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ETNA
Electronic Transactions on Numerical Analysis.
Volume 41, pp. 306-327, 2014.
Copyright  2014, Kent State University.
ISSN 1068-9613.
Kent State University
http://etna.math.kent.edu
FINITE ELEMENT APPROXIMATION OF VISCOELASTIC FLOW IN A
MOVING DOMAIN∗
JASON HOWELL†, HYESUK LEE‡, AND SHUHAN XU‡
Abstract. In this work the problem of a viscoelastic fluid flow in a movable domain is considered. A numerical
approximation scheme is developed based on the Arbitrary Lagrangian-Eulerian (ALE) formulation of the flow
equations. The spatial discretization is accomplished by the finite element method, and the discontinuous Galerkin
method is used for stress approximation. Both first and second order time-stepping schemes satisfying the geometric
conservation law (GCL) are derived and analyzed, and numerical experiments that support the theoretical results are
presented.
Key words. Viscoelastic fluid flow, moving boundary, finite elements, fluid-structure interaction.
AMS subject classifications. 65M60, 65M12.
1. Introduction. In this paper we consider a viscoelastic fluid flow problem posed in a
moving spatial domain. Such problems arise in modeling the interaction of fluid flows with an
elastic medium, which is of great interest in many industrial and biomechanical applications,
including the flow of blood in medium-to-large arteries. In such situations, the physical
problem of interest exhibits significant two-way interaction between the fluid and the solid
structure. The accurate and efficient computer simulation of such fluid-structure interaction
(FSI) problems is of paramount importance to researchers working in these applications, and
much effort is dedicated to producing good algorithms [8, 28, 34, 37, 40].
Motivation for the work presented here stems from recent advances in computing Newtonian and quasi-Newtonian fluid flows in moving domains, including methods designed for
fluid motion within deformable elastic structures. The fluid equations and structure equations are most commonly posed from different perspectives in continuum mechanics: the
Eulerian frame of reference for the fluid equations and the Lagrangian frame of reference for
elastic structures. With this discrepancy in mind, the Arbitrary Lagrangian-Eulerian (ALE)
method was developed in the 1980s to allow for the coupled fluid-structure problem to be
posed in a single framework [11, 24]. In [32], Nobile employed the ALE formulation to first
derive methods for a Newtonian fluid flow governed by the Navier-Stokes equations in a moving domain, and then coupled this formulation with an elastic structure. Several subsequent
works discuss different aspects of the Newtonian fluid-structure interaction problem, including boundary conditions [13, 14, 33], numerical stability [7, 14], and fixed-point methods for
the coupled fluid-structure problem [10]. Other researchers have also shown convergence results for the ALE formulation of the Stokes problem [30], and other related problems [1, 19].
However, many industrial and biological fluids of considerable interest do not behave
as Newtonian fluids. One example of great interest is blood. The study of hemodynamical
flows yields constitutive models that are non-Newtonian in nature, exhibiting both shearthinning and viscoelastic behavior [17, 20, 41]. Several recent investigations show that the
non-Newtonian characteristics of blood can have significant impact on the characteristics of
blood flow [3, 4, 25, 29, 31], and algorithms that capture the behavior of such non-Newtonian
fluids in moving domains and deformable elastic structures are desirable. There are existing works which derive and implement numerical methods for simulation of such problems,
∗ Received September 1, 2012. Accepted April 29, 2014. Published online on October, 6, 2014. Recommended
by S. Brenner.
† Department of Mathematics, College of Charleston, Charleston, SC 29424 ([email protected]).
‡ Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975
({hklee,shuhanx}@clemson.edu).
306
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VISCOELASTIC FLOW IN MOVING DOMAIN
including those employing the ALE approach [25, 29], hybrid finite element/finite volume
approaches [31], and the immersed boundary method [9, 39]. However, a detailed numerical
analysis, including theoretical stability results for time-stepping schemes, of such methods
applied to non-Newtonian problems is, in general, lacking from the current literature.
In [27], Lee investigated numerical approximation of an unsteady flow problem governed
by a quasi-Newtonian model in a moving domain, where a boundary velocity is given by a
known function. A variational formulation of the problem by the Arbitrary Lagrange Eulerian
method was derived and a priori error estimates for the semi-discrete and fully discrete ALE
formulations were obtained. Lee also examined several temporal discretization schemes for
stability and accuracy, where the theoretical results were supported by numerical tests.
In the same spirit as [27], the objective of the work presented here is to develop and
analyze a finite element method for the time-dependent Johnson-Segalman viscoelastic fluid
flow model (of which the Oldroyd-B model is a special case) set in a movable domain. This
work serves as an intermediate step between the aforementioned works and the development
and verification of algorithms for the simulation of a viscoelastic fluid in a deformable elastic
structure, and the extension of that to the shear-thinning viscoelastic case. Specifically, the
problem is posed in a moving spatial domain, and the ALE formulation of the conservation
equations is utilized to pose the equations in a reference domain. The spatial discretization
is accomplished via the finite element method, and we employ discontinuous approximations
for the fluid stress. First and second-order time-stepping schemes satisfying the geometric
conservation law (GCL) are derived, and theoretical stability results are shown.
This paper is organized as follows. In Section 2, we describe the model problem and
introduce an ALE formulation. We consider finite element approximations of the ALE formulation in Section 3 and in Section 4 time discretization schemes are discussed and analyzed. Finally, we present numerical results in Section 5 that support the theoretical results
and exhibit stability of the algorithms developed here. Concuding remarks can be found in
Section 6.
