Finite Element Approximation of the FENE-P Model

FINITE ELEMENT APPROXIMATION OF THE FENE-P MODEL

arXiv:1704.00886v1 [math.NA] 4 Apr 2017

´ JOHN W. BARRETT AND SEBASTIEN BOYAVAL Abstract. We extend our analysis on the Oldroyd-B model in Barrett and Boyaval [1] to consider the finite element approximation of the FENE-P system of equations, which models a dilute polymeric fluid, in a bounded domain D ⊂ Rd , d = 2 or 3, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conformation tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics (d = 2) or a reduced version, where the tangential component on each simplicial edge (d = 2) or face (d = 3) is linear. In case (b) the velocity field is approximated by continuous piecewise quadratics or the mini-element. We show that both of these types of schemes, based on the backward Euler type time discretization, satisfy a free energy bound, which involves the logarithm of both the conformation tensor and a linear function of its trace, without any constraint on the time step. Furthermore, for our approximation (b) in the presence of an additional dissipative term in the stress equation, the so-called FENE-P model with stress diffusion, we show (subsequence) convergence in the case d = 2, as the spatial and temporal discretization parameters tend to zero, towards global-intime weak solutions of this FENE-P system. Hence, we prove existence of global-in-time weak solutions to the FENE-P model with stress diffusion in two spatial dimensions.

Keywords: FENE-P model, entropy, finite element method, convergence analysis, stress diffusion, existence of weak solutions AMS Subject Classification: 35Q30, 65M12, 65M60, 76A10, 76M10, 82D60 1. Introduction 1.1. The FENE-P model. We consider the standard FENE-P model for a dilute polymeric fluid. The fluid, confined to an open bounded domain D ⊂ Rd (d = 2 or 3) with a Lipschitz boundary ∂D, is governed by the following non-dimensionalized system for a given b ∈ R>0 : (P) Find u : (t, x) ∈ [0, T ) × D 7→ u(t, x) ∈ Rd , p : (t, x) ∈ DT := (0, T ) × D 7→ p(t, x) ∈ R and σ : (t, x) ∈ [0, T ) × D 7→ σ(t, x) ∈ Rd×d S,>0,b such that ε ∂u (1.1a) + (u · ∇)u = −∇p + (1 − ε)∆u + div (A(σ) σ) + f on DT , Re ∂t Wi (1.1b) div u = 0 on DT , (1.1c) (1.1d)

∂σ A(σ) σ + (u · ∇)σ = (∇u)σ + σ(∇u)T − ∂t Wi u(0, x) = u0 (x) 0

(1.1e)

σ(0, x) = σ (x)

(1.1f)

u=0

where (1.2)

−1 tr(φ) I − φ−1 A(φ) := 1 − b

on DT , ∀x ∈ D,

∀x ∈ D,

on (0, T ) × ∂D; n o d×d ∀φ ∈ Rd×d := ψ ∈ R : tr(ψ) < b . S,>0 S,>0,b

Here Rd×d denotes the set of symmetric Rd×d matrices, and Rd×d S S,>0 the set of symmetric positive d×d d×d definite R matrices. In addition, I ∈ RS,>0 is the identity, and tr(·) denotes trace. The unknowns in (P) are the velocity of the fluid, u, the hydrostatic pressure, p, and the symmetric conformation tensor of the polymer molecules, σ. The latter is linked to the symmetric polymeric ε A(σ) σ. In addition, f : (t, x) ∈ DT 7→ f (t, x) ∈ extra-stress tensor τ through the relation τ = Wi 1

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´ JOHN W. BARRETT AND SEBASTIEN BOYAVAL

Rd is the given density of body forces acting on the fluid; and the following given parameters are dimensionless: the Reynolds number Re ∈ R>0 , the Weissenberg number Wi ∈ R>0 , the elasticto-viscous viscosity fraction ε ∈ (0, 1), and the FENE-P parameter b > 0 (related to a maximal admissible extensibility of the polymer molecules within the fluid). For the sake of simplicity, we will limit ourselves to the no flow boundary conditions (1.1f). Finally, we denote ∇u(t, x) ∈ Rd×d ∂ui , and (div σ)(t, x) ∈ Rd the vector field with the velocity gradient tensor field with [∇u]ij = ∂x j Pd ∂σij [div σ]i = j=1 ∂xj . For data f ≡ 0, divergence free u0 ∈ [L2 (D)]d , and σ 0 , which is symmetric positive def0 ) ) ∈ L1 (D), then the existence of a global-in-time inite for a.e. x ∈ D, satisfying ln(1 − tr(σ b ∞ 2 d 2 weak solution u ∈ L (0, T ; [L (D)] ) ∩ L (0, T, [H01 (D)]d ), σ ∈ L∞ (0, T ; [L∞ (D)]d×d ) and τ ∈ L2 (0, T ; [L2(D)]d×d ) to (P), (1.1a–f), was proved in Masmoudi [20]. In this work, we consider finite element approximations of the FENE-P system (P) and the corresponding model with stress diffusion, (Pα ), which is obtained by adding the dissipative term α ∆σ for a given α ∈ R>0 to the right-hand side of (1.1c) with an additional no flux boundary condition for σ on ∂D. This paper extends the results in Barrett and Boyaval [1], where finite element approximations of the corresponding Oldroyd-B models, where A(σ) = I − σ −1 , were introduced and analysed. In fact, the convergence proof of the finite element approximation of the Oldroyd-B model with stress diffusion for d = 2 in [1] provided the first existence proof of global-in-time weak solutions for this system. Note that A(σ) = I − σ −1 is the formal limit of (1.2) for infinite extensibility; that is, b → ∞. The model (Pα ) has been considered computationally in Sureshkumar and Beris [24]. We recall also that El-Kareh and Leal [14] showed the existence of a weak solution to a modified stationary FENE-P system of equations, which included stress diffusion, but there an additional regularization was also present in their modified system and played an essential role in their proof. We stress that the dissipative term α ∆σ in (Pα ) is not a regularization, but can be physically motivated through the centre-of-mass diffusion in the related microscopic-macroscopic polymer model, though with a positive α ≪ 1, see Barrett and S¨ uli [3], [5], Schieber [23] and Degond and Liu [13]. Barrett and S¨ uli have introduced, and proved the existence of global-in-time weak solutions for d = 2 and 3 to, microscopic-macroscopic dumbbell models of dilute polymers with center-of-mass diffusion in the corresponding Fokker–Planck equation for a finitely extensible nonlinear elastic (FENE) spring law or a Hookean-type spring law, see [4] and [6]. Recently, Barrett and S¨ uli [8] have proved rigorously that the macroscopic Oldroyd-B model with stress diffusion is the exact closure of the microscopic-macroscopic Hookean dumbbell model with center-of-mass diffusion for d = 2, when the existence of global-in-time weak solutions to both models can be proved. In addition, Barrett and S¨ uli [7] have introduced and analysed a finite element approximation for the FENE microscopic-macroscopic dumbbell model with center-of-mass diffusion. From a physical viewpoint, the FENE-P model is more realistic than the Oldroyd-B model because it accounts for the finite-extensibility of the polymer molecules in the fluid through the non-dimensional parameter b > 0. From a mathematical viewpoint, compared to the Oldroyd-B model where the nonlinear terms are only the material derivative terms (like (∇u)σ), the FENE−1 in the definition P model has an additional singular nonlinearity due to the factor 1 − tr(·) b of A(·), which necessitates a careful mathematical treatment. Hence, this paper is not a trivial extension of [1]. In fact, the latter additional nonlinearity is exactly what makes the FENEP model closer to the physics of polymers than the Oldroyd-B model, and thus also to many other macroscopic models based on different constitutive relations that have been developed by physicists for polymers. We note the FENE-P system is the approximate macroscopic closure of the FENE microscopic-macroscopic dumbbell model, whereas the Oldroyd-B system is the exact macroscopic closure of the Hookean microscopic-macroscopic dumbbell model. Hence, the microscopic-macroscopic dumbbell models corresponding to Oldroyd-B and FENE-P, only the spring laws differ; see e.g. Bird et al. [9] and Renardy [22] for a more complete review of the differences between the Oldroyd-B and the FENE-P models from the physical viewpoint, and for


