Numerical simulation of blood flows through a porous interface

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DOI:10.1051/m2an:2008031 www.esaim-m2an.org

NUMERICAL SIMULATION OF BLOOD FLOWS THROUGH A POROUS INTERFACE

Miguel A. Fernández

, Jean-Frédéric Gerbeau

and Vincent Martin

Abstract. We propose a model for a medical device, called a stent, designed for the treatment of cerebral aneurysms. The stent consists of a grid, immersed in the blood flow and located at the inlet of the aneurysm. It aims at promoting a clot within the aneurysm. The blood flow is modelled by the incompressible Navier-Stokes equations and the stent by a dissipative surface term. We propose a stabilized finite element method for this model and we analyse its convergence in the case of the Stokes equations. We present numerical results for academical test cases, and on a realistic aneurysm obtained from medical imaging.

Mathematics Subject Classification. 65M60, 74K25, 76D05, 76Z05.

Received February 5, 2007.

Published online August 12, 2008.

Introduction

This work is motivated by the numerical simulation of a new medical device³designed for the treatment of cerebral aneurysms located on bifurcations of arteries. This device consists of a wire metal mesh tube, called a stent. Contrary to the usual stents – which are typically used to keep arteries open and which are located on the vessel wall – this stent is immersed in the blood flow (Fig. 1). The purpose of this device is to reduce the flux within the aneurysm in order to occlude it by a clot. For practical reasons, a portion of the stent is also present in front of collateral arteries, with a risk of adverse effect in the blood flows. The motivations of modelling are, first, quantify the desired reduction of vorticity and shear stress in the aneurysms and, second, the non-desired reduction of blood flows in collateral branches.

For reasons that will be developed in Section 1, the geometrical details of the stent wires are ignored in this study. Our model thus consists in an homogenized porous interface immersed in the flow. From the mathematical standpoint, the flow is assumed to be governed by the incompressible Navier-Stokes equations and the stent is modelled by a dissipative surface term added to the left-hand side of the equations.

This additional dissipative term induces a jump of the stress across the stent surface which may raise numerical issues. On the one hand, continuous approximations of the pressure may give very inaccurate results, on the other hand, discontinuous approximations of the pressure typically lead to expensive simulations. To circumvent

Keywords and phrases.Stabilized finite element, sieve problem, blood flow, terminal aneurysm, stent, fluid-structure interaction.

1 INRIA Paris-Rocquencourt, BP 105, 78153 Le Chesnay, France.[email protected];

[email protected]

2University of Technology of Compiègne, LMAC, GI, Royallieu, BP 20529, 60205 Compiègne, France. [email protected] 3The LyLyk^c device, by Cardiatis.

Article published by EDP Sciences c EDP Sciences, SMAI 2008

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Stentγ

Inlet

Outlet Outlet

Aneurysm

Figure 1. Example of a two-dimensional domainΩwith a stent (“LyLyk” device by Cardiatis) represented by a lineγ. The part of the stent that lies on the arterial wall (vertical artery) is not represented.

these issues, we propose to use stabilized finite elements, with continuous pressure, such asP1/P1elements, and to introduce a “fissure” in the mesh on the stent surface, in order to allow for the pressure to be discontinuous at the interfaceonly. The proposed stabilization formulation is made of two contributions: a standard residual based stabilization giving a L²-control on the pressure gradient; an interface based stabilization providing a L²-control on the jump of the pressure. In Section 2, a complete convergence analysis and an optimal error estimate are presented for the Stokes equations.

Numerical results are presented in Section3. The theoretical convergence rate of Section 2is confirmed on an academical example where the solution is known in a closed-form. We also propose a validation test on a simple two dimensional configuration. We end the paper with results on realistic three dimensional geometries obtained from medical imaging. In that case, we also take into account the fluid-structure interaction of the blood with the wall artery.

1. Motivations and modelling

The purpose of the stent considered in this work is to treat intra-cranial terminal aneurysm located at an artery bifurcation (see e.g. [25]). Contrary to the stents commonly used to treat stenoses or side aneurysms, this stent is closed at one end. Moreover, it is characterized by very thin wires (40μm), very small windows (100 μm) and a multilayer structure. It has the shape of a small, finely woven metallic socket, whose tip is intended to be inserted inside the aneurysm, while the sleeve is in contact with the main artery. The part inside the aneurysm is finely braided, in order to reduce blood flow, and hopefully cause a thrombosis inside the aneurysm. The lateral part is more coarsely braided, in order to let the blood flow into the two daughter arteries.