2. Model equations and ALE formulation. Let Ωt be a bounded domain at time t
in R2 with the Lipschitz continuous boundary Γt , where Γt is a moving boundary. Movement of Γt is described by the boundary position function h : Γ0 × [0, T ] → Γt such that
Γt = h(t, Γ0 ). Consider the viscoelastic model equations,
µ
¶
∂σ
σ+λ
(2.1)
+ u · ∇σ + ga (σ, ∇u) − 2 α D(u) = 0 in Ωt ,
∂t
µ
¶
∂u
(2.2)
+ u · ∇u − ∇ · σ − 2(1 − α) ∇ · D(u) + ∇p = f in Ωt ,
ρ
∂t
div u = 0
(2.3)
in Ωt ,
where σ denotes the extra stress tensor, u the velocity vector, p the pressure of the fluid, ρ the
density of the fluid, and λ is the Weissenberg number defined as the product of the relaxation
time and a characteristic strain rate. Assume that p has zero mean value over Ωt . In (2.1)
and (2.2), D(u) := (∇u + ∇uT )/2 is the rate of the strain tensor, α a number such that
0 < α < 1 which may be considered as the fraction of viscoelastic viscosity, and f the body
force. In (2.1), ga (σ, ∇u) is defined by
ga (σ, ∇u) :=
1+a
1−a
(σ∇u + ∇uT σ) −
(∇u σ + σ∇uT )
2
2
for a ∈ [−1, 1]. Note that (2.1) reduces to the Oldroyd-B model for the case a = 1.
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J. HOWELL, H. LEE, AND S. XU
Initial and boundary conditions for u and σ are given as follows:
u(x, 0) = u0
in Ω0 ,
σ(x, 0) = σ 0
in Ω0 ,
u = uBC
on Γt ,
σ = σ BC on Γtin ,
R
where Γt uBC · n dΓt = 0 and Γt in is the inflow boundary.
In the present paper, the constitutive equation (2.1) is slightly modified for numerical
analysis of the governing equations. It is well known that the ga term in the constitutive
equation presents a difficulty when analyzing viscoelastic flow equations. Therefore, we will
consider a nearby problem in which the ga term is linearized with the given velocity b(x):
µ
¶
∂σ
σ+λ
(2.4)
+ u · ∇σ + ga (σ, ∇b) − 2 α D(u) = 0 in Ωt ,
∂t
for the constitutive equation, where the following assumption is made for b:
b ∈ H1 (Ωt ),
∇ · b = 0,
kbk∞ ≤ M,
k∇bk∞ ≤ M .
It should be noted that the flow here is not assumed to be creeping (i.e., slow) as in
[12]. Therefore, the convective term u · ∇u is retained in the momentum equation (2.2). We
also assume the homogeneous boundary condition for the stress function, i.e., σ BC = 0 to
simplify the analysis. The non-homogeneous case, σ BC 6= 0, can be treated in the similar
way for the non-homogeneous velocity boundary condition. In the application of a fluidstructure interaction system, the inflow part on the moving boundary changes from time to
time. This is due to an interface condition, which makes numerical studies of the system
extremely challenging, not only due to the change of inflow boundaries, but also because
of a lack of boundary information on the stress. There are several different ways suggested
to implement a stress boundary condition in the literature. One possible way would be to
compute the boundary value of stress using velocity information as given in [18] and use this
as a stress condition for the next (time or sub-) iteration.
We use the Sobolev spaces W m,p (D) with norms k · km,p,D if p < ∞, k · km,∞,D if
p = ∞. Denote the Sobolev space W m,2 by H m with the norm k · km,D . The corresponding
space of vector-valued or tensor-valued functions is denoted by Hm .
The Arbitrary Lagrangian Eulerian (ALE) [11] method is one of the most widely used
numerical schemes for simulating fluid flows in a moving domain. In the ALE formulation, a
one-to-one coordinate transformation is introduced for the fluid domain, and the fluid equations can be rewritten with respect to a fixed reference domain. Specifically, we define the
time-dependent bijective mapping Ψt which maps the reference domain Ω0 to the physical
domain Ωt :
Ψt : Ω0 → Ωt ,
Ψt (y) = x(y, t) ,
where y and x are the spatial coordinates in Ω0 and Ωt , respectively. The coordinate y is often
called the ALE coordinate. Using Ψt , the weak formulation of the flow equations in Ωt can be
recast into a weak formulation defined in the reference domain Ω0 . Thus the model equations
in the reference domain can be considered for numerical simulation and the transformation
function Ψt needs to be determined at each time step as a part of the computations.
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For a function φ : Ωt × [0, T ] → R, its corresponding function φ = φ ◦ Ψt in the ALE
setting is defined as
φ : Ω0 → R,
φ(y, t) = φ(Ψt (y), t).
The time derivative in the ALE frame is also given as
∂φ
∂φ
|y (x, t) =
(y, t).
∂t
∂t
∂φ
|y : Ωt × [0, T ] → R,
∂t
Using the chain rule, we have
∂φ
∂φ
|y =
|x +w · ∇x φ,
∂t
∂t
(2.5)
∂φ
where w := ∂x
∂t |y is the domain velocity. In (2.5), ∂t |y is the so-called ALE derivative of
φ. The flow equations (2.2), (2.3) and (2.4) can be then written in the ALE formulation as
¶
µ
∂σ
(2.6) σ + λ
|y +(u − w) · ∇x σ + ga (σ, ∇x b) − 2 α Dx (u) = 0 in Ωt ,
∂t
µ
¶
∂u
(2.7) ρ
|y +(u − w) · ∇x u − ∇x · σ − 2(1 − α) ∇x · Dx (u) + ∇x p
∂t
= f in Ωt ,
∇x · u = 0
(2.8)
in Ωt ,
where Dx (u) = (∇x u + ∇x uT )/2. Note that all spatial derivatives involved in (2.6)–(2.8),
including the divergence operator, are with respect to x. Throughout the paper we will use
Dx (·) and ∇x only when they need to be clearly specified. Otherwise, D(·), ∇ will be used
as Dx (·), ∇x , respectively.
For the variational formulation of the flow equations (2.6)–(2.8) in the ALE framework,
define function spaces for the reference domain:
U0 := H10 (Ω0 ) ,
Q0 := L20 (Ω0 ) = {q ∈ L2 (Ω0 ) :
R
Ω0
q dΩ = 0} ,
Σ0 := {τ ∈ L2 (Ω0 ) : τij = τji , τ = 0 on Γtin , (b · ∇)τ ∈ L2 (Ω0 )} .
The function spaces for Ωt are then defined as
Ut := {v : Ωt × [0, T ] → R2 , v = v ◦ Ψ−1
t for v ∈ U0 } ,
Qt := {q : Ωt × [0, T ] → R, q = q ◦ Ψ−1
t for p ∈ Q0 } ,
Σt := {τ : Ωt × [0, T ] → R2×2 , τ = τ ◦ Ψ−1
t for τ ∈ Σ0 } .
If the ALE mapping Ψt satisfies the regularity conditions [15, 21, 30]
Ψt ∈ W2,∞ (Ωt ),
2,∞
Ψ−1
(Ωt ),
t ∈W
then
(2.9)
(v, q, τ ) ∈ Ut × Qt × Σt
⇐⇒ (v, q, τ ) = (v ◦ Ψt , q ◦ Ψt , τ ◦ Ψt ) ∈ U0 × Q0 × Σ0 .
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Based on the regularity condition of the ALE function (2.9), we assume that the domain
velocity is bounded, i.e.,
kwk1,Ωt < C
(2.10)
for C > 0.
For given uBC ∈ H1/2 (Γt ), there exists u∗ ∈ H1 (Ωt ) such that
u∗ |Γt = uBC ,
∇ · u∗ = 0
and
|(v · ∇u∗ , v)| ≤ ǫkvk21
(2.11)
∀v ∈ Ut
e=0
e +u∗ , we have u
e |Γt = 0, ∇· u
for any ǫ > 0 [38]. Writing the velocity function as u = u
and the variational formulation for (e
u, p, σ) in the ALE framework is given by: find (e
u, p, σ)