3

other macroscopic models with more nonlinear effects than the Oldroyd-B model, e.g. the Giesekus model and the Phan–Thien Tanner model. Similarly to [1], our analysis in the present paper exploits the underlying free energy of the system, see Wapperom and Hulsen [26] and Hu and Lelièvre [19]. In particular, the finite element approximation of (Pα ) has to be constructed extremely carefully to inherit this free energy structure, and requires the approximation of tr(σ) as a new unknown. It is definitely not our goal to review all the macroscopic models used in rheology, although similar studies could probably be pursued for other macroscopic models endowed with a free energy. We will point out the main differences with Barrett and Boyaval [1], and we thus hope to sufficiently suggest how our technique could be adapted to any nonlinear model with a free energy. We believe that our approach contributes to a better understanding of the numerical stability of the models used in computational rheology, where numerical instabilities sometimes termed “High-Weissenberg Number Problems”, see HWNP in Owens and Phillips [21], still persist. Indeed, as exposed in Boyaval et al. [11], our point is that to make progress in this area one should identify sufficiently general rules for the derivation of good discretizations of macroscopic models such that they retain the dissipative structure of weak solutions to the system, at least in some benchmark flows. The outline of this paper is as follows. First, we end this section by introducing our notation and some auxiliary results. In Section 2 we review the formal free energy bound for the FENE-P system (P). In Section 3 we introduce our regularization Gδ of of G ≡ ln, which appears in the definition of the free energy of the FENE-P system (P). We then introduce a regularized problem (Pδ ), and show a formal free energy bound for it. In Section 4, on assuming that D is a polytope for ease of exposition, we introduce our finite element approximation of (Pδ ), namely (P∆t δ,h ), based on approximating the pressure and the symmetric conformation tensor by piecewise constants; and the velocity field with continuous piecewise quadratics or a reduced version, where the tangential component on each simplicial edge (d = 2) or face (d = 3) is linear. Using the Brouwer fixed point theorem, we prove existence of a solution to (P∆t δ,h ) and show that it satisfies a discrete regularized free energy bound for any choice of time step; see Theorem 4.1. We conclude by showing that, in ∆t the limit δ → 0+ , these solutions of (P∆t δ,h ) converge to a solution of (Ph ) with the approximation of the conformation tensor being positive definite and having a trace strictly less than b. Moreover, this solution of (P∆t h ) satisfies a discrete free energy bound; see Theorem 4.2. Next, in Section 5 we introduce the FENE-P system with stress diffusion, (Pα ), where the dissipative term α ∆σ has been added to the right-hand side of (1.1c). We then introduce the corresponding regularized version (Pα,δ ), and show a formal free energy bound for it. In Section 6 we introduce our finite element approximation of (Pα,δ ), namely (P∆t α,δ,h ), based on approximating the velocity field with continuous piecewise quadratics or the mini element, and the pressure, the symmetric conformation tensor and its trace by continuous piecewise linears. Here we assume that D is a convex polytope and that the finite element mesh consists of quasi-uniform non-obtuse simplices. Using the Brouwer fixed point theorem, we prove existence of a solution to (P∆t α,δ,h ) and show that it satisfies a discrete regularized free energy bound for any choice of time step; see Theorem 6.1. In Section 7 we prove, in the case d = 2, (subsequence) convergence of the solutions of (P∆t α,δ,h ), as the regularization parameter, δ, and the spatial, h, and temporal, ∆t, discretization parameters tend to zero, to global-in-time weak solutions of (Pα ); see Theorem 7.3. This existence result for (Pα ) is new to the literature. 1.2. Notation and auxiliary results. The absolute value and the negative part of a real number s ∈ R are denoted by |s| := max{s, −s} and [s]− = min{s, 0}, respectively. We adopt the following notation for inner products (1.3a)

(1.3b)

v · w := φ : ψ :=

d X i=1

v i w i ≡ v T w = wT v

d X d X i=1 j=1

φij ψ ij ≡ tr φT ψ = tr ψ T φ

∀v, w ∈ Rd , ∀φ, ψ ∈ Rd×d ,


4

(1.3c)

∇φ :: ∇ψ :=

d X d X i=1 j=1

∀φ, ψ ∈ Rd×d ;

∇φij · ∇ψ ij

where ·T and tr (·) denote transposition and trace, respectively. The corresponding norms are (1.4a) (1.4b)

1

1

kvk := (v · v) 2 ,

k∇vk := (∇v : ∇v) 2

kφk := (φ : φ) ,

k∇φk := (∇φ :: ∇φ)

1 2

1 2

∀v ∈ Rd ;

∀φ ∈ Rd×d .

We will use on several occasions that tr(φ) = tr(φT ) and tr(φψ) = tr(ψφ) for all φ, ψ ∈ Rd×d , and (1.5a)

φχT : ψ = χφ : ψ = χ : ψφ

(1.5b)

kψφk ≤ kψk kφk

(1.5c)

kφ vk ≤ kφk kvk

∀φ, ψ ∈ Rd×d , χ ∈ Rd×d , S

∀φ, ψ ∈ Rd×d ,

∀φ ∈ Rd×d , v ∈ Rd .

For any φ ∈ Rd×d , there exists a decomposition S (1.6)

φ = OT DO

⇒

tr (φ) = tr (D) ,

⇒

tr (g(φ)) = tr (g(D)) ,

where O ∈ Rd×d is an orthogonal matrix and D ∈ Rd×d a diagonal matrix. Hence, for any g : R → R, one can define g(φ) ∈ Rd×d as S (1.7)

g(φ) := OT g(D)O

where g(D) ∈ Rd×d is the diagonal matrix with entries [g(D)]ii = g(Dii ), i = 1, . . . , d. Although S the diagonal decomposition (1.6) is not unique, (1.7) uniquely defines g(φ). We note for later purposes that (1.8)

2

d−1 (tr(|φ|))2 ≤ kφk ≤ (tr(|φ|))2

∀φ ∈ Rd×d . S

One can show via diagonalization, see e.g. [1] for details, that for all concave function g ∈ C 1 (R), it holds (1.9)

(φ − ψ) : g ′ (ψ) ≥ tr (g(φ) − g(ψ)) ≥ (φ − ψ) : g ′ (φ)

∀φ, ψ ∈ Rd×d , S

where g ′ denotes the first derivative of g. If g ∈ C 1 (R) is convex, the inequalities in (1.9) are ) and any concave or reversed. It follows from (1.9) and (1.3b) that for any φ ∈ C 1 ([0, T ]; Rd×d S convex g ∈ C 1 (R) d dφ ′ dφ ′ (1.10) tr (g(φ)) = tr g (φ) = : g (φ) ∀t ∈ [0, T ]. dt dt dt Of course, a similar result holds for spatial derivatives. Furthermore, the results (1.9) and (1.10) d×d 1 hold true when C 1 (R) and Rd×d are replaced by C 1 (R>0 ) and Rd×d S S,>0 or C (0, b) and RS,>0,b . 0,1 Finally, one can show that if g ∈ C (R) with Lipschitz constant gLip , then (1.11)

kg(φ) − g(ψ)k ≤ gLip kφ − ψk

∀φ, ψ ∈ Rd×d . S

R We adopt the standard notation for Sobolev spaces, e.g. H 1 (D) := {η : D 7→ R : D [ |η|2 + k∇ηk2 ] dx < ∞} with H01 (D) being the closure of C0∞ (D) for the corresponding norm k · kH 1 (D) . We denote the associated semi-norm as | · |H 1 (D) . The topological dual of the Hilbert space H01 (D), with pivot space L2 (D), will be denoted by H −1 (D). Such function spaces are naturally extended when the range R is replaced by Rd , Rd×d and Rd×d ; e.g. H 1 (D) becomes [H 1 (D)]d , S d×d 1 d×d 1 [H (D)] and [H (D)]S , respectively. For ease of notation, we write the corresponding norms and semi-norms as k · kH 1 (D) and | · |H 1 (D) , respectively, as opposed to e.g. k · k[H 1 (D)]d and | · |[H 1 (D)]d , respectively. We denote the duality pairing between H −1 (D) and H01 (D) as h·, ·iH01 (D) , and we similarly write h·, ·iH01 (D) for the duality pairing between e.g. [H −1 (D)]d and [H01 (D)]d . For d×d 1 notational convenience, we introduce also convex sets such as [H 1 (D)]d×d : S,>0 := {φ ∈ [H (D)]S d×d d×d d×d d×d 1 1 φ ∈ RS,>0 a.e. in D}, and [H (D)]S,>0,b := {φ ∈ [H (D)]S : φ ∈ RS,>0,b a.e. in D}.


5

In order to analyse (P), we adopt the notation (1.12) W :=

[H01 (D)]d ,

2

Q := L (D),

Z V := v ∈ W : q div v dx = 0 ∀q ∈ Q , D

S := [L∞ (D)]d×d , S

S>0 := [L∞ (D)]d×d S,>0

and S>0,b := {φ ∈ S>0 : tr(φ) < b a.e. in D} .

Throughout the paper C will denote a generic positive constant independent of the regularization parameter δ and the mesh parameters h and ∆t. Finally, we recall the Poincaré inequality Z Z 2 (1.13) kvk dx ≤ CP k∇vk2 dx ∀v ∈ W, D

D

where CP ∈ R>0 depends only on D.

2. Formal free energy bound for the problem (P) In this section we recall from Hu and Lelièvre [19] the free energy structure of problem (P). Let F (u, σ) denote the free energy associated with a solution (u, p, σ) to problem (P), where we define Z Z Re tr(φ) ε F (v, φ) := (2.1) b ln 1 − + tr (ln(φ) + I) dx kvk2 dx − 2 D 2Wi D b

∀(v, φ) ∈ [L2 (D)]d × S⋆ R 2 with S⋆ ⊂ S>0,b such that F (·, ·) is well-defined. Here the first term Re 2 D kvk corresponds to the usual kinetic energy term, and the second term, which is nonnegative, is a relative entropy term. Moreover, on noting that ln is a concave function on R>0 , we observe Z Z Re ε 2 (2.2) F (v, φ) ≥ kvk dx + tr (φ − ln(φ) − I) dx ∀(v, φ) ∈ [L2 (D)]d × S⋆ , 2 D 2Wi D

where the right-hand side is the free energy of the Oldroyd-B model under the same no flow boundary conditions, see e.g. [19] and [1]. Clearly, diagonalization yields that the relative entropy term of this Oldroyd-B model is nonnegative. Of course, the I term in the relative entropy for FENE-P and Oldroyd-B plays no real role, and just means that the minimum relative entropy for Oldroyd-B is zero and is obtained by φ = I. Finally, we note that tr (ln(φ)) is rewritten as ln (det(φ)) in [19], which once again is easily deduced from diagonalization. Proposition 2.1. With f ∈ L2 (0, T ; [H −1 (D)]d ) let (u, p, σ) be a sufficiently smooth solution to problem (P), (1.1a–f ), such that σ(t, ·) ∈ S⋆ for t ∈ (0, T ). Then the free energy F (u, σ) satisfies for a.a. t ∈ (0, T ) Z Z d ε (2.3) tr (A(σ))2 σ dx = hf , uiH01 (D) , k∇uk2 dx + F (u, σ) + (1 − ε) 2 dt 2Wi D D where the third term on the left-hand side is positive, via diagonalization, on recalling (1.2).