In general, in finite element studies of stents in blood flows, each wire of the stent is meshed (see for instance [2,26]). This approach requires an important work to generate the mesh, and is computationally heavy.

Its interest is to provide a precise description of the local flow alterations caused by the stent. In our case, this approach is very expensive due to the very small windows of the stent and the complex multilayer structure.

Moreover from the modelling viewpoint, it would be questionable to claim that one solves a resolution of 40μm while neglecting the red blood cells (8μm diameter) which occupy about 50% of the blood volume. This is why we preferred to model the stent “macroscopically” by a mean porous surface immersed in the flow. Note that

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Γ

Ω

Γ

∂γ

n=n1=−n2

γ Ω

Figure 2. The domainΩdecomposed into two subdomainsΩ₁ andΩ₂, separated by the stentγ.

we neglect any other physical aspects of the stent: deformation of its structure, interaction with the vessel wall, mechanical stability of the stent in the artery, etc.

Let Ω be a simply connected smooth domain in Rⁿ, n = 2 or 3, and Γ = ∂Ω its regular boundary. We suppose (see Fig.1) that the stent can be represented by a regular surface γimmersed inΩ.

Let uandpbe the velocity and the pressure of the fluid. The strain rate tensor D and the Cauchy stress tensorTassociated withu, pare defined by

D(u) =1 2

∇u+∇u

andT(u, p) =−pI+ 2μD(u),

whereμis the dynamic viscosity of the fluid,uis the velocity,pthe pressure andIis the identity matrix inRⁿ. The blood is assumed to be governed by the incompressible Navier-Stokes equations. This is a commonly accepted hypothesis in the large vessels we are considering. To model the stent, we propose to add a dissipative surface term to the conservation of momentum equations. Thus, the model formally reads: find uandpsuch that

ρf

∂u

∂t +u· ∇u

−divT(u, p) +R_γuδγ = f in Ω, divu = 0 in Ω,

(1.1) where ρf is the density of the fluid, and f is the body force density, Rγ is a symmetric and positive definite tensor that represents the dissipation due to the stent, and δγ is the Dirac measure on the stent surface γ. In other words, forv∈[H¹(Ω)]ⁿ,

Rγuδγ,v=

γ

Rγu·vdγ.

A comment on this model is in order: the 3D Navier-Stokes equations give a precise information on local scales while the stent is governed by a rather rough model. This discrepancy may be considered as a weakness of the model. We nevertheless believe that our choice, in spite of its obvious limitations, may be relevant: on the one hand, we indeed need the local information provided by the Navier-Stokes equations (e.g. to evaluate the wall shear stress which is known to be important in hemodynamics), on the other hand, we cannot afford to resolve a space resolution of a 40μm wire and we are mainly interested in global effects induced by the presence of the stent (e.g. vorticity reduction in the aneurysm, flux modification in the branches).

2. Numerical analysis in the case of Stokes flows

In this section, to simplify the numerical analysis and to explain how to compute the permeability tensor from homogenization results, we make two important assumptions: first we assume the fluid to be governed by the stationary Stokes equations; second we suppose that the stent is represented by a hyperplaneγ (see Fig.2)

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that divides the domainΩinto two connected subdomains

Ω_f = Ω\γ= Ω₁∪Ω₂, Ω₁∩Ω₂=∅.

We denote byni the outward normal onγ viewed as a part of theΩ_i boundary, for i= 1,2. We denote by Γ_i the part of the boundary ofΩ_i in common with the boundary ofΩ:

Γ_i=∂Ω_i∩Γ, i= 1,2. We also introduce

ui=u|Ωi, pi=p|Ωi, fi=f|Ωi, fori= 1,2.

We will consider the usual Sobolev spacesH^m(O),m≥0, for a given bounded open setO ⊂R^d,1≤d≤n. In particular, we haveL²(O) =H⁰(O). The scalar product inL²(O)is denoted by(·,·)O and its norm by · 0,O. The closed subspacesH₀¹(O), consisting of functions inH¹(O)with zero trace on∂O, andL²₀(O), consisting of function inL²(O)with zero mean inO, will also be used. We define the following notations:

M =L²(Ω), M˜ =L²₀(Ω), V= [H₀¹(Ω)]ⁿ,

equipped with their usual norms · 0,Ω and · 1,Ω. The subscript Ωwill in general be omitted, when the norm is taken over the whole domainΩ. We assume that the right-hand sidef ∈[L²(Ω)]ⁿ.

