.1, Page 7), the following result holds:

Theorem 1. There exists a unique solution to the problem (25)–(27).

2.3. Finite Element Discretization

Let and be shape-regular and quasiuniform partition of and , respectively, both consisting of affine elements with maximal element diameter . The two partitions may be nonmatching at the interface . For the discretization of the fluid velocity and pressure, we choose finite element spaces and , which are assumed to be inf-sup stable. Examples of such spaces include the MINI elements, the Taylor-Hood elements, and the conforming Crouzeix-Raviart elements. For the discretization of the porous medium problem, we choose and to be any of well-known inf-sup stable mixed finite element spaces, such as the Raviart-Thomas or the Brezzi-Douglas-Marini spaces. The global spaces are:

We employ a conforming Lagrangian finite element space to approximate the structure displacement. Note that the finite element spaces , , and satisfy the prescribed homogeneous boundary conditions on the external boundaries. For the discrete Lagrange multiplier space, we take

The semidiscrete continuous-in-time problem reads given and , for , find , , , , , and such that for all , , , , , and ,

We will take and to be suitable projections of the initial data and .

We introduce the errors for all variables:

The following results hold cf. [23]:

Theorem 2 (A priori error estimation). There exists a unique solution in of the discrete problem (36)–(38), and if the solution of the continuous problem (25)–(27) is smooth enough, then we have:

For , we can subtract (36)–(38) to (25)–(27) to obtain the Galerkin orthogonality relation for all :

3. A Posteriori Error Analysis

In order to solve the Stokes-Biot model problem by efficient adaptive finite element methods, reliable and efficient a posteriori error analysis is important to provide appropriated indicators. In this section, we first define the local and global indicators (Section 3.1), and then, the lower and upper error bounds are derived (Sections 3.4 and 3.5).

3.1. Residual Error Estimators

The general philosophy of residual error estimators is to estimate an appropriate norm of the correct residual by terms that can be evaluated easier and that involve the data at hand. To this end, define the exact element residuals:

Definition 3 (Exact Element Residuals). Let and be an arbitrary finite element function. The exact element residuals over a triangle or tetrahedra and over are defined for all by:

As it is common, these exact residuals are replaced by some finite-dimensional approximation called approximate element residual , , , , . This approximation is here achieved by projecting and on the space of piecewise constant functions in and piecewise functions in , more precisely for all we take

While for all , we take and as the unique element of , respectively such that: respectively,

Thereby, we define the approximate element residuals.

Definition 4 [Approximate Element Residuals]. Let and be an arbitrary finite element function. Then, the approximate element residuals are defined for all by:

Next, introduce the gradient jump in normal direction by where is the identity matrix of .

Definition 5 [Residual Error Estimators]. Let be the finite element solution of the problem (36)–(38) in . Then, the residual error estimator is locally defined by where

The global residual error estimator is given by

Furthermore, denote the local approximation terms by where

The global approximation term is defined by

Remark 6. The residual character of each term on the right-hand sides of (48)–(51) is quite clear since if would be the exact solution of (2)–(10), then they would vanish.

3.2. Analytical Tools

3.2.1. Inverse Inequalities

In order to derive the lower error bounds, we proceed similarly as in [29, 33] (see also [34]), by applying inverse inequalities, and the localization technique based on simplex-bubble and face-bubble functions. To this end, we recall some notation and introduce further preliminary results. Given , and , we let and be the usual simplexe-bubble and face-bubble functions, respectively (see (1.5) and (1.6) in [28]). In particular, satisfies , , , and . Similarly, , , and . We also recall from [35] that, given , there exists an extension operator that satisfies and . A corresponding vectorial version of , that is, the componentwise application of , is denoted by . Additional properties of , , and are collected in the following lemma (see [35]):

Lemma 7. Given , there exist positive constants depending only on and shape regularity of the triangulations (minimum angle condition), such that for each simplexe and there hold

Lemma 8 (Continuous trace inequality). There exists a positive constant depending only on such that

3.2.2. Clément Interpolation Operator

In order to derive the upper error bounds, we introduce the Clément interpolation operator that approximates optimally nonsmooth functions by continuous piecewise linear functions:

In addition, we will make use of a vector valued version of , that is,

, which is defined componentwise by The following lemma establishes the local approximation properties of (and hence of ), for a proof see ([36], Section 3).

