Analysis of a stabilized finite element method for fluid flows through a porous interface

HAL Id: inria-00543014

https://hal.inria.fr/inria-00543014

Submitted on 5 Dec 2010

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Analysis of a stabilized finite element method for fluid flows through a porous interface

Alfonso Caiazzo, Miguel Angel Fernández, Vincent Martin

To cite this version:

Alfonso Caiazzo, Miguel Angel Fernández, Vincent Martin. Analysis of a stabilized finite element method for fluid flows through a porous interface. Applied Mathematics Letters, Elsevier, 2011, 24 (12), pp.2124-2127. �10.1016/j.aml.2011.06.012�. �inria-00543014�

a p p o r t

d e r e c h e r c h e

ISSN0249-6399ISRNINRIA/RR--7475--FR+ENG

Thème BIO

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Analysis of a stabilized finite element method for fluid flows through a porous interface

Alfonso Caiazzo — Miguel A. Fernández — Vincent Martin

N° 7475

December 2010

Unité de recherche INRIA Rocquencourt

Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

Analysis of a stabilized finite element method for fluid flows through a porous interface

Alfonso Caiazzo^∗ , Miguel A. Fern´andez^† , Vincent Martin^‡

Th`eme BIO — Syst`emes biologiques Projet REO

Rapport de recherche n°7475 — December 2010 — 8 pages

Abstract: This work is devoted to the numerical simulation of an incompress- ible fluid through a porous interface, modeled as a macroscopicresistive inter- face term in the Stokes equations. We improve the results reported in [M2AN 42(6):961–990, 2008], by showing that the standard Pressure Stabilized Petrov- Galerkin (PSPG) finite element method is stable, and optimally convergent, without the need for controlling the pressure jump across the interface.

Key-words: Stokes equation, porous interface, stabilized finite element method.

∗INRIA, REO project-team and Weierstrass Institute,a.caiazzo@googlemail.com.

†INRIA, REO project-team,Miguel.Fernandez@inria.fr.

‡UTC Compi`egne and INRIA, REO project-team,Vincent.Martin@utc.fr.

Analyse d’une m´ethode d’´el´ements finis stabilis´ee pour les ´ecoulements `a travers une

interface poureuse

R´esum´e : Ce travail concerne la simulation num´erique d’un fluide incompress- ible `a travers une interface poureuse, mod´elis´ee par un terme d’interface r´esistif macroscopique dans les ´equations de Stokes. Nous am´eliorons les r´esultats de [M2AN 42(6):961–990, 2008], en montrant que la m´ethode d’´el´ements finis sta- bilis´ee classique PSPG (Pressure Stabilized Petrov-Galerkin) est stable, et con- verge de fa¸con optimale, sans contrˆole additionnel sur le saut de pression `a l’interface.

Mots-cl´es : Equation de Stokes, interface poureuse, m´ethode d’´el´ements finis´ stabilis´ee.

Stabilized FEM for fluid flows through a porous interface 3

1 Introduction

We consider a regular domain Ω⊂R^d,d= 2 or 3, and a porous interface defined by a hyperplane domain Γ ⊂ R^d−1, dividing Ω in two connected subdomains as Ω = Ω1∪Γ∪Ω2. We denote by n1, n2 the outgoing normals from each subdomain Ωi at the interface, with n1 =−n2, and we define n = n1. The fluid velocity u and pressure pare governed by the following modified Stokes equations [3]:

∇p−µ∆u+rΓδΓu=f in Ω,

divu= 0 in Ω, (1)

with a homogeneous Dirichlet condition on∂Ω. In (1), the symbolµstands for the fluid viscosity,f for a given volume force,δΓfor the Dirac measure on Γ, and rΓ>0 is a given interface resistance, related to the permeability and porosity of the interface. Without loss of generality,rΓ is assumed to be a constant scalar.

Problem (1) can be reformulated equivalently as a two-domain Stokes prob- lem, complemented with the interface conditions

JuK=0, Jµ∇u·n−pnK=−rΓu on Γ, (2) whereJqK^def= q1|Γ−q2|Γ denotes the jump across Γ andqi

def= q|^Ωi (i= 1,2).

In [3], problem (1) was discretized with an extension of the PSPG stabilized method (see [4]): an additional consistent term (based on (2)) was introduced to control the interface pressure jump. Numerical evidence showed, however, that this term did not improve noticeably the behavior of the numerical solution with respect to a standard PSPG stabilized formulation [3]. The aim of this note is to show that, indeed, stability and optimal accuracy can be derived without the need for this extra interface stabilization term (which is convenient in practice).