2.1. Problem setting

The Stokes counterpart of problem (1.1) reads in a two-domain formulation:

−μΔui+∇pi = fi inΩ_i, i= 1,2, divui = 0 inΩ_i, i= 1,2,

u₁ = u₂ onγ, μ∇u1· n1−p1n1+μ∇u2·n2−p2n2 = −Rγu onγ,

ui = 0 onΓ_i, i= 1,2,

(2.1)

where homogeneous Dirichlet boundary conditions onΓ_i, i= 1,2 have been chosen for simplicity. Letv in V andvi =v|_Ωi, i= 1,2. Multiplying the first equations of (2.1) byvi, i= 1,2, integrating by parts and adding the results, we readily obtain

2 i=1

(μ∇ui,∇vi)_Ω

i−(pi,divvi)_Ω

i−(μ∇ui· ni−pini,vi)_γ = (f,v)_Ω. (2.2) Using (2.1)₄ and the regularity ofuand v, we obtain a variational formulation of problem (2.1): find u∈V andp∈M˜ such that

(μ∇u,∇v)_Ω+ (Rγu,v)_γ−(divv, p)_Ω = (f, v)_Ω ∀v∈V,

(divu, r)_Ω = 0 ∀r∈M .˜ (2.3)

With the above assumptions on the data, we can state the following result.

Proposition 2.1. Problem (2.3)has a unique solution.

Proof. We define the bilinear forma(·,·)on[H₀¹(Ω)]ⁿ×[H₀¹(Ω)]ⁿ by a(u,v) = (μ∇u,∇v)_Ω+ (Rγu,v)_γ.

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Oε

rε ε

Figure 3. Planar Dirichlet sieve: regularly spaced obstacles where the flow is zero.

This bilinear form is continuous (consequence of the trace theorem) and coercive (consequence of the Poincaré inequality and the positiveness of Rγ). This result is therefore a direct consequence of the standard theory of

the Stokes equations (see [19]).

2.2. Microstructure and non-scalar resistivity tensor

The heuristic model proposed above can be rigorously justified in some cases. Moreover, an explicit form of the resistivity tensor Rγ can be obtained from extra computations, thus linking the microstructure of the sieve to its macroscopic impact. In this work, we did not use directly these results since we based our resistance tensor on simpler experimental data. Nevertheless, we find interesting to give some ideas about the non-scalar tensors that could be derived from a known microscopic structure.

Consider the 3D Stokes flow through a planar sieve made of a set of periodically spaced 2D identical obstacles and assume no-slip boundary conditions on the obstacles (see Fig.3). Then, three main asymptotic behaviors can be observed depending upon the limitα= lim_ε→0(rε/ε²)of the ratio of the radiusrεof the obstacles over the square of the spatial periodicity ε:

• when α= 0, the obstacles are asymptotically not seen by the flow: the limit problem is the Stokes problem in the whole domain;

• when α = +∞, the obstacles become asymptotically a wall, and the limit problem consists of two independent Stokes problems with no-slip conditions on the interface;

• when0< α <+∞, the limit problem is precisely (2.1).

In that last case, the resistivity tensor Rγ can be computed from the following cell-problem. LetObe the unit obstacle in[−1,1]×[−1,1]. We denote byek thek-th canonical vector ofR³and we define (w^k, q^k), for k= 1,2,3, as the solution of the Stokes problem:

⎧⎨

⎩

−Δw^k+∇q^k = 0 in R³\O, divw^k = 0 in R³\O,

w^k = e^k onO, (2.4)

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withlim_|y|→∞w^k(y) = 0,lim_|y|→∞∇w^k(y) = 0andlim_|y|→∞q^k(y) = 0. Then the permeability tensor is given by:

Rγ =αμ 4

R³\O∇w^k· ∇w^ldy

k,l=1,2,3

.

It can be proved that Rγ is symmetric and positive definite. The proofs of these statements can be found in [1,4]. We also refer the reader to [10,11,23].

2.3. Stabilized finite element approximation

The classical Galerkin method applied to (2.3) requires the fulfillment of a inf-supcondition [19] between the velocity and pressure spaces, which leads to formulations involving mixed interpolations [3]. From the computational point of view, it is more convenient to deal with equal order velocity-pressure interpolations, but in that case, stability has to be enforced in another way. A standard approach consists in usingstabilizedfinite element methods where some terms are added to the standard Galerkin formulation in order to enhance the stability of the method (see e.g.[5,9,20,28]). For instance, in [20,28] stability in achieved by adding a residual based term which gives L²-control of the gradient of the pressure. Optimal error estimates, for arbitrary polynomial order k ≥ 1, can be derived assuming that the solution is smooth enough, typically (u, p) ∈ [H^k+1(Ω)]ⁿ×H^k(Ω). In this work, we extend this approach to our specific situation.