Proof. Multiplying the Navier-Stokes equation (1.1a) with u and the stress equation (1.1c) with ε 2Wi A(σ), summing and integrating over D yields, after using integrations by parts, the boundary condition (1.1f) and the incompressibility property (1.1b) in the standard way, that # −1 Z " Re ∂kuk2 tr(σ) ε 1− (2.4) σ : ∇u dx + (1 − ε)k∇uk2 + 2 ∂t Wi b D # Z " ∂σ ε A(σ) σ : A(σ) dx + + (u · ∇)σ + 2Wi D ∂t Wi Z ε T : A(σ) dx = hf , uiH01 (D) . − (∇u) σ + σ (∇u) 2Wi D


6

It follows from the chain rule and (1.10) that tr(σ) ∂ ∂σ (2.5) − tr (ln(σ)) . + (u · ∇)σ : A(σ) = + (u · ∇) −b ln 1 − ∂t ∂t b

On integrating (2.5) over D, the (u · ∇) term on the right-hand side vanishes as u(t, ·) ∈ V. On noting (1.2), (1.5a), (1.3b) and (1.1b), we obtain that −1 tr(σ) T σ (∇u) + (∇u) σ : A(σ) = 2 1 − (2.6) σ : ∇u. b

Hence, on combining (2.4)–(2.6) and noting a trace property, we obtain the desired free energy equality (2.3). For later purposes, we note the following. ε Remark 2.1. The step in the above proof of testing (1.1c) with 2Wi A(σ) is equivalent to testing ε −1 (1.1c) with − 2Wi σ and testing the corresponding trace equation

tr (A(σ) σ) ∂ tr(σ) + (u · ∇) tr(σ) = 2∇u : σ − ∂t Wi −1 tr(σ) ε , and adding. with 2Wi 1 − b (2.7)

on DT

Recall that in the limit b → ∞ the FENE-P model formally converges to the Oldroyd-B model. It is thus interesting to note that when b → ∞, the free energy equality (2.3) formally converges to the corresponding free energy equality for the Oldroyd-B model, on recalling (2.2). Finally, we note the following result. Corollary 2.1. Under the assumptions of Proposition 2.1 it follows that Z Z 1−ε ε (2.8) tr (A(σ))2 σ dx dt sup F (u(t, ·), σ(t, ·)) + k∇uk2 dx dt + 2 2 2Wi t∈(0,T ) DT DT 1 + CP 2 ≤ 2 F (u0 , σ 0 ) + kf kL2 (0,T ;H −1 (D)) . 2(1 − ε) Proof. One can bound the term hf , uiH01 (D) in (2.3), using the Cauchy-Schwarz and Young inequalities for ν ∈ R>0 , and the Poincaré inequality (1.13), by (2.9)

hf , uiH01 (D) ≤ kf kH −1 (D) kukH 1 (D) ≤

ν2 1 2 kf k + kuk2H 1 (D) −1 H (D) 2ν 2 2

ν2 1 2 2 kf kH −1 (D) + (1 + CP ) k∇ukL2 (D) . 2 2ν 2 Combining (2.9) and (2.3) with ν 2 = (1 − ε)/(1 + CP ), and integrating in time yields the result (2.8). ≤

3. Formal free energy bound for a regularized problem (Pδ ) 3.1. A regularization. Let G : s ∈ R>0 7→ ln s ∈ R denote the logarithm function, whose domain of definition can be straightforwardly extended to the set of symmetric positive definite matrices using (1.6) and (1.7). We define the following concave C 1,1 (R) regularization of G based on a given parameter δ ∈ (0, 1): ( G(s) ∀s ≥ δ, Gδ : s ∈ R 7→ s (3.1) ⇒ Gδ (s) ≥ G(s) ∀s ∈ R>0 . + G(δ) − 1 ∀s ≤ δ δ We define also the following scalar functions (3.2)

βδ (s) := (G′δ (s))

Hence, we have that (3.3)

−1

∀s ∈ R

βδ : s ∈ R 7→ max{s, δ}

β(s) := (G′ (s))

and and

−1

β : s ∈ R>0 7→ s.

∀s ∈ R>0 .


7

For later purposes, we note the following results concerning these functions. Lemma 3.1. For any φ, ψ ∈ Rd×d , η ∈ R and for any δ ∈ (0, 1) we have that S βδ (φ)G′δ (φ) = G′δ (φ)βδ (φ) = I, 2 tr (ηI − G′δ (φ)) βδ (φ) ≥ 0,

(3.4a) (3.4b) (3.4c)

tr (φ − Gδ (φ) − I) ≥ 0,

(φ − ψ) : [G′δ (ψ)] ≥ tr (Gδ (φ) − Gδ (ψ)) ,

(3.4d)

2

− (φ − ψ) : [G′δ (φ) − G′δ (ψ)] ≥ δ 2 kG′δ (φ) − G′δ (ψ)k .

(3.4e)

In addition, if δ ∈ (0, 12 ] we have that ( 1 2 kφk (3.5) tr (φ − Gδ (φ)) ≥ 1 2δ k[φ]− k

and

φ : (I − G′δ (φ)) ≥ 21 kφk − d.

Proof. All the results are proved in Lemma 2.1 in [1], except (3.4b) and this can be easily proved via diagonalization. We introduce the following regularization of A, (1.2), for any δ ∈ (0, 12 ]: η (3.6) I − G′δ (φ) ∀(φ, η) ∈ Rd×d × R. Aδ (φ, η) := G′δ 1 − S b

In addition to Lemma 3.1 we will also make use of the following result, which is similar to (3.5). Lemma 3.2. For any s ∈ R, b ∈ R>0 and δ ∈ (0, 12 ], we have that s 1 −b Gδ 1 − − s ≥ [|s| − 3b]+ , (3.7a) b 2 s G′δ 1 − (3.7b) − 1 s ≥ [ |s| − b ]+ . b

Proof. On recalling (3.1), we first note from the concavity of Gδ that s ≥ −b Gδ (1) + s G′δ (1) = s ∀s ∈ R. (3.8) − b Gδ 1 − b From the scalar version of (3.5), we have that s s s 1 1− − Gδ 1 − ≥ 1 − (3.9) b b 2 b We note that

b 1 − 2

(3.10)

s −b= b

(

b s s − s ≥ 1 − − b. −b Gδ 1 − b 2 b

⇒

1 2 (|s| − 3b) − 12 (s + b) ≥ 21 (|s|

if s ≥ b, − 3b)

if s ≤ b.

Combining (3.8)–(3.10) yields the desired result (3.7a). We now consider (3.7b). If 1 − sb ≤ δ, i.e. s ≥ b(1 − δ), then 1 s −1 s= (3.11) − 1 s ≥ |s|. G′δ 1 − b δ If 1 − (3.12)

s b

≥ δ, i.e. s ≤ b(1 − δ), then

s2 s ≥0 −1 s= G′δ 1 − b b−s

and

Combining (3.11) and (3.12) yields the desired result (3.7b).

s2 ≥ |s| − b. b−s


8

3.2. The regularized problem (Pδ ). Using the regularizations Gδ , βδ and Aδ introduced above, we consider the following regularization of (P) for a given δ ∈ (0, 12 ]: (Pδ ) Find uδ : (t, x) ∈ [0, T ) × D 7→ uδ (t, x) ∈ Rd , pδ : (t, x) ∈ DT 7→ pδ (t, x) ∈ R and σ δ : (t, x) ∈ [0, T ) × D 7→ σ δ (t, x) ∈ Rd×d such that S (3.13a) ε ∂uδ + (uδ · ∇)uδ = −∇pδ + (1 − ε)∆uδ + div (Aδ (σ δ , tr(σ δ )) βδ (σ δ )) + f Re ∂t Wi (3.13b) div uδ = 0 (3.13c) ∂σ δ Aδ (σ δ , tr(σ δ )) βδ (σ δ ) + (uδ · ∇)σ δ = (∇uδ )βδ (σ δ ) + βδ (σ δ )(∇uδ )T − ∂t Wi (3.13d) uδ (0, x) = u0 (x) ∀x ∈ D,

(3.13e)

σ δ (0, x) = σ 0 (x)

(3.13f)

uδ = 0

on DT , on DT , on DT ,

∀x ∈ D,

on (0, T ) × ∂D.