such that
¶
µ
∂σ
∗
u + u − w) · ∇)σ + ga (σ, ∇b), τ
|y +((e
(2.12) (σ, τ )Ωt + λ
∂t
Ωt
(2.13)
−2α (D(e
u), τ )Ωt = 2α (D(u∗ ), τ )Ωt ∀τ ∈ Σt ,
µ
¶
∂e
u
e · ∇u∗ + u∗ · ∇e
ρ
|y +(e
u − w) · ∇e
u+u
u, v
+ (σ, D(v))Ωt
∂t
Ωt
= (f , v)Ωt − ρ
(2.14)
e )Ωt = 0
(q, ∇ · u
µ
+2(1 − α)(D(e
u), D(v))Ωt − (p, ∇ · v)Ωt
¶
∂u∗
|y +(u∗ − w) · ∇u∗ , v
∂t
Ωt
−2(1 − α)(D(u∗ ), D(v))Ωt ∀v ∈ Ut ,
∀q ∈ Qt .
e = 0 and u
e |Γt = 0, the convective terms in (2.12)–(2.13) can
Using integration by parts, ∇ · u
be written as
1
[((∇ · w)σ, σ)Ωt − ((w · n)σ, σ)Γt ] ,
2
1
e )Ωt .
= ((∇ · w)e
u, u
2
((e
u − w) · ∇)σ, σ)Ωt =
e )Ωt
((e
u − w) · ∇)e
u, u
Note that if w = 0,
(e
u · ∇σ, σ)Ωt = 0,
e )Ωt = 0 .
(e
u · ∇e
u, u
In order to simplify expressions, throughout this paper we will use the bilinear form At
defined by
(2.15)
u), τ )Ωt
At ((e
u, σ), (v, τ )) := (σ, τ )Ωt + λ (ga (σ, ∇b), τ )Ωt − 2α (D(e
u), D(v))Ωt .
+ 2α (σ, D(v))Ωt + 4α(1 − α) (D(e
Since
(ga (σ, ∇b), τ )Ωt ≤ 4k∇bk∞,Ωt kσk0,Ωt kτ k0,Ωt ≤ 4M kσk0,Ωt kτ k0,Ωt ,
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311
uk1,Ωt ,
we have, using the Poincaré inequality kD(e
u)k0,Ωt ≥ Cp ke
At ((e
u, σ), (e
u, σ)) ≥ (1 − 4λM ) kσk20,Ωt + 4α(1 − α)kD(e
u)k20,Ωt
≥ (1 − 4λM ) kσk20,Ωt + 4α(1 − α)Cp2 ke
uk21,Ωt
(2.16)
and
e ), (τ , v)) ≤ (1 + 4λM ) kσk0,Ωt kτ k0,Ωt + 2αkD(e
u)k0,Ωt kτ k0,Ωt
At ((σ, u
u)k0,Ωt kD(v)k0,Ωt
+2αkσk0,Ωt kD(v)k0,Ωt + 4α(1 − α)kD(e
uk1,Ωt )(kτ k0,Ωt + kvk1,Ωt ) .
≤ C(kσk0,Ωt + ke
Therefore, At is coercive and continuous if λ M is small so that 1 − 4λM > 0. This would
be the case when the ratio of a time scale for the fluid memory to a time scale of the flow is
small and the fluid has a small effect of elasticity.
A variational formulation called a conservative form [32] is derived based on the fact that
the test function space can be mapped into a time-independent space using Ψ−1
t . In order to
derive a conservative variational formulation, consider the Reynolds transport formula [36]
Z
Z
Z
d
∂φ
∂φ
φ(x, t) dV =
|y +φ∇x · w dV =
|x +w · ∇x φ + φ∇x · w dV
dt V (t)
∂t
V (t)
V (t) ∂t
for a function φ : V (t) → R, where V (t) ⊂ Ωt such that V (t) = Ψt (V0 ) with V0 ⊂ Ω0 . If v
is a function from Ωt to R and v = v ◦ Ψ−1
t for v : Ω0 → R, we have that
∂v
|y = 0 ,
∂t
(2.17)
and therefore
d
dt
(2.18)
d
dt
Z
Z
v dΩ =
Ωt
φv dΩ =
Ωt
Z
v ∇x · w dΩ ,
Ωt
Z
Ωt
µ
∂φ
|y +φ∇x · w
∂t
¶
v dΩ .
Then, applying (2.18) to (2.12)–(2.14), we have the following variational formulation: find
(e
u, p, σ) for each t ∈ (0, T ] such that
µ
d
(2.19) (σ, τ )Ωt + λ (σ, τ )Ωt + λ −σ(∇ · w) + ((e
u + u∗ − w) · ∇)σ
dt
¶
+ga (σ, ∇b), τ
− 2α(D(e
u), τ )Ωt = 2α (D(u∗ ), τ )Ωt ∀τ ∈ Σt ,
Ωt
(2.20) ρ
d
e · ∇u∗ + u∗ · ∇e
u(∇ · w) + ((e
u − w) · ∇)e
u+u
u, v)Ωt
(e
u, v)Ωt + ρ (−e
dt
+2(1 − α)(D(e
u), D(v))Ωt + (σ, D(v))Ωt − (p, ∇ · v)Ωt
¶
µ ∗
∂u
|y +(u∗ − w) · ∇u∗ , v
= (f , v)Ωt − ρ
∂t
Ωt
−2(1 − α)(D(u∗ ), D(v))Ωt ∀v ∈ Ut ,
e )Ωt = 0 ∀q ∈ Qt .
(2.21) (q, ∇ · u
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The ALE weak formulation (2.19)–(2.21) is conservative in the sense that if we take a subsetRV (t) ⊂ Ωt with Lipschitz continuous boundary, f = 0 and v = constant, then
d
e
e
dt V (t) u dV is given in terms of boundary integrals only. Therefore, the variation of u
over V (t) is due only to boundary terms [15].
In order to define the ALE mapping Ψt , we consider the boundary position function
h : Γ0 × [0, T ] → Γt . The ALE mapping then may be determined by solving the Laplace
equation,
∆y x(y) = 0
x(y) = h(y)
in Ω0 ,
on Γ0 .
This method is called the harmonic extension technique, where the boundary position function h is extended onto the whole domain [15]. When the problem is posed as a fluid-structure
interaction problem, other equations such as a linear elastic problem and a parabolic system
also can be used to obtain the domain velocity [15, 21].
3. Finite element approximation. The spatial discretization of the viscoelastic flow
Suppose Th,0 is a triangulation of Ω0 such that
problem follows that of [2].
Ω0 = {∪K : K ∈ Th,0 }. Assume that there exist positive constants c1 , c2 such that
c1 ρK ≤ hK ≤ c2 ρK ,
where hK is the diameter of K, ρK is the diameter of the greatest ball included in K, and
h = maxK∈Th,0 hK .
Let Pk (K) denote the space of polynomials of degree less than or equal to k on
K ∈ Th,0 . We define finite element spaces for the approximation of (u, p) in Ω0 :
Uh,0 := {v ∈ U0 ∩ (C 0 (Ω))2 : v|K ∈ P2 (K)2 , ∀K ∈ Th,0 } ,
Qh,0 := {q ∈ Q0 ∩ C 0 (Ω) : q|K ∈ P1 (K), ∀K ∈ Th,0 } .
The stress σ is approximated in the discontinuous finite element space of piecewise linear
functions,
Σh,0 := {τ ∈ Σ0 : τ |K ∈ P1 (K)2×2 , ∀K ∈ Th,0 } .
Let Nue := dim(Uh,0 ) and {ϕi : ϕi ∈ Uh,0 for i ∈ Nue } be a set of basis functions for Uh,0 .
Similarly, let Nσ := dim(Σh,0 ) and {ψ i : ψ i ∈ Σh,0 for i ∈ Nσ } be a set of basis functions
for Σh,0 . The finite element spaces defined above satisfy the standard approximation properties; see [6] or [22]. It is also well known that the Taylor-Hood pair (Uh,0 , Qh,0 ) satisfies the
inf-sup (or LBB) condition,
inf
sup
06=qh ∈Qh,0 06=vh ∈Uh,0
(qh , ∇ · vh )
≥C,
kvh k1 kqh k0
where C is a positive constant independent of h.
We consider a discrete mapping Ψh,t : Ω0 → Ωt approximated by Pl Lagrangian finite
elements such that Ψh,t (y) = xh (y, t). The finite element spaces for Ωt are then defined as
Uh,t := {vh : Ωt × [0, T ] → R2 , vh = vh ◦ Ψ−1
h,t for vh ∈ Uh,0 } ,
Qh,t := {qh : Ωt × [0, T ] → R, qh = q h ◦ Ψ−1
h,t for q h ∈ Qh,0 } ,
Σh,t := {σ h : Ωt × [0, T ] → R2×2 , σ h = σ h ◦ Ψ−1
t for σ h ∈ Σh,0 } .
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313
e h , σ h are expressed as a combination of basis functions multiThe approximate solutions u
plied by time-dependent coefficients, i.e.,
X
X
e h (x, t) =
(3.1)
u
u
ei (t)ϕi (x, t) ,
σ h (x, t) =
σi (t)ψi (x, t) ,
i∈Nu
e
i∈Nσ
−1
where ϕi := ϕi ◦ Ψ−1
t and ψi := ψ i ◦ Ψt .
For the discrete ALE mapping, define the set
Xh := {x ∈ H1 (Ω0 ) : x|K ∈ Pl (K)2 , ∀K ∈ Th,0 } .
If we denote the ith basis function of Xh by ϕ̂i , then the discrete ALE mapping Ψh,t provides
the discrete coordinate function for x as
X
xh (y, t) = Ψh,t (y) =
xi (t)ϕ̂i (y) ,
i∈N X
where N X is the set of nodal points of Xh . Then the discrete domain velocity wh is defined
by
wh (x, t) =
∂xh
|y (Ψ−1
h,t (x), t) .
∂t
In order to analyze the convective term (e
u · ∇e
u, v)Ωt in the finite element space, we
define the trilinear form
b(e
u, w, v)Ωt :=
1
u · ∇v, w)Ωt ] .
[(e
u · ∇w, v)Ωt − (e
2
e = 0, we obtain
Using Green’s theorem and ∇ · u