2 Finite element formulation

Let {Th}0<h≤1 be a regular family of quasi-uniform triangulations of Ω, con- forming with the interface Γ. The corresponding triangulation of the interface is denoted byGh and we seth^def= maxT∈ThhT, where hT is the diameter of the elementT. We introduce the spacesV ^def= [H₀¹(Ω)]^d, Q^def= L²₀(Ω), and the finite element spaces of degreek≥1,V^k_h and N_h^k, equal order approximations ofV andQ:

V^k_h^def=

vh∈(C⁰(Ω))^d| vh|T ∈(Pk)^d ∀T∈ Th ∩V, N_h^kdef=

qh|Ωi ∈ C⁰(Ωi), i= 1,2| qh|T ∈Pk ∀T ∈ Th ∩Q.

(3)

Note that the space N_h^k of discrete pressures allows discontinuity at the inter- face Γ. As underlined in [3], this is of utmost importance to get a correct approximation of the solution without excessive mesh refinement. Additionally, we introduce the spacesV0

def= {v∈V|v|^Γ = 0}andV_0,h^{k def}= V0∩V^k_h.

RR n°7475

4 Caiazzo, Fern´andez & Martin

Let us consider the two following bilinear forms A^rδ^Γ x_h,y_hdef

= (µ∇u_h,∇v_h)−(ph,divv_h) + (rΓu_h,v_h)_Γ+ (divu_h, qh) +δ X

T∈Th

h²_T

µ (µ∆uh+∇ph,∇qh)_T, Bδ^rΓ xh,y_hdef

= A^rδ^Γ xh,y_h

−δ X

E∈Gh

hE

µ (Jµ∇uh·n−phnK+rΓuh,JqhnK)_E for allxh= (uh, ph) andy_h= (vh, qh) inV^k_h×N_h^k andδ >0 is a stabilization parameter. The discrete formulation proposed and analyzed in [3] is based on B^rδ^Γ. In this note, we consider the numerical analysis of the standard PSPG formulation

A^rδ^Γ x_h,y_h

= (f,v_h) ∀y_h∈V^k_h×N_h^k. (4)

3 Stability analysis

Let us consider the mesh-dependent energy norm

|||(uh, ph)|||²_h^def= µk∇uhk²0,Ω+rΓkuhk²0,Γ+δ X

T∈Th

h²_T

µ k∇phk²0,T +1

µkphk²0,Ω.

Note that, unlike in [3], this norm provides no control on the interface pressure jump. We address now the stability of (4) in the||| · |||h norm.

By applying the inverse inequality (see [1])

k∆vhk0,T ≤c∆h⁻¹k∇vhk0,T, vh∈V^k_h, and the Schwarz and Young inequalities to the term P

T∈Thh²_T(∆uh,∇ph)_T, we get the following coercivity estimate.

Proposition 3.1 Let δbe such that0< δc²_∆≤1. Then A^rδ^Γ(xh,x_h)≥ µ

2k∇u_hk²0,Ω+rΓku_hk²0,Γ+ξ² 2 ≥ 1

2

|||(uh, ph)|||²h− 1

µkphk²0,Ω

(5) for allxh= (uh, ph)∈V^k_h×Q^k_h, withξ^{2 def}= δP

T∈Th

h²T

µ k∇phk²0,T.

The stability and the optimal convergence are stated in the following result.

Proposition 3.2 Under the assumption of Proposition 3.1 there holds:

(i)there exists a constant β=β(δ,_r^µ_Γ)independent of h, such that inf

xh∈V^kh×Q^kh

sup

yh∈V^kh×Q^kh

A^rδ^Γ(xh,y_h)

|||xh|||_h|||y_h|||_h ≥β. (6) Moreover, if δ≪1 we haveβ ∼δ, andβ=O(µ/rΓ) for rΓ/µ≫1;

INRIA

Stabilized FEM for fluid flows through a porous interface 5

(ii)let(uh, ph)be the solution of (4)and assume that(u, p), the solution of (1), is such that u_i ∈

H^k+1(Ωi)d

, pi ∈H^k(Ωi), i = 1,2. The following estimate holds

|||(u−uh, p−ph)|||h≤c h^k X

i=1,2

kukk+1,Ωi+kpkk,Ωi

, (7)

wherec=c(rΓ, δ, µ)is a positive constant, independent ofh.