3.3. Formal free energy bound for (Pδ ). In this section, we extend the formal energy results (2.3) and (2.8) for (P) to problem (Pδ ). We will assume throughout that (3.14) f ∈ L2 0, T ; [H −1 (D)]d , u0 ∈ H := {w ∈ [L2 (D)]d : div w = 0 a.e. in D, w · n∂D = 0 on ∂D}, 0 0 with σmin kξk2 ≤ ξ T σ 0 (x) ξ ≤ σmax kξk2 tr σ 0 (x) ≤ b⋆ for a.e. x in D;

σ 0 ∈ S>0

and

∀ξ ∈ Rd

for a.e. x in D

0 0 where n∂D is normal to ∂D, b⋆ , σmin , σmax ∈ R>0 with b⋆ < b. Let Fδ (uδ , σ δ , tr(σ δ )) denote the free energy associated with a solution (uδ , pδ , σ δ ) to problem (Pδ ), where we define Z Z h i η ε Re 2 (3.15) Fδ (v, φ, η) := b Gδ 1 − + tr (Gδ (φ) + I) dx kvk dx − 2 D 2Wi D b

∀(v, φ, η) ∈ [L2 (D)]d × S × L1 (D).

Note that the second term in Fδ has been regularized in comparison with F in (2.1). Similarly to (2.2), we have, on noting (3.8), the inequality (3.16) Z Z Re ε Fδ (v, φ, tr(φ)) ≥ kvk2 dx + tr (φ − Gδ (φ) − I) dx ∀(v, φ) ∈ [L2 (D)]d × S, 2 D 2Wi D where the right-hand side in (3.16) is the free energy of the corresponding regularized Oldroyd-B model, see [1] and note (3.4c). It also follows from (3.1) and (3.14) that (3.17)

Fδ (u0 , σ 0 , tr(σ 0 )) ≤ F (u0 , σ0 ).

Proposition 3.1. Let δ ∈ (0, 21 ] and (uδ , pδ , σ δ ) be a sufficiently smooth solution to problem (Pδ ), (3.13a–f ). Then the free energy Fδ (uδ , σ δ , tr(σ δ )) satisfies for a.a. t ∈ (0, T ) Z d (3.18) k∇uδ k2 dx Fδ (uδ , σ δ , tr(σ δ )) + (1 − ε) dt D Z ε 2 + tr (A (σ , tr(σ ))) β (σ ) dx = hf , uδ iH01 (D) , δ δ δ δ δ 2Wi2 D where the third term on the left-hand side is nonnegative from (3.6) and (3.4b).

Proof. Similarly to the proof of Proposition 2.1, we multiply the regularized Navier-Stokes equaε Aδ (σ δ , tr(σ δ )), sum and tion (3.13a) by uδ and the regularized stress equation (3.13c) with 2Wi integrate over D, use integrations by parts, the boundary condition (3.13f) and the incompressibility property (3.13b). This yields the desired result (3.18) on noting the following analogues of


9

(2.5) and (2.6) ∂σ δ (3.19) + (uδ · ∇)σ δ : Aδ (σ δ , tr(σ δ )) ∂t ∂ tr(σ δ ) = − tr(Gδ (σ δ )) + (uδ · ∇) −b Gδ 1 − ∂t b and (3.20)

tr(σ δ ) ′ βδ (σ δ ) : ∇uδ . βδ (σ δ ) (∇uδ ) + (∇uδ ) βδ (σ δ ) : Aδ (σ δ , tr(σ δ )) = 2 Gδ 1 − b T

Here we have recalled (3.6) and (1.10) for (3.19), and (1.5a), (1.3b), (3.4a) and (3.13b) for (3.20). Similarly to Remark 2.1, we note the following. Remark 3.1. The step in the above proof of testing the regularized stress equation (3.13c) with ε ε ′ 2Wi Aδ (σ δ , tr(σ δ )) is equivalent to testing (3.13c) with − 2Wi Gδ (σ δ ) and testing the corresponding regularized trace equation ∂ tr(σ δ ) tr(Aδ (σ δ , tr(σ δ )) βδ (σ δ )) + (uδ · ∇) tr(σ δ ) = 2∇uδ : βδ (σ δ ) − ∂t Wi ε δ) , and adding. G′δ 1 − tr(σ with 2Wi b (3.21)

on DT

Corollary 3.1. Under the assumptions of Proposition 3.1 it follows that Z 1−ε (3.22) k∇uδ k2 dx dt sup Fδ (uδ (t, ·), σ δ (t, ·), tr(σ δ (t, ·))) + 2 t∈(0,T ) DT Z ε + tr (Aδ (σ δ , tr(σ δ )))2 βδ (σ δ ) dx dt 2 2Wi DT 1 + CP 2 0 0 kf kL2 (0,T ;H −1 (D)) . ≤ 2 F (u , σ ) + 2(1 − ε)

Proof. The proof of (3.22) follows from (3.18) in the same way as (2.8) follows from (2.3), and in addition noting (3.17). 4. Finite element approximation of (Pδ ) and (P) 4.1. Finite element discretization. We now introduce a finite element discretization of the problem (Pδ ), which satisfies a discrete analogue of (3.18). The time interval [0, T ) is split into intervals [tn−1 , tn ) with ∆tn = tn − tn−1 , n = 1, . . . , NT . We set ∆t := maxn=1,...,NT ∆tn . We will assume throughout that the domain D is a polytope. We define a regular family of meshes {Th }h>0 with discretization parameter h > 0, which is built from partitionings of the domain D into regular open simplices so that NK

D = Th := ∪ Kk k=1

with

max

k=1,...,NK

hk ≤ C. ρk

Here ρk is the diameter of the largest inscribed ball contained in the simplex Kk and hk is the diameter of Kk , so that h = maxk=1,...,NK hk . For each element Kk , k = 1, . . . , NK , of the mesh Th let {Pik }di=0 denotes its vertices, and {nki }di=0 the outward unit normals of the edges (d = 2) or faces (d = 3) with nki being that of the edge/face opposite vertex Pik , i = 0, . . . , d. In addition, let {ηik (x)}di=0 denote the barycentric coordinates of x ∈ Kk with respect to the vertices {Pik }di=0 ; that is, ηik ∈ P1 and ηik (Pjk ) = δij , i, j = 0, . . . , d. Here Pm denote polynomials of maximal degree E m in x, and δij the Kronecker delta notation. Finally, we introduce ∂Th := {Ej }N j=1 as the set of internal edges Ej of triangles in the mesh Th when d = 2, or the set of internal faces Ej of tetrahedra when d = 3.


10

We approximate the problem (Pδ ) by the problem (P∆t δ,h ) based on the finite element spaces As is standard, we require the discrete velocity-pressure spaces Wh0 × Q0h ⊂ W × Q satisfy the discrete Ladyshenskaya-Babuˇska-Brezzi (LBB) inf-sup condition Z q div v dx D (4.1) inf0 sup ≥ µ⋆ > 0, q∈Qh v∈W0 kqkL2 (D) kvkH 1 (D) h

Wh0 × Q0h × S0h .

see e.g. [16, p114]. In the following, we set Wh0 := Wh2 ⊂ W if d = 2 or Wh2,− ⊂ W if d = 2 or 3,

(4.2a)

Q0h := {q ∈ Q : q |Kk ∈ P0

(4.2b) (4.2c)

and

k = 1, . . . , NK } ⊂ Q

S0h := {φ ∈ S : φ |Kk ∈ [P0 ]d×d S

k = 1, . . . , NK } ⊂ S;

where Wh2 := {v ∈ [C(D)]d ∩ W : v |Kk ∈ [P2 ]d

(4.3a)

Wh2,−

(4.3b)

d

d

k = 1, . . . , NK },

:= {v ∈ [C(D)] ∩ W : v |Kk ∈ [P1 ] ⊕ span{ς ki }di=0

k = 1, . . . , NK }.

Here, for k = 1, . . . , NK and i = 0, . . . , d d Y

ς ki (x) = nki

(4.4)

ηjk (x)

j=0,j6=i

We introduce also Vh0

(4.5)

for x ∈ Kk .

Z 0 0 q div v dx = 0 ∀q ∈ Qh , := v ∈ Wh : D

which approximates V. It is well-known that the choices (4.2a,b) satisfy (4.1), see e.g. [12, p221] for Wh0 = Wh2 and d = 2, and Chapter II, Sections 2.1 (d = 2) and 2.3 (d = 3) in [16] for Wh0 = Wh2,− . Moreover, these particular choices of S0h and Q0h have the desirable property that tr(φ) 0 0 + tr (Gδ (φ)) ∈ Q0h , (4.6) φ ∈ Sh ⇒ Aδ (φ, tr(φ)) ∈ Sh and b Gδ 1 − b which makes it a straightforward matter to mimic the free energy inequality (3.18) at a discrete level. Since S0h is discontinuous, we will use the discontinuous Galerkin method to approximate the advection term (uδ · ∇)σ δ in the following. Then, for the boundary integrals, we will make use of the following definitions (see e.g. [15, p267]). Given v ∈ Wh0 , then for any φ ∈ S0h (or Q0h ) and for any point x that is in the interior of some Ej ∈ ∂Th , we define the downstream and upstream values of φ at x by φ+v (x) = lim+ φ(x + ρ v(x))

(4.7)

ρ→0

and

φ−v (x) = lim− φ(x + ρ v(x)); ρ→0

respectively. In addition, we denote by φ+v (x) + φ−v (x) , 2 the jump and mean value, respectively, of φ at the point x of boundary Ej . From (4.7), it is clear that the values of φ+v |Ej and φ−v |Ej can change along Ej ∈ ∂Th . Finally, it is easily deduced that NK Z NE Z X X +v |v · n|[[q1 ]]→v q2 ds = − (4.9) (v · nKk ) q1 q2+v ds ∀v ∈ Wh0 , q1 , q2 ∈ Q0h ; [[φ]]→v (x) = φ+v (x) − φ−v (x)

(4.8)

j=1

Ej

k=1

and

v

{φ} (x) =

∂Kk

where n ≡ n(Ej ) is a unit normal to Ej , whose sign is of no importance, and nKk is the outward unit normal vector of boundary ∂Kk of Kk . We note that similar ideas appear in upwind schemes; e.g. see Chapter IV, Section 5 in [16] for the Navier-Stokes equations.


