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Parameter-robust methods for the Biot-Stokes interfacial coupling without

Lagrange multipliers

Wietse M. Boon

a,1

, Martin Hornkjøl

, Miroslav Kuchta

c,2,∗

, Kent-Andr´e Mardal

b,c,3

, Ricardo

Ruiz-Baier

d,e,4

KTH Royal Institute of Technology, Lindstedtsv¨agen 25, 114 28, Stockholm, Sweden

Department of Mathematics, University of Oslo, Norway

Simula Research Laboratory, 0164 Oslo, Norway

School of Mathematics, Monash University, 9 Rainforest Walk, Melbourne 3800 VIC, Australia

Institute of Computer Science and Mathematical Modelling, Sechenov University, Moscow, Russian Federation

Abstract

In this paper we advance the analysis of discretizations for a ﬂuid-structure interaction model of the mono-

lithic coupling between the free ﬂow of a viscous Newtonian ﬂuid and a deformable porous medium separated

by an interface. A ﬁve-ﬁeld mixed-primal ﬁnite element scheme is proposed solving for Stokes velocity-

pressure and Biot displacement-total pressure-ﬂuid pressure. Adequate inf-sup conditions are derived, and

one of the distinctive features of the formulation is that its stability is established robustly in all material

parameters. We propose robust preconditioners for this perturbed saddle-point problem using appropriately

weighted operators in fractional Sobolev and metric spaces at the interface. The performance is corroborated

by several test cases, including the application to interfacial ﬂow in the brain.

Keywords: Transmission problem, Biot-Stokes coupling, total pressure, mixed ﬁnite elements, operator

preconditioning, brain poromechanics.

2000 MSC: 65N60, 76S05, 74F10, 92C35.

1. Introduction

1.1. Scope

We address the construction of appropriate monolithic solvers for multiphysics ﬂuid-poromechanical cou-

plings interacting through an interface. Particular attention is payed to tracking parameter dependence of

the continuous and discrete formulations so that the resulting numerical methods are robust with respect to

typical scales in material constants spanning over many orders of magnitude. We adopt a multi-domain ap-

proach, where appropriate conditions for the coupling through the shared interface need to be imposed. We

use the conditions proposed in [1] (although, other forms and dedicated phenomena could be incorporated

∗

Corresponding author

Email addresses: wietse@kth.se (Wietse M. Boon), marhorn@math.uio.no (Martin Hornkjøl), miroslav@simula.no

(Miroslav Kuchta), kent-and@math.uio.no (Kent-Andr´e Mardal), ricardo.ruizbaier@monash.edu (Ricardo Ruiz-Baier)

WMB was supported by the Dahlquist Research Fellowship.

MK acknowledges support from the Research Council of Norway (NFR) grant 303362.

KAM acknowledges support from the Research Council of Norway, grant 300305 and 301013.

RRB acknowledges support from the Monash Mathematics Research Fund S05802-3951284 and from the Ministry of Science

and Higher Education of the Russian Federation within the framework of state support for the creation and development of

World-Class Research Centers “Digital biodesign and personalised healthcare” No. 075-15-2020-926.

Preprint submitted to Computer Methods in Applied Mechanics and Engineering November 11, 2021

arXiv:2111.05653v1 [math.NA] 10 Nov 2021

without much eﬀort, such as ﬂuid entry resistance [2, 3]). The recent literature contains various numer-

ical methods for (Navier-)Stokes/Biot interface formulations including mixed, double mixed, monolithic,

segregated, conforming, non-conforming, and DG discretizations [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

In [12, 16] (and starting from the Biot-Stokes equations advanced in [5, 17]) the authors rewrite the

poroelasticity equations using displacement, ﬂuid pressure and total pressure (also as in the poromechanics

formulations from [18, 19, 20]). Since ﬂuid pressure in the poroelastic domain has suﬃcient regularity, no

Lagrange multipliers are needed to enforce the coupling conditions, which resembles the diﬀerent formulations

for Stokes-Darcy advanced in [21, 22, 23, 24]. Another advantage of the three-ﬁeld Biot formulation is its

robustness with respect to the Lam´e constants of the poroelastic structure. This robustness is of particular

importance when we test the ﬂow response to changes in the material properties of the skeleton and when the

solid is nearly incompressible. The work [12] focuses on the stability analysis and its precise implications on

the asymptotics of the interface conditions when the permeability depends on porosity heterogeneity, whereas