It is important to notice that, due to the presence of the interface resistive term, one cannot expect more than(u, p)∈[H¹(Ω)]ⁿ×L²(Ω) as global regularity of the solution of (2.3). As a result, some additional cares have to be taken in order to approximate (2.3) by finite elements. On the other hand it seems reasonable to assume that

(ui, pi)∈[H^k+1(Ω_i)]ⁿ×H^k(Ω_i), i= 1,2. (2.5) In this section, we propose a conforming stabilized finite element method, that allows equal order interpo- lations, and for which we can prove optimal error estimates under this reduced regularity. The key ingredient consists in combining the techniques of [20,28] with an interface based stabilization allowing pressure disconti- nuities through the interfaceγ.

2.3.1. Preliminaries

In what follows, we shall assumeΩto be a Lipschitz-continuous domain in Rⁿ (n= 2or 3) with a polyhe- dral boundary∂Ω and outward pointing normaln. We shall also assume the viscosityμ to be constant. Let {Th}_0<h≤1denotes a regular family of triangulations of the domainΩ(in the sense of [7]). For each triangula- tionTh, the subscripth∈(0,1]refers to the level of refinement of the triangulation, which is defined by

h= max

T∈ThhT,

withhT, the diameter ofT. We shall assume for the sake of simplicity the quasi-uniformity of the triangulation, i.e. there exist two positive constants Cmin andCmax such that

CminhT ≤h≤CmaxhT, ∀T ∈ Th.

Moreover, for all 0 < h ≤1, the triangulation Th is supposed to be conforming with the interface γ. Let G_h be the set of inter-element boundaries of T_h (faces in 3D, edges in 2D) lying onγ. For a given piecewise continuous functionϕ, the jumpϕnover an edgeE∈ G_his defined by

ϕn(x) = lim

t→0⁺

ϕ(x+tn¹_E)n¹_E+ϕ(x+tn²_E)n²_E ,

where x ∈ E, and n¹_E (respectively n²_E) is the unit normal vector to E pointing outward Ω₁ (respectively outwardΩ₂).

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We consider equal order approximations of order k ≥ 1 for the velocity and the pressure. Both velocity and pressure approximations will be continuous at inter-element boundaries, except for the pressure that will be discontinuous on the faces in 3D (on the edges in 2D) of the interfaceγ. Thus, we introduce the following velocity and pressure discrete spaces:

V^k_h =

vh∈ C⁰( ¯Ω) : ∀T ∈ Th vh|T ∈(Pk)ⁿ

∩V, M_h^k =

qh∈M˜ : p|Ωi ∈ C⁰( ¯Ω_i), i= 1,2, ∀T ∈ Th qh|T ∈Pk

.

Finally, fori= 1,2, letM_h,i^k the space of the restrictions of elements ofM_h^k toΩ_i, M_h,i^k =

qh|Ωi : qh∈M_h^k .

In order to approximate (non-smooth) functions of [H₀¹(Ω)]ⁿ with n≥2, which are not in the domain of the Lagrange interpolation operator, we consider the Scott-Zhang interpolation operatorSZ_h^k ontoV^k_h (see [24]).

The following error estimate then holds (see [24] or [13], p. 70),

u−SZ_h^ku_0,Ω≤c0hu_1,Ω. (2.6)

In particular, using a trace inequality (see [8,27]), we also have

u−SZ_h^ku0,E ≤C0h^1/2u1,Ω. (2.7)

Let I_h^k be the Lagrange interpolation operator onto V^k_h. Thus, we have the following estimates (see [13]), for T ∈ Th, and0≤m≤2,2≤l≤k+ 1:

u−I_h^kum,T ≤c1h^l−m||u||l,T u∈[H^l(T)]ⁿ. (2.8) Analogously, using a trace inequality, forE∈ Gh (E=T1∩T2, T1, T2∈ Th), we also have

u−I_h^ku0,E ≤c2h^l−1/2

i=1,2

ul,Ti u|Ti∈[H^l(Ti)]ⁿ, i= 1,2, u∈[H¹(T1∪T2)]ⁿ. (2.9)

Fori= 1,2letJ_h,i^k :L²(Ω_i)→M_h,i^k be theL²-projection ontoM_h,i^k . Thus, for0≤m≤1,1≤l≤k+ 1, we have pi−Jh,ipim,Ωi ≤c3h^l−mpil,Ωi pi∈H^l(Ω_i),

pi−Jh,ipi0,γ≤c4h^l−1/2pil,Ωi pi∈H^l(Ω_i). (2.10) We then introduce the (global) operatorJ_h^k:L²(Ω)→M_h^k, defined by