11

4.2. A free energy preserving approximation (P∆t δ,h ) of (Pδ ). For any source term f ∈ 2 −1 d L 0, T ; [H (D)] , we define the following piecewise constant function with respect to the time variable Z tn 1 f (t, ·) dt, t ∈ [tn−1 , tn ), n = 1, . . . , NT . (4.10) f ∆t,+ (t, ·) = f n (·) := ∆tn tn−1 It is easily deduced that for n = 1, . . . , NT n X

(4.11a) (4.11b)

∆tm kf m krH −1 (D) ≤

m=1 ∆t,+

and

f

→f

Z

0

tn

kf (t, ·)krH −1 (D) dt

for any r ∈ [1, 2],

strongly in L2 (0, T ; [H −1 (D)]d ) as ∆t → 0+ .

Throughout this section we choose u0h ∈ Vh0 to be the L2 projection of u0 onto Vh0 and σ 0h ∈ S0h to be the L2 projection of σ 0 onto S0h . Hence, we have that Z 1 (4.12a) σ 0 dx, k = 1, . . . , NK , ku0h kL2 (D) ≤ ku0 kL2 (D) , σ 0h |Kk = |Kk | Kk where |Kk | is the measure of Kk ; and it immediately follows from (3.14) that 0 0 σmin kξk2 ≤ ξ T σ 0h |Kk ξ ≤ σmax kξk2 ∀ξ ∈ Rd , Z 1 tr(σ 0h ) |Kk = tr(σ 0 ) dx ≤ k tr(σ 0 )kL∞ (Kk ) ≤ b⋆ < b. |Kk | Kk

(4.12b) (4.12c)

1 We are now ready to introduce our approximation (P∆t δ,h ) of (Pδ ) for δ ∈ (0, 2 ]: ∆t 0 0 0 0 0 0 (Pδ,h ) Setting (uδ,h , σ δ,h ) = (uh , σ h ) ∈ Vh × (Sh ∩ S>0,b ) as defined in (4.12a), then for n = 1, . . . , NT find (unδ,h , σ nδ,h ) ∈ Vh0 × S0h such that for any test functions (v, φ) ∈ Vh0 × S0h

(4.13a) Z " Re

n−1 unδ,h − uδ,h

∆tn

D

!

·v+

+ (1 − ε)∇unδ,h (4.13b) Z " D

n−1 σ nδ,h − σ δ,h

∆tn

+

!

NE Z X j=1

i Re h n−1 n−1 · ∇)v (uδ,h · ∇)unδ,h · v − unδ,h · (uδ,h 2 # ε n n n : ∇v + Aδ (σ δ,h , tr(σ δ,h )) βδ (σ δ,h ) : ∇v dx = hf n , viH01 (D) , Wi

# n n n A (σ , tr(σ )) β (σ ) : φ δ δ δ,h δ,h δ,h dx : φ − 2 (∇unδ,h ) βδ (σ nδ,h ) : φ + Wi

Ej

n−1 n +un−1 uδ,h · n [[σ δ,h ]]→un−1 : φ δ,h ds = 0. δ,h

In deriving (P∆t δ,h ), we have noted (1.5a) and that Z Z (4.14) v · [(z · ∇)w] dx = − w · [(z · ∇)v] dx D

D

∀z ∈ V,

∀v, w ∈ [H 1 (D)]d ,

and we refer to [15, p267] and [11] for the consistency of our approximation of the stress advection term. We note that on replacing Aδ (σ nδ,h , tr(σ nδ,h )) with I − G′δ (σ nδ,h ) then (P∆t δ,h ), (4.13a,b), collapses to the corresponding finite element approximation of Oldroyd-B studied in [1], see (3.12a,b) there. Before proving existence of a solution to (P∆t δ,h ), we first derive a discrete analogue of the energy bound (3.18) for (P∆t ), which uses the elementary equality δ,h (4.15)

2s1 (s1 − s2 ) = s21 − s22 + (s1 − s2 )2

∀s1 , s2 ∈ R.


12

4.3. Energy bound for (P∆t δ,h ). Proposition 4.1. For n = 1, . . . , NT , a solution unδ,h , σ nδ,h ∈ Vh0 × S0h to (P∆t δ,h ), (4.13a,b), if it exists, satisfies Z n−1 n−1 n−1 Fδ (unδ,h , σ nδ,h , tr(σ nδ,h )) − Fδ (uδ,h , σ δ,h , tr(σ δ,h )) Re n−1 2 + (4.16) kun − uδ,h k dx ∆tn 2∆tn D δ,h Z Z 2 ε n n n A (σ , tr(σ )) β (σ ) dx tr + (1 − ε) k∇unδ,h k2 dx + δ δ δ,h δ,h δ,h 2Wi2 D D Z (1 − ε) 1 + CP ≤ hf n , unδ,h iH01 (D) ≤ kf n k2H −1 (D) . k∇unδ,h k2 dx + 2 2(1 − ε) D Proof. Similarly to (3.20), we have that (4.17) Z

D

∇unδ,h βδ (σ nδ,h ) : Aδ (σ nδ,h , tr(σ nδ,h )) dx =

tr(σ nδ,h ) G′δ 1 − βδ (σ nδ,h ) : ∇unδ,h dx, b D

Z

where we have noted (3.6), (1.3b), (3.4a) and (4.5). Then, similarly to the proof of Proposition 3.1, ε Aδ (σ nδ,h , tr(σ nδ,h )) ∈ S0h in (4.13b) and obtain, on we choose v = unδ,h ∈ Vh0 in (4.13a) and φ = 2Wi noting (4.15), (3.4a), (4.5) and (4.17), that (4.18) n

hf , unδ,h iH01 (D) ≥

Z " D

n−1 2 kunδ,h k2 − kuδ,h k

Re 2

ε + 2Wi

∆tn

Z

D

n−1 σ nδ,h − σ δ,h

∆tn

!

+

n−1 2 kunδ,h − uδ,h k

∆tn

!

#

+ (1 − ε)k∇unδ,h k2 dx

: Aδ (σ nδ,h , tr(σ nδ,h )) dx

Z 2 ε tr Aδ (σ nδ,h , tr(σ nδ,h )) βδ (σ nδ,h ) dx + 2 2Wi D NE Z +un−1 ε X n−1 n δ,h + ds. uδ,h · n [[σ δ,h ]]→un−1 : Aδ (σ nδ,h , tr(σ nδ,h )) δ,h 2Wi j=1 Ej

It follows from (3.6), (3.4d) and the concavity of Gδ that

(4.19) n−1 σ nδ,h − σ δ,h : Aδ (σ nδ,h , tr(σ nδ,h )) tr(σ nδ,h ) n−1 n−1 n ′ ≥ tr(σ δ,h ) − tr(σ δ,h ) Gδ 1 − + tr(Gδ (σ δ,h ) − tr(Gδ (σ nδ,h )) b ! ! n−1 tr(σ δ,h ) tr(σ nδ,h ) n−1 n − b Gδ 1 − + tr Gδ (σ δ,h ) + tr Gδ (σ δ,h ) . ≥ b Gδ 1 − b b Similarly to (4.19), we have, on recalling (3.6), (4.7) and (4.8), that (4.20) [[σ nδ,h ]]→un−1 δ,h

:

+un−1 δ,h Aδ (σ nδ,h , tr(σ nδ,h ))

≥ −[[b Gδ

tr(σ nδ,h ) 1− b

n−1 Finally, we note from (4.9) and as uδ,h ∈ Vh0 that for all q ∈ Q0h

(4.21) NE Z X j=1

Ej

NK Z X n−1 uδ,h · n [[q]]→un−1 ds = − δ,h

k=1

∂Kk

+ tr Gδ (σ nδ,h ) ]]→un−1 . δ,h

NK Z X n−1 uδ,h · nKk q ds = − k=1

Kk

n−1 q div uδ,h dx = 0.


13

Combining (4.18)–(4.21) yields the first desired inequality in (4.16). The second inequality in (4.16) follows immediately from (2.9) with ν 2 = (1 − ε)/(1 + CP ). 4.4. Existence of a solution to (P∆t δ,h ). n−1 , σ n−1 ) ∈ Vh0 × S0h and for any time step Proposition 4.2. Let δ ∈ (0, 12 ], then, given (uδ,h δ,h ∆tn > 0, there exists at least one solution unδ,h , σ nδ,h ∈ Vh0 × S0h to (P∆t δ,h ), (4.13a,b).