Lemma 9. There exist constants , independent of , such that for all there hold where and .

3.2.3. Helmholtz Decomposition

Lemma 10 ([7]). There exists such that every can be decomposed as , where , , and

3.3. Error Equation

We set and .

Let and . For , we define the operator by

Then, the continuous problem (25)–(27) is equivalent to find such that for all , we have:

We define the discrete version by the same way: find such that for all ,

Since for all and , , then from (62), we obtain

Now, we set,

Then, from (64), we can write: where

3.4. Reliability of the A Posteriori Error Estimator

The first main result is given by the following theorem:

Theorem 11 (Upper Error Bound). Let be the exact solution and be the finite element solution. Then, there exist a positive constant such that the error is bounded globally from above by:

Proof. Let . We consider the residue Equation (65), and we set . We take with and . As , then by Lemma 3.4, admits the decomposition where and with and . We consider with and . Thus, . Now, we recall: Therefore, integrate by parts element by element we may write: where Thus, from Equations (66) and (70), we obtain Coercivity of operator leads to inf-sup condition: By consequently, the identities (65) and (72), inf-sup condition of operator (73), Cauchy-Schwarz inequality, estimation of Lemma 10, and the approximation properties of Lemma 9 imply the required estimate and finish the proof.

3.5. Efficiency of the A Posteriori Error Estimator

For , we set where

The error estimator is considered efficient if it satisfies the following theorem.

Theorem 12 (Lower Error Bound). Let be the exact solution and be the finite element solution. Then, there exist a positive constant such that the error is bounded locally from below for all by: where is a finite union of neighborning elements of and is the product norm.

Proof. The lower bound are proved using standard elementwise integration by parts, namely, error equation of Section 3.3 (i.e., identity (65) and Equation (72)) and some a inverse estimates of Lemma 7 (Cf. [37] for details).

4. Discussion

In this paper, we have discussed a posteriori error estimates for a finite element approximation of the Stokes-Biot system. A residual type a posteriori error estimator is provided, that is both reliable and efficient.

There are many issues to be addressed in this area such as the other types of a posteriori error estimates and related implementation of the adaptive finite element methods.

Further, it is well known that an internal layer appears at the interface as the permeability tensor degenerates; in that case, anisotropic meshes have to be used in this layer (see for instance [10]). Hence, we intend to extend our results to such anisotropic meshes.

5. Nomenclature

(i) bounded domain(ii) The poroelastic medium domain(iii)(iv)(v) (vi) (resp. ): The unit outward normal vector along (resp. )(vii): The fluid velocity in (viii): The fluid pressure in (ix): The fluid velocities in (x): The fluid pressure in (xi)In , the curl of a scalar function is given as usual by (xii)In , the of a vector function is given as usual by , namely, (xiii): The space of polynomials of total degree not larger than (xiv): Triangulation of (xv): The corresponding induced triangulation of , (xvi)For any , is the diameter of , and is the diameter of the largest ball inscribed into (xvii)(xviii): The set of all the edges or faces of the triangulation(xix): The set of all the edges () or faces () of an element (xx)(xxi): The set of all the vertices of an element (xxii)(xxiii)For , (xxiv)For , we associate a unit vector such that is orthogonal to and equals to the unit exterior normal vector to , (xxv)For , is the jump across in the direction of (xxvi)In order to avoid excessive use of constants, the abbreviations and stand for and , respectively, with positive constants independent of , or (xxvii)(xxviii)(xxix)(xxx)(xxxi): Product norm on (xxxii): Local product norm on .

Data Availability

There are no data underlying the findings in this paper to be shared.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Houedanou Koffi Wilfrid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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