(J_h^kp)|Ωi=J_h,i^k (p|Ωi), i= 1,2. From (2.10), it then follows that for0≤m≤1,1≤l≤k+ 1,

p−J_h^kp_m,Ωi ≤c3h^l−mp_l,Ωi p∈L²(Ω), p|_Ωi∈H^l(Ω_i), i= 1,2, p−J_h^kp0,γ ≤c4h^l−1/2

i=1,2

pl,Ωi p∈L²(Ω), p|Ωi∈H^l(Ω_i), i= 1,2. (2.11)

Using the quasi-uniformity of the mesh, the following inverse and trace estimates hold (see [8,13]),

Δvh0,T≤c5h⁻¹∇vh0,T vh∈V^k_h, (2.12)

∇vh·n0,E≤c6h^−1/2vh1,T1∪T2 vh∈V^k_h. (2.13)

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Finally, we also have the standard Sobolev trace inequality

u|γ0,γ ≤cγu1,Ω u∈V.

2.3.2. A stabilized finite element method

Our finite element approximation for problem (2.3) reads: find(uh, ph)∈V^k_h×M_h^k such that Bδ,Rγ

(uh, ph),(vh, qh)

=Fδ

(vh, qh)

, ∀(vh, qh)∈V^k_h×Mh^k, (2.14) with

Bδ,Rγ

(u, p),(v, q)

= (μ∇u,∇v)Ω+ (Rγu,v)γ−(p,divv)Ω+ (q, divu)Ω

+δ

T∈Th

h²

μ (−μΔu+∇p,∇q)_T

−δ

E∈Gh

h μ

μ∇u·n−pn+Rγu,qn

E,

Fδ

(v, q)

= (f,v)Ω+δ

T∈Th

h²

μ (f, ∇q)_T,

(2.15)

andδ >0 is a parameter independent ofhand that will be determined in Theorem2.7.

The stabilization is made of two contributions: a residual based stabilization [20,28] giving aL²-control on the pressure gradient, and an interface based stabilization providingL²-control on the jumps of the pressure. Both stabilizing terms seem necessary to establish an inf-sup condition independent of the discretization parameter (see Prop.2.3).

The following proposition states the strong consistency of the discrete formulation (2.14).

Proposition 2.2. Assume that(u, p)∈[H¹(Ω)]ⁿ×L²(Ω)and let(uh, ph)∈V^k_h×M_h^k be the solution of (2.14).

Then

Bδ,Rγ

(u−uh, p−ph),(vh, qh)

= 0, ∀(vh, qh)∈V^k_h×M_h^k. (2.16) Proof. It is a direct consequence of the strong consistency of the standard Galerkin method, theγ-conformity of the triangulationThand the fact that the solution(u, p)satisfies

f+μΔu− ∇p= 0, in [D(Ω_i)]ⁿ, i= 1,2, μ∇u·n−pnγ+Rγu= 0, in [H^−1/2(γ)]ⁿ. In particular, notice that these expressions and the regularity off yield

μΔu− ∇p∈[L²(T)]ⁿ, ∀T ∈ Th, μ∇u·n−pnE+Rγu∈[L²(E)]ⁿ, ∀E∈ Gh,

so that thebroken integralsin (2.15) are well defined.

2.3.3. Stability

In what follows we assume that there exist two positive constantsrmin andrmax such that

rmin|y|²≤y^TRγ(x)y≤rmax|y|², ∀x∈γ, ∀y∈Rⁿ. (2.17)

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The convergence of the discrete solution will be stated in terms of the following mesh-dependent norm on V^k_h×M_h^k:

|||(u, p)|||²_h=|μ^1/2u|²_1,Ω+r^1/2_minu²_0,γ+δ

T∈Th

h²

μ∇p²_0,T+δ

E∈Gh

h

μp²_0,E+μ^−1/2p²_0,Ω, (2.18) for all(u, p)∈V_h^k×M_h^k.

The main stability result for our method is stated in the following theorem:

Theorem 2.3. Assume that

0< δ≤ 1

c²₅+ 2c²₆, δh≤ μ rmin

2r_max² · (2.19)

Then, there exists a constantβ =β(δ, rmin, rmax, μ)>0 such that

(uh,ph)∈Vinf^k_h×M_h^k sup

(vh,qh)∈V^k_h×M_h^k

Bδ,Rγ

(uh, ph),(vh, qh)

|||(u_h, ph)|||_h|||(v_h, qh)|||_h ≥β.

Moreover,β =O(δ), ifδ1, andβ =O μ

rmin

, if rmin

μ 1.