u, w, v)Ωt
(e
u · ∇w, v)Ωt = b(e
and
(3.2)
b(e
u, v, v)Ωt = 0
∀v ∈ Uh,t .
Since ∇ · u∗ = 0, we also have
(3.3)
b(u∗ , v, v)Ωt = 0
∀v ∈ Uh,t .
The following estimate will be used when analyzing the trilinear term [26]:
(3.4)
uk1,Ωt kwk1,Ωt kvk1,Ωt .
b(e
u, w, v)Ωt ≤ C ke
We introduce some notation in order to analyze an approximate solution of σ by the
discontinuous Galerkin method. Define
∂K − (v) := {x ∈ ∂K, v · n < 0} ,
where ∂K is the boundary of K and n is the outward unit normal to ∂K,
τ ± (v) := lim± τ (x + ǫv(x)) ,
ǫ→0
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and
±
< σ ,τ
±
>h,v :=
X Z
(σ ± : τ ± )|n · v| ds .
∂K − (v)
K∈Th,t
Introduce the operator c(·, ·, ·) defined by
1
c(v − w, σ, τ )Ωt := (((v − w) · ∇)σ, τ )Ωt + (∇ · v σ, τ )Ωt + < σ + − σ − , τ + >h,v−w .
2
e |Γt = 0,
Note that the second term vanishes when ∇ · v = 0. Using integration by parts and u
we have
1
e τ , σ)Ωt
u − w) · ∇)τ , σ)Ωt − (∇ · u
c(e
u − w, σ, τ )Ωt = −(((e
2
+ < σ − , τ − − τ + >h,eu−w +((∇ · w)σ, τ )Ωt − ((w · n)σ, τ )Γt .
Therefore,
1
[((∇ · wh )σ h , σ h )Ωt − ((wh · n)σ h , σ h )Ωt
2
¤
+ < σ h + − σ h − , σ h + − σ h − >h,euh −wh
1
≥ [((∇ · wh )σ h , σ h )Ωt − ((wh · n)σ h , σ h )Γt ] .
2
c(e
uh − wh , σ h , σ h )Ωt =
(3.5)
Also for u∗ such that ∇ · u∗ = 0 and u∗ |Γt 6= 0, we have that
c(u∗ , σ h , σ h ) ≥
(3.6)
1
((u∗ · n)σ h , σ h )Γt .
2
Consider the semi-discrete variational formulation of the fluid problem in the ALE framework: find (e
uh , ph , σ h ) such that
·
d
uh − wh , σ h , τ h )Ωt + c(u∗ , σ h , τ h )Ωt
(σ h , τ h )Ωt + c(e
dt
¸
uh ), τ h )Ωt
−(σ h (∇ · wh ), τ h )Ωt + (ga (σ h , ∇b), τ h )Ωt + (σ h , τ h )Ωt − 2α(D(e
(3.7) λ
= 2α(D(u∗ ), τ h )Ωt
∀τ h ∈ Σh,t ,
·
d
e h , vh )Ωt − (e
(3.8) ρ
uh , vh )Ωt
uh (∇ · wh ), vh )Ωt − (wh · ∇e
uh , u
(e
uh , vh )Ωt + b(e
dt
¸
uh , vh )Ωt
+(e
uh · ∇u∗ , vh )Ωt + (u∗ · ∇e
+2(1 − α)(D(e
uh ), D(vh ))Ωt + (σ h , D(vh ))Ωt + (ph , ∇ · vh )Ωt
µ ∗
¶
∂u
= (f , vh )Ωt − ρ
|y +(u∗ − w) · ∇u∗ , vh
∂t
Ωt
−2(1 − α)(D(u∗ ), D(vh ))Ωt
e h )Ωt = 0 ∀qh ∈ Qh,t .
(3.9) (qh , ∇ · u
∀vh ∈ Uh,t ,
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Using the bilinear form At in (2.15), equations (3.7)–(3.8) are written as
¸
·
d
∗
uh − wh , σ h , τ h )Ωt + c(u , σ h , τ h )Ωt − (σ h (∇ · wh ), τ h )Ωt
(σ h , τ h )Ωt + c(e
λ
dt
·
d
e h , vh )Ωt − (e
+2α ρ
uh , vh )Ωt
uh (∇ · wh ), vh )Ωt − (wh · ∇e
uh , u
(e
uh , vh )Ωt + b(e
dt
¸
e h ), (τ h , vh ))
uh , vh )Ωt + At ((σ h , u
+(e
uh · ∇u∗ , vh )Ωt + (u∗ · ∇e
(3.10)
where
(3.11)
f , (vh , τ h ))Ωt
−2α(ph , ∇ · vh )Ωt = (e
∀(vh , τ h ) ∈ Uh × Σh ,
µ ∗
·
¶
∂u
(e
f , (v, τ ))Ωt := 2α (f , v)Ωt − ρ
|y +(u∗ − w) · ∇u∗ , v
∂t
Ωt
¸
− 2(1 − α)(D(u∗ ), D(v))Ωt + 2α (D(u∗ ), τ )Ωt .
A conditional energy estimate for the solution of the semi-discrete problem (3.9)–(3.10) is
derived in the next theorem.
T HEOREM 3.1. If λM satisfies 1 − 4λM > 0, a solution to the problem (3.9)–(3.10)
satisfies the bound
(3.12)
Z t
λ
2
2
kD(e
uh )k20,Ωt ds
+ kσ h k0,Ωt + 2α(1 − α)Cp
2
0
Z
Z Z
1 − 4λM t
λ t
2
+
((u∗ − wh ) · n)|σ h |2 dΓt ds
kσ h k0,Ωt ds +
2
2 0 Γt
0
λ
≤ α ρ ke
uh,0 k20,Ωt + kσ h,0 k20,Ωt
2
!
° ∗ °2
Z tÃ
°
°
∂u
°
+ ku∗ k41,Ωt + ku∗ k21,Ωt ds ,
kf k2−1,Ωt + °
+C
° ∂t |y °
αρ ke
uh k20,Ωt
0
0,Ωt
e h,0 , σ h,0 are interpolants of u
e 0 and σ 0 in Uh,0 , Σh,0 , respectively.
where u
Proof. In (3.7)–(3.8) we let τ h = ψi , vh = ϕi , where ψi , ϕi are basis functions for
Σh,t and Uh,t , respectively in (3.1). Unlike a standard fixed domain problem, the choice of
e h (or τ h = σ) is not generally acceptable because u
e h and vh may have a different
vh = u
If
we
multiply
(3.8) by u
ei (t) and summing
time evolution in the time derivative term [32].
P
d
ei (t) dt
over Nue , the time derivative term becomes i∈Nue u
(e
uh , ϕi )Ωt and, using (3.1), vh in
e h . We obtain from (2.17), (3.1) that
all other terms can be replaced by u
¸
X
X ·d
d
de
ui (t)
u
ei (t) (e
uh ,
uh , ϕi )Ωt =
(e
uh , u
ei (t)ϕi )Ωt − (e
ϕi )Ωt
dt
dt
dt
i∈Nu
e
i∈Nu
e
=
X de
X
ui (t)
d
uh ,
(e
uh ,
u
ei (t)ϕi )Ωt − (e
ϕi )Ωt
dt
dt
i∈Nu
e
i∈Nu
e
=
d
e h )Ωt
(e
uh , u
dt
X ∂(e
ui (t)ϕi )
− (e
uh ,
|y )Ωt
∂t
i∈Nu
e
∂e
uh
d
uh k20,Ωt − (e
|y )Ωt .
uh ,
= ke
dt
∂t
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2
uh |
uh
1 ∂|e
|y , 1)Ωt , by (2.18),
Since (e
uh , ∂e
∂t |y )Ωt = 2 ( ∂t
(3.13)
X
i∈Nu
e
u
ei (t)
d
d
1 d
1
e h )Ωt
(e
uh , ϕi )Ωt = ke
uh k20,Ωt −
ke
uh k20,Ωt + (e
uh (∇ · wh ), u
dt
dt
2 dt
2
=
1 d
1
e h )Ωt .
ke
uh k20,Ωt + (e
uh (∇ · wh ), u
2 dt
2
By the same argument, we get
X
1 d
1
d
(3.14)
kσ h k20,Ωt + (σ h (∇ · wh ), σ h )Ωt .
σi (t) (σ h , ψi )Ωt =
dt
2 dt
2
i∈Nσ
Therefore, using (2.11), (2.16), (3.2), (3.3), (3.5), (3.6), (3.9), (3.13), and (3.14), equation
(3.10) implies that
¸
·
1
1 d
kσ h k20,Ωt + (((u∗ − wh ) · n)σ h , σ h )Γt
(3.15) λ
2 dt
2
¶
µ
1 d
1
e h )Ωt − ǫke
e h )Ωt − (wh · ∇e
+ 2α ρ
uh k21,Ωt
uh , u
ke
uh k20,Ωt − (e
uh (∇ · wh )), u
2 dt
2
≤ (e
f , (e
uh , σ h ))Ω .
uk2
+ (1 − 4λM )kσ h k2 + 4α(1 − α)C 2 ke
p
0,Ωt
1,Ωt
t
By the Cauchy-Schwarz inequality, Young’s inequality, (2.10) and (3.4),
#
"
° ∗ °2
° ∂u °
2
∗
4
∗
2
°
(3.16) (e
f , (e
uh , σ h ))Ωt ≤ C kf k−1,Ωt + °
+ ku k1,Ωt + ku k1,Ωt
° ∂t °
0,Ωt
+δ1 ke
uh k21,Ωt + δ2 kσ h k20,Ωt .
for arbitrary δ1 , δ2 > 0. Now the estimates (3.15), (3.16) and the identity
1
e h )Ωt = − ((∇ · wh ))e
e h )Ωt ,
(wh · ∇e
uh , u
uh , u
2
imply that
αρ
λ d
d
ke
uh k20,Ωt +
kσ h k20,Ωt + (4α(1 − α)Cp2 − 2α ρǫ − δ1 )kD(e
uh )k20,Ωt
dt
2 dt
λ
+ (1 − 4λM − δ2 )kσ h k20,Ωt + (((u∗ − wh ) · n)σ h , σ h )Γt
2
"
#
° ∗ °2
° ∂u °
2
∗ 4
∗ 2
°
°
≤ C kf k−1,Ωt + °
+ ku k1,Ωt + ku k1,Ωt .
∂t °
0,Ωt
2α(1−α)C 2
The bound (3.12) follows by letting ǫ = 2αρ+1 p , δ1 = ǫ, δ2 =
over (0, t).
R EMARK 3.2. Note that the boundary integral in (3.12),
Z
((u∗ − wh ) · n)|σ h |2 dΓt ,
1−4λM
2
and integrating
Γt
∗
is nonnegative if (u − wh ) · n ≥ 0. Since u∗ − wh represents the relative velocity of the
fluid, under the assumption of a homogeneous stress condition on the inflow boundary, where
(u∗ − wh ) · n < 0, this term may be deleted. Therefore, an unconditional stability estimate
can be obtained. This is also the case for the stability estimate, (4.6) below, of the fully
discretized problem by a first-order scheme.
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4. Time discretization. In the implemention of the ALE method there is a condition
on the time integration scheme, referred to as the geometric conservation law (GCL), which
is considered to be related to the consistency of numerical solutions [5, 15, 16, 32]. The