Proof. We introduce the following inner product on the Hilbert space Vh0 × S0h Z (4.22) [w · v + ψ : φ] dx ∀(w, ψ), (v, φ) ∈ Vh0 × S0h . ((w, ψ), (v, φ))D = D

Given

n−1 n−1 (uδ,h , σ δ,h )

∈

Vh0

× S0h ,

let F : Vh0 × S0h 7→ Vh0 × S0h be such that for any (w, ψ) ∈ Vh0 × S0h

(4.23) (F (w, ψ), (v, φ))D ! Z " n−1 i w − uδ,h Re h n−1 n−1 (uδ,h · ∇)w · v − w · (uδ,h · ∇)v := Re ·v+ ∆tn 2 D ε Aδ (ψ, tr(ψ)) βδ (ψ) : ∇v + (1 − ε)∇w : ∇v + Wi # ! n−1 ψ − σ δ,h Aδ (ψ, tr(ψ)) βδ (ψ) : φ dx : φ − 2 ((∇w) βδ (ψ)) : φ + + ∆tn Wi NE Z X n−1 n−1 n − hf , viH01 (D) + ∀(v, φ) ∈ Vh0 × S0h . uδ,h · n [[ψ]]→un−1 : φ+uδ,h ds j=1

δ,h

Ej

We note that a solution (unδ,h , σ nδ,h ) to (4.13a,b), if it exists, corresponds to a zero of F ; that is, (4.24) F (unδ,h , σ nδ,h ), (v, φ) D = 0 ∀(v, φ) ∈ Vh0 × S0h .

0 0 In addition, it is easily deduced that the mapping F is continuous. For any (w, ψ) ∈ Vh × Sh , on ε choosing (v, φ) = w, 2Wi Aδ (ψ, tr(ψ)) , we obtain analogously to (4.16) that

(4.25) ε F (w, ψ), w, Aδ (ψ, tr(ψ)) 2Wi D Z n−1 n−1 n−1 Fδ (w, ψ, tr(ψ)) − Fδ (uδ,h , σ δ,h , tr(σ δ,h )) Re n−1 2 + kw − uδ,h k dx ≥ ∆tn 2∆tn D Z Z 1−ε ε 1 + CP 2 2 + kf n kH −1 (D) . k∇wk2 dx + tr (A (ψ, tr(ψ))) β (ψ) dx − δ δ 2 2 2(1 − ε) 2Wi D D

Let us now assume that for any γ ∈ R>0 , the continuous mapping F has no zero (unδ,h , σ nδ,h ) satisfying (4.24), which lies in the ball (4.26) Bγ := (v, φ) ∈ Vh0 × S0h : k(v, φ)kD ≤ γ ; where

(4.27)

1

k(v, φ)kD := [((v, φ), (v, φ))D ] 2 =

Z

D

[ kvk2 + kφk2 ] dx

21

.

Then for such γ, we can define the continuous mapping Gγ : Bγ 7→ Bγ such that for all (v, φ) ∈ Bγ (4.28)

Gγ (v, φ) := −γ

F (v, φ) . kF (v, φ)kD

14


By the Brouwer fixed point theorem, Gγ has at least one fixed point (wγ , ψ γ ) in Bγ . Hence it satisfies

(wγ , ψ γ ) = Gγ (w γ , ψ γ ) = γ. (4.29) D D

On noting (4.2c) and (4.29), we have that Z Z 1 kψ γ k2 dx = µ2h kψ γ k2 dx ≤ µ2h γ 2 , (4.30) kψγ k2L∞ (D) ≤ mink∈NK |Kk | D D 1

where µh := [1/(mink∈NK |Kk |)] 2 . Then (3.15), (3.16), (3.5), (4.30) and (4.29) yield that (4.31)

Fδ (w γ , ψ γ , tr(ψ γ )) Z Z tr(ψ γ ) ε Re 2 −b Gδ 1 − − tr Gδ (ψ γ ) + I dx kwγ k dx + = 2 D 2Wi D b Z Z Re ε ≥ kwγ k2 dx + tr ψ γ − Gδ (ψ γ ) − I dx 2 D 2Wi D Z Z Re ε ≥ kψγ k dx − 2d|D| kwγ k2 dx + 2 D 4Wi D Z Z εd|D| ε Re kψγ kL∞ (D) kψγ k dx − kwγ k2 dx + ≥ 2 D 4Wi µh γ 2Wi D Z Re εd|D| ε ≥ min kwγ k2 + kψγ k2 dx − , 2 4Wi µh γ 2Wi D ε εd|D| Re , . γ2 − = min 2 4Wi µh γ 2Wi

Hence for all γ sufficiently large, it follows from (4.25), (4.31) and (3.4b) that ε Aδ (ψ γ , tr(ψ γ )) ≥ 0. (4.32) F (w γ , ψ γ ), wγ , 2Wi D

On the other hand as (wγ , ψ γ ) is a fixed point of Gγ , we have that ε F (wγ , ψ γ ), wγ , Aδ (ψ γ , tr(ψ γ )) (4.33) 2Wi

D i

F (wγ , ψ γ ) Z h ε D ψ γ : Aδ (ψ γ , tr(ψ γ )) dx. kwγ k2 + =− γ 2Wi D

It follows from (3.6), (3.7b), (3.5) and similarly to (4.31), on noting (4.30) and (4.29), that Z h i ε kwγ k2 + (4.34) ψ γ : Aδ (ψ γ , tr(ψ γ )) dx 2Wi D Z h i tr(ψ γ ) ε ε ′ 2 Gδ 1 − − 1 tr(ψ γ ) + ψ γ : I − G′δ (ψ γ ) dx = kwγ k + 2Wi b 2Wi D Z Z ε kψγ k dx − 2d|D| kwγ k2 dx + ≥ 4Wi D D ε εd|D| ≥ min 1, γ2 − . 4Wi µh γ 2Wi Therefore on combining (4.33) and (4.34), we have for all γ sufficiently large that ε Aδ (ψ γ , tr(ψ γ )) (4.35) < 0, F (w γ , ψ γ ), wγ , 2Wi D

which obviously contradicts (4.32). Hence the mapping F has a zero in Bγ for γ sufficiently large, and so there exists a solution (unδ,h , σ nδ,h ) to (4.13a,b).


15

Theorem 4.1. For any δ ∈ (0, 12 ], NT ≥ 1 and any partitioning of [0, T ] into NT time steps, there 0 0 NT T exists a solution {(unδ,h , σ nδ,h )}N to (P∆t n=1 ∈ [Vh × Sh ] δ,h ), (4.13a,b). In addition, it follows for n = 1, . . . , NT that n Z h i 1 X m−1 2 m 2 n n n (4.36) dx Rekum Fδ (uδ,h , σ δ,h , tr(σ δ,h )) + δ,h − uδ,h k + (1 − ε)∆tm k∇uδ,h k 2 m=1 D Z n 2 ε X m m m + ∆t tr A (σ , tr(σ )) β (σ ) dx m δ δ δ,h δ,h δ,h 2Wi2 m=1 D ≤ Fδ (u0h , σ 0h , tr(σ 0h )) + ≤ F (u0h , σ0h ) + which yields that (4.37)

max

n=0,...,NT

Z

D

n 1 + CP X ∆tm kf m k2H −1 (D) 2(1 − ε) m=1

1 + CP kf k2L2 (0,tn ;H −1 (D)) ≤ C, 2(1 − ε)

kunδ,h k2 + kσnδ,h k + δ −1 k[σnδ,h ]− k + δ −1 [b − tr(σ nδ,h )]− dx ≤ C.

Moreover, for some C(h, ∆t) ∈ R>0 , but independent of δ, it follows that for k = 1, . . . , NK and n = 1, . . . , NT tr(σ nδ,h ) ′ (4.38a) kβδ (σ nδ,h )k ≤ C(h, ∆t) on Kk , Gδ 1 − b tr(σ nδ,h ) (4.38b) on Kk . k[βδ (σ nδ,h )]−1 k ≤ C(h, ∆t) 1 + G′δ 1 − b Proof. Existence of a solution to (P∆t δ,h ) and the first inequality in (4.36) follow immediately from Propositions 4.2 and 4.1, respectively. Similarly to (3.17), the second inequality in (4.36) is a direct consequence of (3.15), (3.1), (4.12b,c) and (4.11a). Finally, the third inequality in (4.36) follows from (4.12a–c) and (3.14). It follows from (4.36) and (3.16) that Z Z ε Re (4.39) kunδ,h k2 dx + tr σ nδ,h − Gδ (σ nδ,h ) − I dx ≤ C, n = 1, . . . , NT . 2 D 2Wi D The first three bounds in (4.37) then follow immediately from (4.39) and (3.5). Next we note that (3.4c), (3.15) and (4.36) yield that Z tr(σ nδ,h ) tr(σ nδ,h ) 1− b (4.40) − Gδ 1 − dx b b D Z tr(σ nδ,h ) n + tr Gδ (σ δ,h ) + I − b dx ≤ C. b Gδ 1 − ≤− b D The last bound in (4.37) is then simply obtained by using a scalar version of (3.5). Next, we deduce from (4.36), (3.6) and (3.4b) that for n = 1, . . . , NT 2 0 ≤ tr Aδ (σ nδ,h , tr(σ nδ,h )) βδ (σ nδ,h ) ≤ C(h, ∆t) on Kk , (4.41) k = 1, . . . , NK .