Proof. Using the definition ofBδ,Rγ and the notations

x=

δ

T∈Th

h²

μ∇ph²_{T 1/2}

, y=

δ

E∈Gh

h

μph²_{E 1/2}

, (2.20)

we have

Bδ,Rγ

(uh, ph),(uh, ph)

≥ |μ^1/2uh|²_1,Ω+r^1/2_minuh²_0,γ+x²+y²+δ

T∈Th

h²(−Δuh,∇ph)_T

+δ

E∈Gh

h

μ(μ∇uh·n+Rγuh,−phn)E,

for all(uh, ph)∈V^k_h×M_h^k. Using the inverse estimates (2.12) and (2.13), it follows that Bδ,Rγ

(uh, ph),(uh, ph)

≥ |μ^1/2uh|²_1,Ω+r^1/2_minuh²_0,γ+x²+y²−δ^1/2c5(μ|uh|²_1,Ω)^1/2x

−δ^1/2c6(μ|uh|²_1,Ω)^1/2y−

δr²_maxh rminμ

_1/2

r^1/2_minuh_0,γy

≥

1−δ c²₅

2 +c²₆

|μ^1/2uh|²_1,Ω+

1−δr²_maxh rminμ

r^1/2_minuh²_0,γ +1

2x²+1 2y². Finally, using (2.19), we get

Bδ,Rγ

(uh, ph),(uh, ph)

≥ 1 2

|μ^1/2uh|²_1,Ω+r^1/2_minuh²_0,γ+x²+y²

, (2.21)

for all(uh, ph)∈V^k_h×M_h^k, which provides the coercivity of the stabilized bi-linear form Bδ,Rγ.

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Let now(uh, ph)∈V^k_h×M_h^k fixed. There exists av∈Vand a constantcΩ, which only depends onΩ, such that divv =−ph and v1,Ω≤ cΩph0,Ω, see for instance [19]. Let vh =SZ_h^kv. Using the H¹-stability of SZ_h^k (see [24] or [13], p. 71) we have

vh1,Ω≤cSZv1,Ω≤c_Ωph0,Ω.

Using partial integration element-wise and the continuity ofphin Ω₁andΩ₂, one obtains Bδ,Rγ

(uh, ph),(vh,0)

= (μ∇uh,∇vh)_Ω+ (Rγuh,vh)_γ−(ph,div v)_Ω

−

T∈Th

(∇ph, v−vh)_T +

E∈Gh

(phn,v−vh)_E.

Therefore, using Cauchy-Schwarz and approximation (2.6)-(2.7), one obtains Bδ,Rγ

(uh, ph),(vh,0)

≥ ph²_0,Ω− |μ^1/2uh|_1,Ω|μ^1/2vh|_1,Ω−rmax

r^1/2_minr_min^1/2uh_0,γvh_0,γ

−

⎡

⎣c0

T∈Th

h²∇ph²_{0,T 1/2}

+C0

E∈Gh

hph²_{0,E 1/2}⎤

⎦v1,Ω.

It then follows that Bδ,Rγ

(uh, ph),(vh,0)

≥ ph²_0,Ω−c_Ωμ^1/2|μ^1/2uh|1,Ωph0,Ω−c_Ωcγrmax

r_min^1/2r^1/2_minuh0,γph0,Ω

−μ δ

_1/2

c0cΩxph_0,Ω−μ δ

_1/2

C0cΩyph_0,Ω

≥ ph²_0,Ω−cδμ^1/2

|μ^1/2uh|1,Ω+r^1/2_minuh0,γ+x+y

ph0,Ω

≥1

2ph²_0,Ω−2c²_δμ

|μ^1/2uh|²_1,Ω+r^1/2_minuh²_0,γ+x²+y²

, (2.22)

where

cδ = max

c_Ω, c_Ωcγ rmax

μ^1/2r^1/2_min, δ^−1/2c0cΩ, δ^−1/2C0cΩ

.