GCL requires a numerical time discretization scheme to simulate a uniform flow exactly on
a moving domain. The GCL in the finite element ALE framework suggests that a quadrature
rule should be chosen so that the time integration
(4.1)
Z
Ωtn+1
vh dΩ −
Z
vh dΩ =
Z
tn+1
tn
Ωtn
Z
vh ∇ · wh dΩ dt ≈
Ωt
Z
tn+1
p(s) ds
tn
is performed exactly, where p(t) is a polynomial for t ∈ [tn , tn+1 ] of degree k × d − 1, where
d is the dimension [35]. For example, a quadrature formula with the degree of precision 1
or higher satisfies the GCL if d = 2 and piecewise linear elements are used for the ALE
mapping Ψt . Thus, the mid-point rule or the trapezoidal rule satisfies (4.1).
It was reported in [23] that when a temporal discretization not satisfying the GCL is
applied to a moving mesh problem, its accuracy may not be as high as the scheme on a fixed
domain. The authors also pointed out that a higher-order method not satisfying the GCL
tends to lose more accuracy than a lower-order method. However, the effect of the GCL on
stability was not clearly verified analytically and numerically. Some other studies on the GCL
condition applied to the ALE finite element formulation also can be found in [5, 16].
We will investigate the stability of fully discretized systems by first-order and secondorder methods, respectively. Throughout this section we use un to denote unh , an approximation of uh (tn ), to simplify our notation. The standard first-order method is given below.
e n+1 , pn+1 ) satisfying
A LGORITHM 4.1 (First-order non-GCL). Find (σ n+1 , u
i
h
£ n+1
u
− wn+1 , σ n+1 , τ )Ωtn+1
λ (σ n+1 , τ )Ωtn+1 − (σ n , τ )Ωtn + λ ∆t c(e
+c(u∗ n+1 , σ n+1 , τ )Ωtn+1 − (σ n+1 (∇ · wn+1 ), τ )Ωtn+1
¤
+(ga (σ n+1 , ∇bn+1 ), τ )Ωtn+1
¤
£
+∆t (σ n+1 , τ )Ωtn+1 − 2α(D(e
un+1 ), τ )Ωtn+1
= 2α ∆t(D(u∗ n+1 ), τ )Ωtn+1 ∀τ ∈ Σh,t ,
£ n+1
¤
£
e n+1 , v)Ωtn+1
ρ (e
u
, v)Ωtn+1 − (e
un , v)Ωtn + ∆t ρ b(e
un+1 , u
−(e
un+1 (∇ · wn+1 ), v)Ωtn+1 − (wn+1 · ∇e
un+1 , v)Ωtn+1
i
un+1 , v)Ωtn+1
+(e
un+1 · ∇u∗ n+1 , v)Ωtn+1 + (u∗ n+1 · ∇e
£
+∆t 2(1 − α)(D(e
un+1 ), D(v))Ωtn+1 + (σ n+1 , D(v))Ωtn+1
¤
+(pn+1 , ∇ · v)Ωtn+1
·
= ∆t (f n+1 , v)Ωtn+1 −
µ
¶
∂u∗ n+1
|y +(u∗ n+1 − wn+1 ) · ∇u∗ n+1 , v
∂t
Ωtn+1
¸
−2(1 − α)(D(u∗ n+1 ), D(v))Ωtn+1 ∀v ∈ Uh,t ,
ρ
e n+1 )Ωtn+1 = 0
(q, ∇ · u
∀q ∈ Qh,t .
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The above scheme, however, does not satisfy (4.1). If we apply the mid-point rule
(4.2)
Z
Z
vh dΩ −
Ωtn+1
vh dΩ =
Z
tn+1
tn
Ωtn
Z
vh ∇ · wh dΩ dt
Ωt
≈ ∆t
Z
vh ∇ · wh ds
Ω
n+ 1
2
t
for time integration, the first-order scheme is given below.
e n+1 , pn+1 ) satisfying
A LGORITHM 4.2 (First-order GCL). Find (σ n+1 , u
i
h
(4.3)
λ (σ n+1 , τ )Ωtn+1 − (σ n , τ )Ωtn
h
1
+λ ∆t c(e
un+1 − wn+ 2 , σ n+1 , τ )Ω n+ 1 + c(u∗ n+1 , σ n+1 , τ )Ω n+ 1
2
t
−(σ
h
+∆t (σ
n+1
(∇ · w
n+ 21
), τ )Ω n+ 1 + (ga (σ
t
n+1
n+1
u
, τ )Ω n+ 1 − 2α(D(e
t
2
2
), τ )Ω n+ 1
t
2
= 2α ∆t(D(u∗ n+1 ), τ )Ω n+ 1 ,
n+1
, ∇b
), τ )Ω n+ 1
2
t
i
i
2
t
(4.4)
2
t
n+1
h
£ n+1
¤
e n+1 , v)Ω n+ 1
ρ (e
u
, v)Ωtn+1 − (e
un , v)Ωtn + ∆t ρ b(e
un+1 , u
t
n+1
−(e
u
(∇ · w
n+ 21
), v)Ω n+ 1 − (w
t
n+ 21
2
n+1
· ∇e
u
, v)Ω n+ 1
t
i2
n+1
∗ n+1
· ∇e
u
, v)Ω n+ 1
+ (u
2
+(e
un+1 · ∇u∗ n+1 , v)Ω n+ 1
2
2
t
t
£
n+1
n+1
, D(v))Ω n+ 1
+∆t 2(1 − α)(D(e
u
), D(v))Ω n+ 1 + (σ
2
2
t
t
¤
n+1
+(p
, ∇ · v)Ω n+ 1
2
t
·
1
= ∆t (f n+ 2 , v)Ω n+ 1
−ρ
µ
t
∗ n+1
∂u
∂t
2
|y +(u∗ n+1 − wn+ 2 ) · ∇u∗ n+1 , v
1
¸
−2(1 − α)(D(u∗ n+1 ), D(v))Ω n+ 1 ,
t
(4.5)
e n+1 )Ω n+ 1 = 0 ,
(q, ∇ · u
t
2
for all (τ , v, q) ∈ Σh,t × Uh,t × Qh,t .
2
¶
Ω
n+ 1
2
t
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T HEOREM 4.3. A solution to the fully discretized system (4.3)–(4.5) satisfies the inequality
(4.6)
λ n+1 2
kσ
k0,Ωtn+1 + αρke
un+1 k21,Ωtn+1
2
n ·
X
2α(1 − α)Cp2 ke
ui+1 k21,Ω
+∆t
i=0
n
+∆t
λX
2 i=0
Z
1 − 4λM i+1 2
kσ k0,Ω 1
i+
2
2
t
((u∗ i+1 − wi+1 ) · n)|σ i+1 |2 dΓ
Γ
i+ 1
2
t
λ 0 2
kσ k0,Ω0 + αρke
u0 k21,Ω0
2
"
n
X
1
kf i+ 2 k2−1,Ω
+C ∆t
≤
i+ 1
2
t
+
i=0
°
°
° ∂u∗ i+1 °2
°
°
|y °
+°
°
° ∂t
i+ 1
2
t
0,Ω
+ku∗ i+1 k41,Ω
i+ 1
2
t
+
¸
1
ti+ 2
i+ 1
2
t
ku∗ i+1 k21,Ω
i+ 1
2
t
#
.
e n+1 and q = pn+1 in (4.3)–(4.5), we obtain
Proof. Letting τ = σ n+1 , v = u
un+1 k21,Ωtn+1
(4.7) λkσ n+1 k20,Ωtn+1 + 2α ρke
h
1
+λ ∆t c(e
un+1 − wn+ 2 , σ n+1 , σ n+1 )Ω n+ 1
t
2
i
1
(∇ · wn+ 2 ), σ n+1 )Ω n+ 1
,σ
,σ
)Ω n+ 1 − (σ
2
2
t
t
h
n+1
n+ 21
n+1
n+1
n+1
n+1
e
e
e
u
(∇ · w
+2∆t αρ b(e
u
,u
,u
)Ω n+ 1 − (e
), u
)Ω n+ 1
∗ n+1
+c(u
n+1
n+1
n+1
2
t
2
t
e n+1 )Ω n+ 1 + (e
e n+1 )Ω n+ 1
un+1 , u
−(wn+ 2 · ∇e
un+1 · ∇u∗ n+1 , u
2
2
t
t
i
n+1
n+1
∗ n+1
e
· ∇e
u
,u
)Ω n+ 1
+(u
1
t
2
e n+1 ), (σ n+1 , u
e n+1 ))Ω n+ 1
+∆t At ((σ n+1 , u
t
2
1
e n+1 )Ωtn + ∆t (e
un , u
f n+ 2 , (un+1 , σ n+1 ))Ω n+ 1 ,
= λ(σ n , σ n+1 )Ωtn + 2α ρ(e
2
t
1
where (e
f n+ 2 , (un+1 , σ n+1 ))Ω n+ 1 is defined as (3.11). By (2.11), (3.2), (3.5), (3.6), we
t
2
have
(4.8)
c(e
un+1 − wn+ 2 , σ n+1 , σ n+1 )Ω n+ 1 + c(u∗ n+1 , σ n+1 , σ n+1 )Ω n+ 1
1
t
1
2
t
−(σ n+1 (∇ · wn+ 2 ), σ n+1 )Ω n+ 1
t
2
1
1
≥ − (σ n+1 (∇ · wn+ 2 ), σ n+1 )Ω n+ 1
2
2
t
1
n+ 21
∗ n+1
) · n)σ n+1 , σ n+1 )Γ n+ 1 ,
+ (((u
−w
2
2
t
2
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J. HOWELL, H. LEE, AND S. XU
and
(4.9)
1
e n+1 )Ω n+ 1
e n+1 , u
e n+1 )Ω n+ 1 − (e
b(e
un+1 , u
un+1 (∇ · wn+ 2 ), u
2
t
− (w
n+ 12
n+1
· ∇e
u
2
t
n+1
e
,u
n+1
)Ω n+ 1 + (e
u
+ (u
· ∇u
2
t
∗ n+1
∗ n+1
n+1
· ∇e
u
n+1
e
,u
)Ω n+ 1
2
t
n+1
e
,u
)Ω n+ 1
t
2
1
1 n+1
e n+1 )Ω n+ 1 − ǫke
u
(∇ · wn+ 2 ), u
≥ − (e
un+1 k1,Ω n+ 1 .
2
2
2
t
t
Therefore, using (2.16), (4.2), (4.8) and (4.9), we obtain a lower bound for the left hand
side of (4.7):
(4.10)
LHS ≥
·
i
λ h n+1 2
kσ
k0,Ωtn+1 + kσ n+1 k20,Ωtn
2
h
+ αρ ke
un+1 k21,Ωtn+1 − ke
un+1 k21,Ωtn
+ ∆t (1 −
4λM )kσ n+1 k20,Ω
n+ 1
2
t
+ (4α(1 −
α)Cp2
−
i
2αρǫ)ke
un+1 k21,Ω
n+ 1
2
t
¸
λ
+ ∆t (((u∗ n+1 − wn+1 ) · n)σ n+1 , σ n+1 )Γ n+ 1 .
2
2
t
On the other hand, we have an upper bound for the right side of (4.7) by (2.10), (3.4), the
Poincaré inequality and Young’s inequality:
(4.11)
RHS ≤
λ n 2
λ
kσ k0,Ωtn + kσ n+1 k20,Ωtn + αρke
un k21,Ωtn + αρke
un+1 k21,Ωtn
2
2
"
° ∗ n+1 °2
° ∂u
°
n+ 21 2
k−1,Ω 1 + °
|y °
+ C ∆t kf
°
°
n+
∂t
2
t
0,Ω
+ ku∗ n+1 k41,Ω
n+ 1
2
t
+ ku∗ n+1 k21,Ω
n+ 1
2
t
+ δ1 ke
un+1 k21,Ω
for δ1 , δ2 > 0. Choosing ǫ =
that
(4.12)
2α(1−α)Cp2
2αρ+1 ,
δ1 = ǫ, δ2 =
λ n+1 2
kσ
k0,Ωtn+1 + αρke
un+1 k21,Ωtn+1
2
·
+∆t 2α(1 −
α)Cp2 ke
un+1 k21,Ω
n+ 1
2
t
n+ 1
2
t
1−4λM
,
2
n+ 1
2
#t
+ δ2 kσ n+1 k20,Ω
n+ 1
2
t
we get from (4.10)–(4.11)
1 − 4λM n+1 2
+
kσ
k0,Ω 1
n+
2
2
t
¸
λ
+∆t (((u∗ n+1 − wn+1 ) · n)σ n+1 , σ n+1 )Γ n+ 1
2
2
t