We remark that the stability and convergence results are essentially the same as the ones given in [3], but without the need for the extra stabilization term.

Theinf-supconstantβ has also the same asymptotic behavior.

Proof. For the sake of conciseness, we prove only point(i). The proof of(ii) follows [3], owing to the stability ofA^rδ^Γ. Letx_h= (uh, ph)∈V^k_h×N_h^k . Given (5), the inf-sup stability ofA^r_δ^Γ requires additional stability estimates needed to control the pressure.

A pressurep ∈ L²₀(Ω) has zero mean in Ω, but this is not true in general for its restriction to Ωi, i= 1,2. Following an argument of [2], we decompose ph∈N_h^k ⊂L²₀(Ω) asph=p⁰_h+p_h, withp⁰_h,i∈L²₀(Ωi) andp_h,i^def= (ph,i,1)Ωi(i.e., p⁰_hhas zero mean over each subdomain and p_h is constant in each subdomain).

The following relations hold:

kphk²0,Ω= p⁰_h

2

0,Ω+kp_hk²0,Ω, p_h,1|Ω1|+p_h,2|Ω2|= 0, kp_hk²0,Ω=p²_h,1|Ω1|+p²_h,2|Ω2|.

(8)

We show how to control separately p⁰_h and p_h. Since p⁰_h,i ∈L²₀(Ωi), i = 1,2, there exists a function v⁰∈V0, such thatv⁰_i ∈[H₀¹(Ωi)]^d, −divv⁰_i =p⁰_h,i and v⁰

_1,Ω ≤cΩ p⁰_h

_0,Ω. We take v⁰_h as the Scott-Zhang interpolant of v⁰ into V^k_0,h, defined separately on each subdomain. Using the properties of the Scott- Zhang operator [1], we also have

v⁰_h

1,Ω ≤ c^′_Ω p⁰_h

0,Ω and

v⁰−v⁰_h 0,Ω ≤ cπh

v⁰

1,Ω. Since v⁰_i,v⁰_i,h ∈ [H₀¹(Ωi)]^d, i = 1,2, and p_h is constant on each subdomain, we have (v⁰−v⁰_h)|Γ = 0, (p_h,divv⁰_h) = 0 and (p_h,divv⁰) = 0.

Hence, using the fact that ph ∈ N_h^k is continuous in Ω1 and Ω2 we obtain, integrating by parts in each subdomain:

−(ph,divv⁰_h) =− p⁰_h,divv⁰

+ p⁰_h,div(v⁰−v⁰_h)

≥ p⁰_h

2

0,Ω−ξcπcΩδ⁻¹²µ¹² p⁰_h

0,Ω,

withξdefined in Proposition 3.1. Using Young’s inequality, this yields A^rδ^Γ xh,(v⁰_h,0)

≥ −c^′_Ωµk∇uhk0,Ω

p⁰_h

_0,Ω−µ δ

¹2

cπcΩξ p⁰_h

_0,Ω+ p⁰_h

2 0,Ω

≥ 1 2

p⁰_h

2

0,Ω−(c^′_Ω)²µ²k∇uhk²0,Ω−µ δc²_πc²_Ωξ².

(9) To handle the constant part of the pressure, we need the following Lemma (whose proof can be found in [2] in a more complex case):

RR n°7475

6 Caiazzo, Fern´andez & Martin

Lemma 3.1 There exist two (non-constant) functions v^α_h ∈ V¹_h, α = 1,2, defined over the whole domainΩsuch that

k∇v^α_hk0,Ω+kv^α_hk0,Γ≤c p_h,α

0,Ωα, Z

Γ

v^α_h,1·n1=− Z

Γ

v^α_h,2·n2=p_h,α|Ωα|.

Let vh

def= v²_h−v¹_h ∈ V^k_h. Since ∇p_h,i = 0 and using (8) and Lemma 3.1, we have

−(p_h,divvh) = X

i=1,2

(p_h,i,(v¹_h−v²_h)·ni)Γ

=p²_h,1|Ω1| −p_h,1p_h,2(|Ω2|+|Ω1|) +p²_h,2|Ω2|

= 2 p²₁|Ω1|+p²₂|Ω2|

= 2kp_hk²0,Ω, and, by applying Lemma 3.1 once more, there follows

−(ph,divvh) =−(p_h,divvh)−(p⁰_h,divvh)