[16] addresses the stability of the semi- and fully discrete problems, and the application to interfacial ﬂow

in the eye. Here, we extend these works by concentrating on deriving robust stability, on designing eﬃcient

block preconditioners (robust with respect to all material parameters) following the general theoretical

formalism from [25], and on the simulation of free ﬂow interacting with interstitial ﬂow in the brain. In

such a context (and in the wider class of problems we consider in this paper), tissue permeability is of

the order of 10

−15

, and the incorporation of tangential interface transmission conditions usually involves

terms that scale inversely proportional to the square root of permeability. Moreover, the solid is nearly

incompressible, making the ﬁrst Lam´e parameter signiﬁcantly larger than the other mechanical parameters

and exhibiting volumetric locking for some types of displacement-based formulations. Other ﬂow regimes

that are challenging include low-storage cases [26]. It is then important that the stability and convergence

of the numerical approximations are preserved within the parameter ranges of interest.

Here we follow [27, 28, 20, 29] and use parameter-weighted norms to achieve robustness. However, as we

will see, combining proper preconditioners for Stokes and Biot single-physics problems is not suﬃcient for the

interface coupled problem. In fact, the condition number of the preconditioned system, although robust in

mesh size, grows like the square root of the ratio between ﬂuid viscosity and permeability. This phenomenon

is demonstrated in Example 2.1, below. That is, the eﬃciency of seemingly natural preconditioners varies

with the material parameters. In order to regain stability with respect to all parameters, we include both an

additional fractional term involving the pressure and a metric term coupling the tangential ﬂuid velocity and

solid displacement at the interface, hereby increasing the regularity at the interface in a proper parameter

dependent manner. This strategy draws inspiration from similar approaches employed in the design of robust

solvers for Darcy and Stokes-Darcy couplings [30, 21, 23].

1.2. Outline

We have organised the contents of this paper in the following manner. The remainder of this section

contains preliminaries on notation and functional spaces to be used throughout the manuscript. Section 2

outlines the main details of the balance equations, stating typical interfacial and boundary conditions,

and restricting the discussion to the steady Biot-Stokes coupled problem. There we also include the weak

formulation and demonstrate that simple diagonal preconditioners based on standard norms do not lead

to robustness over the whole parameter range. This issue is addressed in Section 3 where we show well-

posedness of the system using a global inf-sup argument with parameter weighted operators in fractional

spaces, which in turn assist in the design of robust solvers by operator preconditioning. Section 4 discusses

ﬁnite element discretization of the coupled problem using both conforming and non-conforming elements;

and it also contains numerical experiments demonstrating robustness of the fractional preconditioner and its

feasibility for simpliﬁed simulations of interfacial ﬂow in the brain.

1.3. Preliminaries

Let us consider a spatial domain Ω ⊂ R

, where d = 2, 3, disjointly split into Ω

and Ω

. These sub-

domains respectively represent the region ﬁlled with an incompressible ﬂuid and the elastic porous medium

Ω

1,P

2,P

1,F

2,F

Ω

1,P

2,P

1,F

2,F

Figure 1.1: A conﬁguration of subdomains and boundary partition for the Biot-Stokes coupled problem. (Left) Setup assumed

for the analysis in Theorem 3.1. (Right) Example of another conﬁguration investigated by some of the numerical experiments

in Section 4.4.

(a deformable solid matrix or an array of solid particles). We will denote by n the unit normal vector on

the boundary ∂Ω, and by Σ = ∂Ω

∩ ∂Ω

the interface between the two subdomains, which is assumed

suﬃciently regular. We adopt the convention that on Σ the normal vector points from Ω

to Ω

. We also

deﬁne the boundaries Γ

= ∂Ω

\ Σ and Γ

= ∂Ω

\ Σ. The sub-boundary Γ

is further decomposed be-

tween Γ

and Γ

where we impose, respectively, no slip velocities and zero normal total stresses. Similarly,

we split Γ

into Γ

and Γ

where we prescribe zero traction and clamped boundaries, respectively. For

the analysis, the setup of trace spaces needs that dist(Σ, Γ

) > 0 and that dist(Σ, Γ

) > 0, which can be

satisﬁed if the interface meets the boundary at the Biot displacement and Stokes velocity boundaries (see

Figure 1.1, left), where Γ

= Γ

1,P

∪ Γ

2,P

and Γ

= Γ

1,F

∪ Γ

2,F

. Our numerical tests will also include cases

where the interface intersects boundaries Γ

and Γ

on the Stokes and Biot sides, respectively.