° ∗ n+1 °2
° ∂u
°
1
λ
°
≤ kσ n k2Ωtn + αρke
un k2Ωtn + C ∆t kf n+ 2 k2−1,Ω 1 + °
°
°
n+
2
∂t
2
t
0,Ω n+ 1
2
t
¸
+ku∗ n+1 k41,Ω 1 + ku∗ n+1 k21,Ω 1 .
n+
2
t
n+
2
t
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Summing over all times steps in (4.12), we obtain (4.6).
Next, we consider the second order scheme based on (4.2).
e n+1 , pn+1 ) satisfying
A LGORITHM 4.4 (Second-order GCL). Find (σ n+1 , u
i
h
(4.13) λ (σ n+1 , τ )Ωtn+1 − (σ n , τ )Ωtn
" µ
¶
n+1
e n+1 + u
en
1 σ
u
+ σn
+λ ∆t c
− wn+ 2 ,
,τ
2
2
Ω n+ 1
2
t
µ
µ n+1
¶
¶
n+1
n
n
1 σ
+
σ
σ
+
σ
∗ n+ 2
n+ 12
,
+c u
−
,τ
(∇ · w
), τ
2
2
Ω n+ 1
Ω n+ 1
2
2
t
t
#
¶ ¶
µ µ n+1
1
σ
+ σn
, ∇bn+ 2 , τ
+ ga
2
Ω n+ 1
2
t