≥2kp_hk²0,Ω− p⁰_h

_0,Ωd¹²

∇(v¹_h−v²_h) _0,Ω

≥2kp_hk²0,Ω−d¯c² p⁰_h

2 0,Ω−1

4

p_h,1

0,Ω1+ p_h,2

0,Ω2

2

≥ kp_hk²0,Ω−dc² p⁰_h

2 0,Ω,

where we recall thatddenotes the spatial dimension. Hence, A^rδ^Γ(xh,(vh,0))≥ −2µck∇uhk0,Ωkp_hk0,Ω−2rΓckuhk0,Γkp_hk0,Ω

+kp_hk²0,Ω−dc² p⁰_h

2 0,Ω

≥1

2kp_hk²0,Ω−dc² p⁰_h

2

0,Ω−4c²µ²k∇uhk²0,Ω−4c²r²_Γkuhk²0,Γ. (10) Therefore, by takingy_h= (λv⁰_h+ (1−λ)vh,0), withλ^def= _2(1+dc^1+2dc22) ∈(0,1), and using (9) and (10), we obtain

A^rδ^Γ(xh,y_h)≥ λ

2 −(1−λ)d¯c²

p⁰_h

2

0,Ω+1−λ 2 kp_hk²0,Ω

−µ λ(c^′_Ω)²+ (1−λ)4c²

µk∇uhk²0,Ω

−µ

δλc²_πc²_Ωξ²−(1−λ)4c²r_Γ²kuhk²0,Γ

≥1

4˜ckphk²0,Ω−µc²_max

|||(uh, ph)|||²_h−1

µkphk²0,Ω

,

(11)

where we have introduced ˜c^def= 1+dc², andc²_max^def= max

c^′2_Ω+4c²

,¹_δc²_πc²_Ω,4c^2r_µ^Γ . Equation (11) provides a control on the pressure. To conclude the proof, we

INRIA

Stabilized FEM for fluid flows through a porous interface 7

take a test function zh

def= (1−ω)xh+ωy_h, withω ^def= _{µ+2˜c(1+2µc}^2˜c 2

max) ∈(0,1), and apply (5) and (11), to obtain

A^r_δ^Γ(xh,zh)≥1 2(1−ω)

|||(uh, ph)|||²_h−1

µkphk²0,Ω

+ω 1

4˜ckphk²0,Ω−µc²_max

|||(uh, ph)|||²h− 1

µkphk²0,Ω

≥ 1−ω

2 −ωµc²_max

µk∇uhk²0,Ω+rΓkuhk²0,Γ+ξ² + ω

4˜ckphk²0,Ω

≥ µ

2 µ+ 2˜c(1 + 2µc²_max)|||x_h|||²h.

(12) Moreover, it can be shown thatz_hcan be controlled byx_h as

|||zh|||h≤(1−ω)|||xh|||h+ω |||(v⁰_h,0)|||h+|||(v¹_h,0)|||h+|||(v²_h,0)|||h

≤(1−ω)|||xh|||_h+ωµ¹²c^′_Ω p⁰_h

0,Ω+ω√

2c(µ+rΓ)¹²kp_hk0,Ω

≤(1−ω)|||x_h|||h+ωµ√

2cmax|||x_h|||h

≤

1−ω+ωµ√ 2cmax

|||xh|||h

≤µ1 + 2˜ccmax(2cmax+√ 2) µ+ 2˜c(1 + 2µc²_max) |||xh|||_h.

(13)

Combining (12) and (13) we obtain that the global inf-sup condition (6) follows with a constant

β^def= 1

2 1 + 2˜ccmax(2cmax+√ 2).

The stated asymptotic behavior ofβ follows from the definition ofcmax.

References

[1] S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods. Springer Verlag, 2002.

[2] Erik Burman and Peter Hansbo. A unified stabilized method for Stokes’ and Darcy’s equations. J. Comput. Appl. Math., 198(1):35–51, 2007.

[3] M. A. Fern´andez, J.-F. Gerbeau, and V. Martin. Numerical simulation of blood flows through a porous interface. Math. Model. Num. Anal. (M2AN), 42(6):961–990, 2008.

[4] T.J.R. Hughes, L.P. Franca, and M. Balestra. A new finite element for- mulation for computational fluid dynamics. V. Circumventing the Babuˇska- Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech.

Engrg., 59(1):85–99, 1986.

RR n°7475

8 Caiazzo, Fern´andez & Martin

Contents

1 Introduction 3

2 Finite element formulation 3

3 Stability analysis 4

INRIA

Unité de recherche INRIA Rocquencourt

Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

Éditeur

INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

http://www.inria.fr ISSN 0249-6399