For generic Sobolev spaces X, Y and a scalar ζ > 0, the weighted space ζX refers to X endowed with the

norm ζk · k

. The intersection X ∩ Y provided with the norm kvk

X∩Y

= kvk

+ kvk

, is a Hilbert space

[30, 31].

Vector ﬁelds and vector-valued spaces will be written in boldface. In addition, by L

(Ω) we will denote

the usual Lebesgue space of square integrable functions and H

(Ω) denotes the usual Sobolev space with

weak derivatives of order up to m in L

(Ω), and H

(Ω) denotes its vector counterpart. In addition, L

(Σ)

will denote the space of functions z : Ω

→ R

, i ∈ {F, P }, such that z − n(z · n) ∈ L

(Σ).

An L

(Ω) (as well as L

(Ω)) inner product over a generic bounded domain Ω is denoted as (·, ·)

Ω

. The

symbol h·, ·i

will denote the pairing between the trace functional space H

1/2

(Σ) and its dual H

−1/2

(Σ), and

we will also write h·, ·i to denote other, more general, duality pairings. Moreover, for z ∈ H

(Ω

) ∩ L

(Σ),

its normal and tangential traces deﬁned by bounded surjective maps, will be denoted by T

z ∈ H

1/2

(Σ)

and T

z ∈ L

(Σ), respectively [23]. We also remark that, while H

1/2

(Σ) is a subspace of L

(Σ) as a set, the

tangential trace of z ∈ ζ

(Ω

) ∩ζ

(Σ) is in ζ

1/2

(Σ) ∩ζ

(Σ) and that in our setting it is important

to track the parameter dependence for the sake of robustness.

Pertaining to the poroelastic domain, and assuming momentarily that |Γ

| = 0, we recall from [32,

Section 2.4.2] the deﬁnition of the space

1/2

(Σ) = {η ∈ H

1/2

(Σ) : E

(η) ∈ H

1/2

(∂Ω

)}, (1.1)

supplied with the norm

kηk

1/2,00,Σ

:= kE

(η)k

1/2,∂Ω

where E

: H

1/2

(Σ) → H

1/2

(∂Ω

) denotes the extension-by-zero operator

(η) =

(

η on Σ,

0 on ∂Ω

\ Σ,

∀η ∈ H

1/2

(Σ).

Furthermore, the restriction of ψ ∈ H

−1/2

(∂Ω

) to Σ, deﬁned as

hψ|

, ηi

= hψ, E

(η)i

∂Ω

∀η ∈ H

1/2

(Σ),

belongs to the dual [H

1/2

(Σ)]

of H

1/2

(Σ). Its norm is

kψ|

−1/2,00,Σ

:= sup

06=η∈H

1/2

(Σ)

hψ|

, ηi

kηk

1/2,00,Σ

= sup

06=η∈H

1/2

(Σ)

hψ, E

(η)i

∂Ω

(η)k

1/2,∂Ω

. (1.2)

The boundary setup in Figure 1.1 is such that |Γ

|·|Γ

| > 0, and therefore a modiﬁcation of (1.1)-(1.2)

is required. With that purpose, we note that any η ∈ H

1/2

(Σ) can be continuously extended to ϕ in the

space

1/2

(Σ) := {η ∈ H

1/2

(Σ) : E

(η) ∈ H

1/2

(Γ

)}, (1.3)

where

Σ ∪

1,P

∪

2,P

(see Figure 1.1, left), and

(η) =

(

η on Σ,

0 on Γ

∀η ∈ H

1/2

(Σ).

This treatment can be interpreted as replacing H

1/2

(Σ) by the space of functions η ∈ H

1/2

(Σ) such that

−1/2

η ∈ L

(Σ), where ξ is a suﬃciently regular trace function, positive on Σ, and vanishing only on Γ

(see, for example, [33]).

The norm of the resulting extension ϕ ∈ H

1/2

(Σ) is deﬁned analogously as before,

kϕk

1/2,01,Σ

:= kE

(ϕ)k

1/2,Γ

, (1.4)

and therefore the restriction of a distribution to the interface, ψ|

, is in the dual space [H

1/2

(Σ)]

and its

norm is

kψ|

−1/2,01,Σ

:= sup

06=η∈H

1/2

(Σ)

hψ|

, ηi

kηk

1/2,01,Σ

= sup

06=η∈H

1/2

(Σ)

hψ, E

(η)i

(η)k

1/2,Γ

. (1.5)