µ
µ
µ n+1
¶
¶
¶
n
n+1
n
e
e
σ
+σ
u
+u

− 2α D
+∆t 
,τ
,τ
2
2
Ω
Ω
1
1
n+
2
t
n+
2
t
∗ n+ 21
= 2α ∆t(D(u
), τ )Ω n+ 1 ∀τ ∈ Σh,t ,
t
2
£ n+1
¤
ρ (e
u
, v)Ωtn+1 − (e
uh , v)Ωtn

¶
µ n+1
e n+1 + u
en
e
en u
u
+u

,
,v
+∆t ρ b
2
2
Ω n+ 1
2
t
µ n+1
µ
¶
¶
e
en
e n+1 + u
en
1
u
+u
u
n+ 21
n+ 2
−
− w
), v
·∇
(∇ · w
,v
2
2
Ω
Ω
1
(4.14)
n+
2
t
n+ 1
2
t

¶
µ
¶
n+1
n
1
1
e
e
e n+1 + u
en
u
+
u
u
n+
n+

+
+ u∗ 2 · ∇
· ∇u∗ 2 , v
,v
2
2
Ω n+ 1
Ω n+ 1
2
2
t
t

¶
¶
¶
µ µ n+1
µ n+1
n
n
e
e
u
+u
σ
+σ

, D(v)
, D(v)
+∆t 2(1 − α) D
+
2
2
Ω n+ 1
Ω n+ 1
2
2
t
t
#
µ
+(pn+1 , ∇ · v)Ω n+ 1
t
"
= ∆t (f
n+ 21
, v)Ω n+ 1 − ρ
t
2
Ã
∂u∗
∂t
n+ 21
∗ n+ 12
|y +(u
−w
1
−2(1 − α)(D(u∗ n+ 2 ), D(v))Ω n+ 1
t
(4.15)
µ
q, ∇ ·
e n+1 + u
en
u
2
¶
=0
Ω
n+ 1
2
t
2
2
n+ 12
#
∗ n+ 21
) · ∇u
∀v ∈ Uh,t ,
∀q ∈ Qh,t .
,v
!
Ω
n+ 1
2
t
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A conditional stability result for Algorithm 4.4 is obtained by the same approach as in
the proof of Theorem 4.3. The discretized time derivative terms can be estimated as shown in
[32] for the Navier-Stokes equation. We present the result without a proof.
T HEOREM 4.5. A solution to the fully discretized system (4.13)–(4.15) satisfies the inequality
λkσ n+1 k20,Ωtn+1 + 2αρke
un+1 k21,Ωtn+1
n ·
X
e i k21,Ω
2α(1 − α)Cp2 ke
ui+1 + u
+∆t
i=0

Z
n
X
λ

+∆t
4 Γ
i=0
i+ 1
2
t
1 − 4λM i+1
+
kσ
+ σ i k20,Ω 1
i+
2
2
t
((u∗ i+1 − wi+ 2 ) · n)|σ i+1 + σ i |2 dΓ
1
i+ 1
2
t
Z
µ
αρ i+1
λ i+1
e i |2
(∇ · w
−
)
|σ
− σ i |2 +
|e
u
−u
4
2
Ω i+ 1
2
t

n
X
 i+ 12 2
u0 k21,Ω0 + C ∆t
≤ λkσ 0 k20,Ω0 + 2αρke
k−1,Ω
kf
i+ 12
¸
1
+ku∗ i+ 2 k41,Ω
1
i+ 1
2
t
¶
i+ 1
2
t
i=0
1
ti+ 2
+ ku∗ i+ 2 k21,Ω
dΩ
1
ti+ 2


°
°2
° ∂u∗ i+ 21
°
°
°
+°
|by °
° ∂t
°
0,Ω
i+ 1
2
t
¸
i+ 1
2
t
.
5. Numerical results. In this section we present numerical results of two experiments
for the model equations
(5.1)
¶
∂σ
+ u · ∇σ + ga (σ, ∇u) − 2 α D(u) = f1
σ+λ
∂t
µ
¶
∂u
ρ
+ u · ∇u − ∇ · σ − 2(1 − α) ∇ · D(u) + ∇p = f2
∂t
µ
div u = 0
in Ωt ,
in Ωt ,
in Ωt .
Although the model equations were analyzed with the linearized ga term in (5.1) and the homogeneous stress boundary condition was assumed to simplify the analysis, we approximated
the model equations in the standard setting without such simplifications.
The first experiment is to investigate convergence of algorithms with decreasing grid
sizes and time steps. The second experiment is designed to test stability of the algorithms.
Experiment 1. Numerical experiments were performed using a non-physical example problem with a known exact solution. The initial domain is chosen as Ω0 = {y : y ∈ [0, 1] ×
[0, 1]} at t = 0 and the domain thereafter is defined by
(5.2)
Ωt = {x : x1 = y1 (2 − cos(πt)), x2 = y2 (2 − cos(πt)) for y ∈ Ω0 }.
Using the parameters λ = 0.5, α = 0.5, a = 0, the right hand side functions f1 , f2 were
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appropriately given so that the exact solution is
¸
·

10 sin(2πt + 1)x21 (x1 − 1)2 x2 (2x2 − 1)(x2 − 1)

 u=


−10 sin(2πt + 2)x22 (x2 − 1)2 x1 (2x1 − 1)(x1 − 1)



p = sin(πt + 2) cos(2πx1 )x2 (x2 − 1)