Multiplying (2.21) by(1−ρ)and (2.22) byρ, and adding the results, one obtains Bδ,Rγ

uh, ph

,

(1−ρ)uh+ρvh,(1−ρ)ph

= (1−ρ)Bδ,Rγ

uh, ph

, uh, ph

+ρBδ,Rγ

uh, ph

, vh,0

≥ 1−ρ

2 −2ρc²_δμ μ|uh|²₁+(rmin)^1/2u²_0,γ+x²+y²

+ρμ

2 μ^−1/2ph²_0,Ω. So that, if we take

0< ρ= 1

1 + 4c²_δμ+μ <1, we then obtain that

Bδ,Rγ

uh, ph

,

(1−ρ)uh+ρvh,(1−ρ)ph

≥ μ

2 + 8c²_δμ+ 2μ|||(uh, ph)|||²_h. (2.23)

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Moreover,

|||

(1−ρ)uh+ρvh,(1−ρ)ph

|||

h≤(1−ρ)|||(uh, ph)|||_h+ρ|||(vh,0)|||_h

≤(1−ρ)|||(u_h, ph)|||_h+ρ

μ|v_h|²_1,Ω+r^1/2_minv_h²_0,γ1/2

≤(1−ρ)|||(uh, ph)|||_h+ρ√

2cδμμ^−1/2ph0,Ω

≤μ1 +√

2cδ+ 4c²_δ

1 + 4c²_δμ+μ |||(uh, ph)|||_h. (2.24) Combining (2.23) and (2.24), one obtains the desired inf-sup estimation with

β = 1

2 + 2√

2cδ+ 8c²_δ·

The asymptotic behavior ofβ follows from this equation and the definition ofcδ, so the proof is complete.

Remark 2.4. The proof of the above theorem points out the purpose of the stabilizing term, givingL²-stability of the pressure jumps through the interface (as shown in (2.21)). Indeed, it allows to keep under control the last term of (2.22) and, as a result, to obtain an inf-sup constantβ independent ofh. To the best of our knowledge, there is no clear-cut evidence that such a (uniform) stability result can be achieved without the addition of an (extra) control on the pressure jump.

As a direct consequence of the previous result, we have the following corollary.

Corollary 2.5. There exists a unique solution to problem(2.14).

2.3.4. Convergence analysis

In this paragraph we provide an optimal error estimate under the reduced regularity assumptions (2.5). First, we prove the following approximability result with respect to the mesh-dependent norm ||| · |||_h.

Proposition 2.6. Assume that(u, p)satisfies (2.5). Then, there exists a positive constantc, independent ofh and the physical parameters, such that:

|||(u−I_h^ku, p−J_h^kp)|||_h≤c h^k

μ^1/2+r_min^1/2h^1/2

Nk+1(u) + (1 +δ^1/2)μ^−1/2Nk(p) ,

with the notations

N_k+1(u) =

i=1,2

u_k+1,Ωi, N_k(p) =

i=1,2

p_k,Ωi.

Proof. Using the definition (2.18) and the approximation properties ofI_h^k andJ_h^k (2.8) and (2.11), we have

|||(u−Ih^ku, p−Jh^kp)|||_h=

|μ^1/2(u−Ih^ku)|²_1,Ω+μ^−1/2(p−Jh^kp)²_0,Ω+r_min^1/2(u−Ih^kuh)²_0,γ

+δh² μ

i=1,2

∇(p−Jh^kp)²_0,Ωi+δh

μp−Jh^kp²_0,γ1/2

≤

c1μ^1/2h^k+c2r^1/2_minh^k+1/2

Nk+1(u)

+

c3μ^−1/2h^k+δ^1/2μ^−1/2(c3+c4)h^k

Nk(p)

≤c h^k

μ^1/2+r^1/2_minh^1/2

Nk+1(u) +

μ^−1/2+δ^1/2μ^−1/2

Nk(p) ,

which completes the proof.

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The main result of this paragraph is stated in the next theorem.

Theorem 2.7. Let (u, p) ∈ V×M˜ be the solution of problem (2.3), and (uh, ph) ∈ Vh×M_h^k the solution of problem (2.14). Assume that the hypotheses of Theorem 2.3 hold. Then, there exists a positive constant c, independent ofh, depending on the inf-sup constantβ, such that

|||(u−uh, p−ph)|||_h ≤ c(β)h^k

μ^1/2+δ^1/2μ^1/2+δ^−1/2μ^1/2

+rmax

r^1/2_min h^1/2+δ^1/2μ^−1/2rmaxh

i=1,2

uk+1,Ωi

+

μ^−1/2+δ^1/2μ^−1/2

i=1,2

pk,Ωi

.

Proof. Take the interpolantsv_h=I_h^kuandqh =J_h^kp. Decompose the error into the interpolation and approxi- mation errors,

|||(u−uh, p−ph)|||_h≤ |||(u−vh, p−qh)|||_h+|||(uh−vh, ph−qh)|||_h.