For any δ > 0, βδ (σ nδ,h ) ∈ Rd×d S,>0 and so it follows from (1.3b), (1.4b), (3.6), (3.4a), (1.5b), (4.41), (1.8), (3.3) and (4.37) that

2

′

tr(σ nδ,h ) n

(4.42) Gδ 1 − βδ (σ δ,h ) − I

b

2 1 2 1 2

= Aδ (σ nδ,h , tr(σ nδ,h )) βδ (σ nδ,h ) ≤ Aδ (σ nδ,h , tr(σ nδ,h )) [βδ (σ nδ,h )] 2 [βδ (σ nδ,h )] 2 2 = tr Aδ (σ nδ,h , tr(σ nδ,h )) βδ (σ nδ,h )) tr(βδ (σ nδ,h )) ≤ C(h, ∆t) kβδ (σ nδ,h )k ≤ C(h, ∆t) kσnδ,h k + δ ≤ C(h, ∆t) on Kk , k = 1, . . . , NK , n = 1, . . . , NT .


16

The desired result (4.38a) follows immediately from (4.42). Similarly to (4.42), we have from (4.41), (1.8), (3.6) and (3.4a) that

(4.43) C(h, ∆t) ≥ Aδ (σ nδ,h , tr(σ nδ,h )) βδ (σ nδ,h ) Aδ (σ nδ,h , tr(σ nδ,h ))

2

tr(σ nδ,h ) tr(σ nδ,h )

n −1 n ′ ′ I + [βδ (σ δ,h )] βδ (σ δ,h ) − 2Gδ 1 − ≥ Gδ 1 −

b b on Kk ,

k = 1, . . . , NK ,

n = 1, . . . , NT .

The desired result (4.38b) follows immediately from (4.43) and (4.38a). (P∆t δ,h )

(P∆t h ).

4.5. Convergence of to We now consider the corresponding direct finite element ∆t approximation of (P), i.e. (Ph ) without the regularization δ: 0 0 0 0 (P∆t h ) Given initial conditions (uh , σ h ) ∈ Vh × (Sh ∩ S>0,b ) as defined in (4.12a), then for n = n n 0 0 1, . . . , NT find (uh , σh ) ∈ Vh × Sh such that for any test functions (v, φ) ∈ Vh0 × S0h Z n uh − uhn−1 Re (4.44a) (uhn−1 · ∇)unh · v − unh · (uhn−1 · ∇)v ·v+ Re ∆t 2 n D ε n n n + (1 − ε)∇uh : ∇v + A(σ h ) σ h : ∇v dx = hf n , viH01 (D) , Wi Z n σ h − σ hn−1 A(σ nh ) σ nh : φ n n (4.44b) dx : φ − 2 ((∇uh ) σ h ) : φ + ∆tn Wi D NE Z X n−1 n n−1 u · n [[σ h ]]→un−1 : φ+uh ds = 0. + h j=1

h

Ej

We note that (4.44a,b) and F (unh , σnh ) are only well-defined if σ nh ∈ S0h ∩ S>0,b . We also note that on replacing A(σ nh ) with I − (σ nh )−1 then (P∆t h ), (4.44a,b), collapses to the corresponding finite element approximation of Oldroyd-B studied in [1], see (3.35a,b) there. K Theorem 4.2. For all regular partitionings Th of D into simplices {Kk }N k=1 and all partitionings NT N T T {∆tn }n=1 of [0, T ], there exists a subsequence {{(unδ,h , σ nδ,h )}n=1 }δ>0 , where {(unδ,h , σ nδ,h )}N n=1 ∈ ∆t n n NT 0 0 NT 0 0 NT solves (Pδ,h ), (4.13a,b), and {(uh , σ h )}n=1 ∈ [Vh × Sh ] such that for the subse[Vh × Sh ] quence

(4.45)

unδ,h → unh ,

σ nδ,h → σ nh

as δ → 0+ ,

for

n = 1, . . . , NT .

In addition, for all t ∈ [0, T ] n = 1, . . . , NT , σ nh |Kk ∈ Rd×d S,>0,b , k = 1, . . . , NK ,. Moreover, ∆t 0 0 NT T {(unh , σnh )}N solves (P ), (4.44a,b), and for n = 1, . . . , NT ∈ [V × S ] n=1 h h h Z Z F (unh , σ nh ) − F (uhn−1 , σ hn−1 ) Re + (4.46) k∇unh k2 dx kunh − uhn−1 k2 dx + (1 − ε) ∆tn 2∆tn D D Z Z 1 + CP 1 ε n 2 n 2 n k∇uh k dx + kf n k2H −1 (D) . + tr (A(σ h )) σ h dx ≤ (1 − ε) 2 2(1 − ε) 2Wi2 D D

Proof. For any integer n ∈ [1, NT ], the desired subsequence convergence results (4.45) follow immediately from (4.37), as (unδ,h , σ nδ,h ) are finite dimensional for fixed Vh0 × Sh0 . It also follows from (4.37), (4.45) and (1.11) that [σ nh ]− and [b − tr(σ nh )]− vanish on D, so that σ nh must be non-negative definite and tr(σ nh ) ≤ b a.e. on D. Moreover, on noting this, (4.45), (3.3) and (1.11), we have the following subsequence convergence results

(4.47) If

tr(σ nh )|Kk

(4.48)

βδ (σ nh ) → σ nh ,

βδ (σ nδ,h ) → σ nh

as δ → 0+ .

= b on some simplex Kk , then for the subsequence of (4.45) we have that tr(σ nδ,h )|Kk → b

as δ → 0+ .

In addition, it follows from (1.8), (3.3) and (4.48) for δ sufficiently small that 1 1 b (4.49) on Kk . kβδ (σ nδ,h )k ≥ √ tr βδ (σ nδ,h ) ≥ √ tr σ nδ,h ≥ √ d d 2 d


17

Hence, (4.38a) and (4.49) yield for the subsequence of (4.45) for all δ sufficiently small that tr(σ nδ,h ) (4.50) ≤ C(h, ∆t) on Kk , G′δ 1 − b

but this contradicts (4.48) on recalling (3.2) and (3.3). Therefore, tr(σ nh )|Kk < b on all simplices Kk , and so it follows from (4.45), (3.2) and (3.3) that −1 tr(σ nδ,h ) tr(σ nδ,h ) tr(σ nh ) tr(σ nh ) → ln 1 − , G′δ 1 − → 1− (4.51) Gδ 1 − b b b b as δ → 0+ . The results (4.38b) and (4.51) yield for δ sufficiently small that k[βδ (σ nδ,h )]−1 k ≤ C(h, ∆t)

(4.52)

on Kk ,

k = 1, . . . , NK .

Furthermore, it follows from (4.52), (4.47), (3.2), (3.3) and as [βδ (σ nδ,h )]−1 βδ (σ nδ,h ) = I that the following subsequence result (4.53)

[βδ (σ nδ,h )]−1 = G′δ (σ nδ,h ) → [σ nh ]−1

holds, and so σ nh |Kk ∈

Rd×d S,>0,b ,

Since φ ∈ S0h

n−1 uδ,h ,

uhn−1

as

δ → 0+ .

∈ C(D), it follows from (4.45), (4.7) and (4.8) that for j = 1, . . . , NE and all

(4.55) Z Z n−1 n +un−1 δ,h ds → n−1 · n [[σ ]] : φ uδ,h δ,h →u Ej

δ → 0+

k = 1, . . . , NK . Therefore, we have from (4.45) and (3.1) that Gδ (σ nδ,h ) → ln(σ nh )

(4.54)

as

δ,h

Ej

n−1 n u · n [[σ ]] h

h →un−1 h

n−1

: φ+uh

ds

as δ → 0+ .

Hence using (4.45), (4.47), (4.51) and (4.55), we can pass to the limit δ → 0+ for the subsequence n n NT 0 0 NT in (P∆t solves (P∆t δ,h ), (4.13a,b), to show that {(uh , σ h )}n=1 ∈ [Vh × Sh ] h ), (4.44a,b). Similarly, using (4.45), (4.47), (4.51), (4.53) and (4.54), and noting (3.15) and (2.1), we can pass to the limit δ → 0+ in (4.16) to obtain (4.46). 5. Fene-p model with stress diffusion 5.1. Model (Pα ), (P) with stress diffusion. In this section we consider the following modified version of (P), (1.1a–f), with a stress diffusion term for a given constant α ∈ R>0 : (Pα ) Find uα : (t, x) ∈ [0, T ) × D 7→ uα (t, x) ∈ Rd , pα : (t, x) ∈ DT 7→ pα (t, x) ∈ R and σ α : (t, x) ∈ [0, T ) × D 7→ σ α (t, x) ∈ Rd×d S,>0,b such that ε ∂uα (5.1a) Re + (uα · ∇)uα = −∇pα + (1 − ε)∆uα + div (A(σ α ) σ α ) + f on DT , ∂t Wi (5.1b) div uα = 0 on DT , (5.1c) (5.1d) (5.1e)

A(σ α ) σ α ∂σ α + (uα · ∇)σ α = (∇uα )σ α + σ α (∇uα )T − + α∆σ α ∂t Wi uα (0, x) = u0 (x) ∀x ∈ D, σ α (0, x) = σ 0 (x)

(5.1f)

uα = 0

(5.1g)

(n∂D · ∇)σ α = 0

on DT ,

∀x ∈ D,

on (0, T ) × ∂D,

on (0, T ) × ∂D.