·
¸




 σ = σ11 σ12
σ21 σ22
in the initial domain, where σ11 = 10 sin(2πt + 1)x21 (x1 − 1)2 x2 (2x2 − 1)(x2 − 1),
σ12 = σ21 = −10 sin(2πt + 2)x22 (x2 − 1)2 x1 (2x1 − 1)(x1 − 1) and σ22 = 0. To approximate the flow equations, we used the Taylor-Hood pair for (u, p) and discontinuous
piecewise linear elements for σ. Since the domain is defined by (5.2), we used the exact ALE
mapping. The exact domain velocity given by
·
¸T
x1 π sin(πt) x2 π sin(πt)
(5.3)
w=
.
,
2 − cos(πt) 2 − cos(πt)
In the first test, we computed the L2 , H 1 errors of velocity, and the L2 error of stress
using the fixed number of elements generated by a 26×26 uniform grid and various time steps
1
1
1
1 1 1
, 5 , 10 , 15
, 20
, 30
. With the number of elements chosen, we expected the errors to
∆t = 2.5
be dominated by the time step when large ∆t values are used. Results by Algorithm 4.2 and
Algorithm 4.4 are summarized in Table 5.1 and Table 5.2, respectively. Errors calculated
by Algorithm 4.2 converge superlinearly, and the convergence rates for Algorithm 4.4 are
higher than for Algorithm 4.2 (as expected). The L2 errors of the velocity show quadratic
convergence.
In the second test we computed velocity and stress errors for various grid sizes with a
small fixed ∆t so that the finite element discretization error dominates the total error. The er1
rors computed on different meshes using ∆t = 2000
are presented in Table 5.3 and Table 5.4.
Baranger and Sandri [2] derived the finite element error estimate for the model equations in a
fixed domain as
kσ − σ h k0 + kD(u) − D(uh )k0 ≤ C h3/2 ,
for the (P2 , P1 , P1DG ) (Taylor-Hood, discontinuous linear) elements.
TABLE 5.1
Errors of velocity and stress by Algorithm 4.2 for t = 0.4.
∆t
1/2.5
1/5
1/10
1/15
1/20
1/30
L2 error
.1090 · 10−2
.5140 · 10−3
.1820 · 10−3
.1019 · 10−3
.6905 · 10−4
.4114 · 10−4
velocity
L2 rate
H 1 error
.4877 · 10−2
1.11
.1974 · 10−2
1.50
.6523 · 10−3
1.43
.3801 · 10−3
1.35
.2673 · 10−3
1.28
.1670 · 10−3
H 1 rate
1.30
1.30
1.33
1.22
1.16
stress
L2 error
L2 rate
.1310 · 10−2
.5539 · 10−3
1.24
.1683 · 10−3
1.72
.9393 · 10−4
1.44
.6404 · 10−4
1.33
.3822 · 10−4
1.27
Experiment 2. In this experiment we investigated the numerical stability of Algorithm 4.2
and Algorithm 4.4. In this test, a nonzero initial velocity u and stress σ are prescribed (at
t = 0), and the right-hand side functions f1 , f2 of the equations are set to 0, and the moving
domain boundary is assumed to follow the same specification as given in Experiment 1 (5.3).
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TABLE 5.2
Errors of velocity and stress by Algorithm 4.4 for t = 0.4.
∆t
1/2.5
1/5
1/10
1/15
1/20
1/30
L2 error
.1312 · 10−2
.3172 · 10−3
.6250 · 10−4
.2674 · 10−4
.1490 · 10−4
.6783 · 10−5
velocity
L2 rate
H 1 error
.5575 · 10−2
2.05
.1638 · 10−2
2.34
.2798 · 10−3
2.09
.1437 · 10−3
2.03
.9515 · 10−4
1.94
.5815 · 10−4
H 1 rate
1.77
2.55
1.64
1.44
1.21
stress
L2 error
L2 rate
−2
.1706 · 10
.6081 · 10−3
1.48
.1300 · 10−3
2.23
.5559 · 10−4
2.10
.3380 · 10−4
1.73
.2872 · 10−4
0.40
TABLE 5.3
Errors of velocity and stress by Algorithm 4.2 for t = 0.2.
grid
3×3
5×5
9×9
13 × 13
17 × 17
L2 error
.2645 · 10−4
.8043 · 10−5
.1859 · 10−5
.1139 · 10−5
.9111 · 10−6
velocity
L2 rate
H 1 error
.3244 · 10−3
1.72
.1291 · 10−3
2.11
.2477 · 10−4
1.21
.1112 · 10−4
.78
.7286 · 10−5
H 1 rate
1.33
2.38
1.98
1.47
stress
L2 error
L2 rate
−3
.2853 · 10
.8307 · 10−4
1.78
.1960 · 10−4
2.08
.8538 · 10−5
2.05
.4681 · 10−5
2.09
TABLE 5.4
Errors of velocity and stress by Algorithm 4.4 for t = 0.2.
grid
3×3
5×5
9×9
13 × 13
17 × 17
L2 error
.2778 · 10−4
.9417 · 10−5
.1455 · 10−5
.6049 · 10−6
.3916 · 10−6
velocity
L2 rate
H 1 error
.3483 · 10−3
1.56
.1570 · 10−3
2.69
.2544 · 10−4
2.16
.1230 · 10−4
1.51
.6256 · 10−5
H 1 rate
1.15
2.63
1.79
2.35
stress
L2 error
L2 rate
−3
.3413 · 10
.1006 · 10−3
1.76
.2172 · 10−4
2.21
.1002 · 10−4
1.91
.6734 · 10−5
1.38
If α 6= 1 and if λ and M are not too large (as required by Theorems 4.3 and 4.5), as time
proceeds beyond the initial value, the solution components u and σ are expected to decay,
eventually leading to kuk1,Ωt = kσk0,Ωt = 0. Provided the algorithms presented in Section 4
are stable, the computed approximations should decay as well. The divergence-free initial
velocity, shown in Figure 5.1, is given by
¸
·
10(x41 − 2x31 + x21 )(2x32 − 3x22 + x2 )
,
u=
−10(2x31 − 3x21 + x1 )(x42 − 2x32 + x22 )
the initial stress is σ = 2αD(u), and the initial pressure is p = 0. We use the parameter
values a = 0 and α = 0.5. Computations were performed on a uniform mesh with initial width h = 1/8 and time-step ∆t = 0.025 for the values λ = 0.1 and λ = 1.0, and
were allowed to continue until the norms kuk1,Ωt and kσk0,Ωt were sufficiently small. A
plot of these norms of solution components as time progresses is given for each value of λ
in Figures 5.2 and 5.3, respectively. As is observed in the plots, the computed approximations do decay for both methods. Note that for larger λ the numerical stability degrades,
due to the fact that the hypothesis 1 − 4M λ > 0 of Theorems 4.3 and 4.5 is violated, as
M ≥ k∇u0 k∞ = 0.625.
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F IG . 5.1. Initial velocity profile, Experiment 2.
Norms of Computed Approximations, λ = 0.1
0.16
1
Velocity H norm, First−order method
1
Velocity H norm, Second−order method
2
Stress L error, First−order method
2
Stress L error, Second−order method
0.14
Norm Values
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.5
1
1.5
Time
F IG . 5.2. Norms of computed approximations, λ = 0.1.
Norms of Computed Approximations, λ = 1.0
0.25
2
Stress L norm, First−order method
2
Stress L norm, Second−order method
1
Velocity H norm, First−order method
1
Velocity H norm, Second−order method
Norm Value
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
Time
F IG . 5.3. Norms of computed approximations, λ = 1.0.
8
9
10
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6. Concluding remarks. We have presented a rigorous stability analysis of a finite element method for the ALE formulation of viscoelastic flows in a moving domain. There
have been numerical results reported on viscoelastic flows in deformable domains or fluids
coupled with elastic solids. However, there are few analytical studies in literature for such
problems. This is our initial effort towards an analytical and numerical study of viscoelastic
flows in an elastic solid structure. Our numerical tests support the analytical stability results,
and suggest that the mesh convergence result for a fixed domain problem may still hold in the
case of moving domain problems. Subsequent work will further develop the fluid-structure
interaction model and include numerical experiments that are designed to determine if the
fluid model significantly affects the behavior of the coupled system.
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