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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 ETNA Kent State University http://etna.math.kent.edu 307 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. ETNA Kent State University http://etna.math.kent.edu 308 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. ETNA Kent State University http://etna.math.kent.edu 309 VISCOELASTIC FLOW IN MOVING DOMAIN 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 . ETNA Kent State University http://etna.math.kent.edu 310 J. HOWELL, H. LEE, AND S. XU 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 , ETNA Kent State University http://etna.math.kent.edu VISCOELASTIC FLOW IN MOVING DOMAIN 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 ETNA Kent State University http://etna.math.kent.edu 312 J. HOWELL, H. LEE, AND S. XU 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 } . ETNA Kent State University http://etna.math.kent.edu VISCOELASTIC FLOW IN MOVING DOMAIN 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 ETNA Kent State University http://etna.math.kent.edu 314 J. HOWELL, H. LEE, AND S. XU 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 , ETNA Kent State University http://etna.math.kent.edu 315 VISCOELASTIC FLOW IN MOVING DOMAIN 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 ETNA Kent State University http://etna.math.kent.edu 316 J. HOWELL, H. LEE, AND S. XU 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. ETNA Kent State University http://etna.math.kent.edu 317 VISCOELASTIC FLOW IN MOVING DOMAIN 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 . ETNA Kent State University http://etna.math.kent.edu 318 J. HOWELL, H. LEE, AND S. XU 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 ETNA Kent State University http://etna.math.kent.edu 319 VISCOELASTIC FLOW IN MOVING DOMAIN 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 ETNA Kent State University http://etna.math.kent.edu 320 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 ETNA Kent State University http://etna.math.kent.edu 321 VISCOELASTIC FLOW IN MOVING DOMAIN 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 ETNA Kent State University http://etna.math.kent.edu 322 J. HOWELL, H. LEE, AND S. XU 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 ETNA Kent State University http://etna.math.kent.edu VISCOELASTIC FLOW IN MOVING DOMAIN 323 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). ETNA Kent State University http://etna.math.kent.edu 324 J. HOWELL, H. LEE, AND S. XU 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. ETNA Kent State University http://etna.math.kent.edu 325 VISCOELASTIC FLOW IN MOVING DOMAIN 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 ETNA Kent State University http://etna.math.kent.edu 326 J. HOWELL, H. LEE, AND S. XU 6. Concluding remarks. 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