2. Governing equations and weak formulation

The momentum and mass balance equations for the ﬂow in the ﬂuid cavity are given by Stokes equations

written in terms of ﬂuid velocity u and ﬂuid pressure p

, whereas the non-viscous ﬁltration ﬂow through

the porous skeleton can be described by Darcy’s law in terms of pressure head p

, and the porous matrix

elastostatics are stated in terms of the solid displacement d. The coupled Biot-Stokes equations arising after

a backward Euler semi-discretization in time, with time step ∆t, read

−div[2µ

(u) −p

I] = ρ

g in Ω

, (2.1a)

div u = 0 in Ω

, (2.1b)

−div[2µ

(d) −ϕI] = ρ

f in Ω

, (2.1c)

ϕ −αp

+ λ div d = 0 in Ω

, (2.1d)





∆t

−

(∆t)λ

ϕ −div



∇p

− ρ



= m

in Ω

, (2.1e)

where f is a vector ﬁeld of body loads, g is the gravity acceleration, µ

is the ﬂuid viscosity, (u) =

(∇u + ∇u

) is the strain rate tensor, and (d) =

(∇d + ∇d

) is the inﬁnitesimal strain tensor, ρ

, ρ

are

the density of the ﬂuid and solid, respectively, λ, µ

are the ﬁrst and second Lam´e constants of the solid, κ is

the heterogeneous tensor of permeabilities (satisfying |w|

. w ·κ(x)w a.e. in Ω

and for all w ∈ R

); m

is a source/sink term for the ﬂuid pressure (which also includes pressures in the previous backward Euler

time step); and C

, α are the total storage capacity and Biot-Willis poroelastic coeﬃcient. Here we have

used the total pressure ϕ := αp

− λ div d, as an additional unknown [27, 20].

We furthermore supply boundary conditions as follows

u = 0 on Γ

, (2.2a)

[2µ

(u) −p

I]n = 0 on Γ

, (2.2b)

d = 0 and

∇p

· n = 0 on Γ

, (2.2c)

[2µ

(d) −ϕI]n = 0 and p

= p

on Γ

. (2.2d)

In order to close the system, we consider the classical transmission conditions on Σ accounting for the

continuity of normal ﬂuxes, momentum balance, equilibrium of ﬂuid normal stresses, and the so-called

Beavers-Joseph-Saﬀman condition for tangential ﬂuid forces [7, 2], which in the present setting reduce to

u ·n = (

∆t

d −

∇p

) ·n on Σ, (2.3a)

(2µ

(u) −p

I)n = (2µ

(d) −ϕI)n on Σ, (2.3b)

−n ·(2µ

(u) −p

I)n = p

on Σ, (2.3c)

−n ·(2µ

(u) −p

I)t

γµ

√

(u −

∆t

d) ·t

, 1 ≤ j ≤ d −1 on Σ, (2.3d)

where γ > 0 is the slip rate coeﬃcient depending on the geometry of the domain, and we recall that the

normal n on the interface is understood as pointing from the ﬂuid domain Ω

towards the porous structure

Ω

, while t

, 1 ≤ j ≤ d −1 are orthonormal tangent vectors on Σ, normal to n.

We proceed to test (2.1a)-(2.1e) against suitable smooth functions and to integrate over the corresponding

subdomain. The challenging model parameters are µ

, C

, λ, γ, α, and the magnitude of κ. Therefore, we

will concentrate on the speciﬁc case where ∆t = 1, and the model parameters (in particular κ) are spatially

constant. Following [24], after applying integration by parts wherever adequate and using the transmission

conditions (2.3a)-(2.3d), we arrive at the following remainder on the interface

, (v − w) ·ni

γµ

√

d−1

j=1

h(u −d) ·t

, (v − w) ·t

+ h(u − d) · n, q

which is well-deﬁned and therefore no additional Lagrange multipliers are required to realize the coupling

conditions. Also, in view of the character of the resulting variational forms in combination with the speciﬁ-

cation of boundary conditions (2.2), we deﬁne the Hilbert spaces

(Ω

) = {v ∈ H

(Ω

) : v|

= 0}, H

(Ω

) = {w ∈ H

(Ω

) : w|

= 0},

(Ω

) = {q

∈ H

(Ω

) : q

= 0},

and the product space H

H = H

(Ω

) ×H

(Ω

) ×L

(Ω

) ×L

(Ω

) ×H

(Ω

). (2.4)

Consequently, we have the following weak form for the Biot-Stokes coupling: Find (u, d, p

, ϕ, p

) ∈ W

such that

2µ

((u), (v))

Ω

γµ

√

(u −d), T

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