The first term can be bounded using Proposition2.6, so we only need to estimate|||(uh−vh, ph−qh)|||_h. Using the inf-sup condition (Thm.2.3) and the Galerkin orthogonality (2.16), we have

|||(uh−vh, ph−qh)|||_h ≤ 1

β sup

(wh,rh)∈V^k_h×M_h^k

Bδ,Rγ

(uh−vh, ph−qh), (wh, rh)

|||(wh, rh)|||_h

≤ 1

β sup

(wh,rh)∈V^k_h×M_h^k

Bδ,Rγ

(u−vh, p−qh),(wh, rh)

|||(wh, rh)|||_h ·

To estimate the right-hand side of the above inequality, we take(wh, rh)∈V^k_h×M_h^k arbitrarily, and we bound separately each term ofBδ,Rγ

(u−vh, p−qh),(wh, rh)

, using approximation estimates (2.8) and (2.11). For the viscous term, we have

(μ∇(u−vh),∇wh)_Ω≤c1h^kμ^1/2N_k+1(u)|||(wh, rh)|||_h. The resistive term is treated as follows

(Rγ(u−vh),wh)_γ≤ Rγ1/2(u−vh)0,γRγ1/2wh0,γ

≤c2rmax

r^1/2_min h^k+1/2N_k+1(u)|||(wh, rh)|||_h. For the pressure term, we have

−(p−qh,divwh)_Ω≤c3μ^−1/2h^kNk(p)√

n|||(wh, rh)|||_h. By integration by parts, we obtain

(rh,div(u−vh))_Ω=−

T∈Th

(∇rh, u−vh)_T +

E∈Gh

(u−vh,rhn)_E

≤(c1+c2)h^k Nk+1(u)δ^−1/2μ^1/2|||(wh, rh)|||_h.

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For the stabilization terms, we have

δ

T∈Th

h²

μ (−μΔ(u−vh) +∇(p−qh),∇rh)_T ≤h^k

c1δ^1/2μ^1/2Nk+1(u) +c3δ^1/2μ^−1/2Nk(p)

|||(wh, rh)|||_h.

On the other hand, using the fact that∇u∈H^k(Ω_i)^n×n, i= 1,2 and since

∇(u−vh)·n0,E ≤ ∇(u−vh)0,E≤c2h^k−1/2

i=1,2

∇uk, Ti, E=T1∩T2,

we obtain

δ

E∈Gh

h

μ(μ∇(u−vh)·n,−rhn)_E ≤ c2h^kδ^1/2μ^1/2Nk+1(u)|||(wh, rh)|||_h,

δ

E∈Gh

h

μ(Rγ(u−vh),−rhn)_E ≤ c2h^k+1δ^1/2μ^−1/2rmaxNk+1(u)|||(wh, rh)|||_h,

δ

E∈Gh

h

μ(−(p−qh)n,−rhn)_E ≤ c4h^kδ^1/2μ^−1/2Nk(p)|||(wh, rh)|||_h.

We conclude the proof by summing up all the contributions.

3. Numerical results

This section is devoted to numerical results. In Section 3.1, we assess the theoretical convergence rates stated in Section2.3with an analytical test-case. In Section3.2, we consider a quasi-Poiseuille flow through a porous surface using different discretization spaces. The purpose is to stress the relevance of using discontinuous pressures across the interface and to show that the proposed stabilized finite element provides solutions similar to those obtained with more expensive finite elements. Finally, in Section3.3, we show the results of simulations with a realistic aneurysm geometry, both with rigid and elastic walls.

3.1. Assessment of the convergence rate

We build an analytical solution of the problem in order to assess the convergence rate proved above. Let Ω = (0,2)×(0,1)be the fluid domain divided in two subdomainsΩ₁= (0, L)×(0,1)andΩ₂= (L,2)×(0,1), by the interfaceγ={L} ×(0,1), whereL= 1. Assume that the viscosityμ= 0.04and the resistivity is the scalar matrix Rγ = 100 I. We compute f and the non-homogeneous boundary conditions such that the following functions are solution to problem (2.1):

u¹=

−19.98x+ 10x²

−40.04 + 19.98y+ 40x²−20xy

, u²=

25−69.98x+ 35x² 9.96 + 69.98y−10x²−70xy

,

and

p¹= 800y², p²= 998 + 800y²,

where(uⁱ, pⁱ)denotes the solution in Ω_i. This solution is depicted in Figure4. The functions(uⁱ, pⁱ), i= 1,2, were chosen as polynomials of degree 2 in(x, y), that satisfy the model problem (2.1) and enjoy a large pressure jump at the interface and a moderate jump of the normal derivatives of the velocity. This corresponds to the typical physical conditions we consider in this paper.

Various unstructured triangulations with decreasing mesh parametersh∈ {1/10; 1/20; 1/40; 1/80}have been considered and the results are reported in Figure5.