Hence problem (Pα ) is the same as (P), but with the added diffusion term α∆σ α for the stress equation (5.1c), and the associated Neumann boundary condition (5.1g). Similarly to (Pδ ), (3.13a–f), we introduce a regularization of (Pα,δ ) of (Pα ) mimicking the free energy structure of (Pα ). Moreover, we need to be able to construct a finite element approximation of (Pα,δ ) that satisfies a discrete analogue of this free energy structure. Apart from the obvious addition of the stress diffusion term, there are three other distinct differences. First, one has to deal with

18


the advective term in the stress equation differently, see (5.11) below, to the approach used in (3.19) for the stability bound, as the approach there cannot be mimicked at a discrete level using continuous piecewise linear functions to approximate σ α,δ , the regularization of σ α . Note that one cannot use S0h with the desirable property (4.6) to approximate σ α,δ due to the additional stress diffusion term. Second, as a consequence of this stress advective term, one has to introduce another regularization of tr(σ α ), ̺α,δ , as well as the obvious candidate tr(βδ (σ α,δ )), and solve for this directly, see Remark 5.1 below. Third, it is desirable for the convergence analysis, as δ → 0, to have a uniform L2 (DT ) bound on the extra stress term Aδ (σ α,δ , ̺α,δ ) βδ (σ α,δ ) in the Navier–Stokes equation (5.5a) below. To achieve this we replace Aδ (σ α,δ , ̺α,δ ) βδ (σ α,δ ) there by κδ (σ α,δ , ̺α,δ ) Aδ (σ α,δ , ̺α,δ ) βδ (σ α,δ ), where we define, for δ ∈ (0, min{ 21 , b}], 12 βδb (η) κδ (φ, η) := (5.2) ∀(φ, η) ∈ Rd×d ×R S tr(βδ (φ)) with

(5.3)

βδb : s ∈ R 7→ min{βδ (s), b}.

It follows from (5.2), (5.3), (1.3b) and (1.4b) that (5.4)

1

1

kκδ (φ, η) Aδ (φ, η) βδ (φ)k2 ≤ kκδ (φ, η) [βδ (φ)] 2 k2 kAδ (φ, η) [βδ (φ)] 2 k2 2 ≤ b tr (Aδ (φ, η)) βδ (φ) ∀(φ, η) ∈ Rd×d × R. S

2 Hence a uniform L (DT ) bound on κδ (σ α,δ , ̺α,δ ) Aδ (σ α,δ , ̺α,δ ) βδ (σ α,δ ) follows from a uniform 2 1 L (DT ) on tr (Aδ (σ α,δ , ̺α,δ )) βδ (σ α,δ ) , which will follow from the free energy bound. Although ̺α,δ 6= tr(βδ (σ α,δ )), we will show, in the limit δ → 0, that βδ (σ α,δ ) → σ α and ̺α,δ → tr(σ α ) with σ α (·, ·) ∈ Rd×d S,>0,b , and hence implying that κδ (σ α,δ , ̺α,δ ) → 1. In order to maintain the free energy bound we need to include κδ (σ α,δ , ̺α,δ ) on the right-hand sides of (5.5c,d) below. Therefore, we consider the following regularization of (Pα ) for a given δ ∈ (0, min{ 21 , b}]: (Pα,δ ) Find uα,δ : (t, x) ∈ [0, T ) × D 7→ uα,δ (t, x) ∈ Rd , pα,δ : (t, x) ∈ DT 7→ pα,δ (t, x) ∈ R, σ α,δ : (t, x) ∈ [0, T ) × D 7→ σ α,δ (t, x) ∈ Rd×d and ̺α,δ : (t, x) ∈ [0, T ) × D 7→ ̺α,δ (t, x) ∈ R such S that ∂uα,δ Re (5.5a) + (uα,δ · ∇)uα,δ = −∇pα,δ + (1 − ε)∆uα,δ ∂t ε div (κδ (σ α,δ , ̺α,δ )Aδ (σ α,δ , ̺α,δ ) βδ (σ α,δ )) + Wi +f on DT ,

(5.5b)

div uα,δ = 0

on DT ,

∂σ α,δ + (uα,δ · ∇)βδ (σ α,δ ) = κδ (σ α,δ , ̺α,δ ) (∇uα,δ )βδ (σ α,δ ) + βδ (σ α,δ )(∇uα,δ )T ∂t Aδ (σ α,δ , ̺α,δ ) βδ (σ α,δ ) + α∆σ α,δ on DT , − Wi ̺α,δ ∂̺α,δ (5.5d) − b (uα,δ · ∇)βδ (1 − ) = 2κδ (σ α,δ , ̺α,δ ) ∇uα,δ : βδ (σ α,δ ) ∂t b tr(Aδ (σ α,δ , ̺α,δ ) βδ (σ α,δ )) + α∆̺α,δ on DT , − Wi (5.5e) uα,δ (0, x) = u0 (x) ∀x ∈ D, (5.5c)

(5.5f)

σ α,δ (0, x) = σ 0 (x),

(5.5g)

uα,δ = 0

(5.5h)

̺α,δ (0, x) = tr(σ 0 (x))

∀x ∈ D,

on (0, T ) × ∂D,

(n∂D · ∇)σ α,δ = 0, (n∂D · ∇)̺α,δ = 0 on (0, T ) × ∂D. ̺ ̺ We note from (3.3) that −b ∇βδ 1 − α,δ = ∇̺α,δ if 1 − α,δ ≥ δ, i.e. ̺α,δ ≤ b(1 − δ). We b b remark again, due the required regularization of the advective terms in (5.5c,d), that ̺α,δ 6=


19

tr(σ α,δ ). However, we note that on taking the trace of (5.5c), subtracting (5.5d) and integrating over D, yields, on noting (5.5b,f–h) and (1.3b), that for all t ∈ [0, T ] Z Z (5.6) [tr(σ α,δ (0, ·)) − ̺α,δ (0, ·)] dx = 0. [tr(σ α,δ (t, ·)) − ̺α,δ (t, ·)] dx = D

D

5.2. Formal free energy bound for (Pα,δ ). First, similarly to (3.1), we introduce, for δ ∈ (0, 1), the concave C 1,1 (R>0 ) function ( G(s) ∀s ∈ (0, δ −1 ], (5.7) Hδ : s ∈ R>0 7→ ⇒ Hδ′ (G′δ (s)) = βδ (s) ∀s ∈ R. −1 δ s + G(δ ) − 1 ∀s ≥ δ −1 We have the following analogue of Proposition 3.1. Proposition 5.1. Let α ∈ R>0 , δ ∈ (0, min{ 21 , b}] and (uα,δ , pα,δ , σ α,δ , ̺α,δ ) be a sufficiently smooth solution to problem (Pα,δ ), (5.5a–h). Then the free energy Fδ (uα,δ , σ α,δ , ̺α,δ ) satisfies for a.a. t ∈ (0, T ) Z d (5.8) Fδ (uα,δ , σ α,δ , ̺α,δ ) + (1 − ε) k∇uα,δ k2 dx dt D Z

αεδ 2 ̺α,δ

2 ′ 2 ′ + k∇Gδ (σ α,δ )k + b ∇Gδ 1 −

dx 2Wi D b Z ε tr (Aδ (σ α,δ , ̺α,δ ))2 βδ (σ α,δ ) dx ≤ hf , uα,δ iH01 (D) . + 2 2Wi D

Proof. Similarly to the proof of Proposition 3.18, on noting Remark 3.1, we multiply the Navierε Stokes equation (5.5a) with uα,δ and the stress equation (5.5c) with − 2Wi G′δ (σ α,δ ), the trace ̺α,δ ε ′ equation (5.5d) with 2Wi Gδ 1 − b , sum and integrate over D. This yields, after performing integration by parts and noting (5.5b,g,h), (1.5a), (3.4a) and (3.6), that Z Re ∂kuα,δ k2 (5.9) + (1 − ε)k∇uα,δ k2 dx 2 ∂t D Z ε ∂σ α,δ − + (uα,δ · ∇)βδ (σ α,δ ) : G′δ (σ α,δ ) dx 2Wi D ∂t Z ̺α,δ ̺α,δ ∂̺α,δ ε G′δ 1 − dx − b (uα,δ · ∇)βδ 1 − + 2Wi D ∂t b b Z h ̺α,δ i αε − ∇σ α,δ :: ∇G′δ (σ α,δ ) − ∇̺α,δ · ∇G′δ 1 − dx 2Wi D b Z ε 2 + tr (Aδ (σ α,δ , ̺α,δ )) βδ (σ α,δ ) dx = hf , uα,δ iH01 (D) . 2 2Wi D Using (1.10), we have that

(5.10) ̺α,δ ∂ ̺α,δ ∂ ∂̺α,δ ′ ∂σ α,δ = −b Gδ 1 − . : G′δ (σ α,δ ) = tr (Gδ (σ α,δ )) and Gδ 1 − ∂t ∂t ∂t b ∂t b As βδ (σ α,δ ) ≡ Hδ′ (G′δ (σ α,δ )), on recalling (5.7), we have, on noting uα,δ ∈ V and the spatial version of (1.10), that Z Z (5.11) βδ (σ α,δ ) : (uα,δ · ∇)G′δ (σ α,δ ) dx − (uα,δ · ∇)βδ (σ α,δ ) : G′δ (σ α,δ ) dx = D D Z = (uα,δ · ∇) tr (Hδ (G′δ (σ α,δ ))) dx = 0. D

Similarly to (5.11), we have that

(5.12) Z h Z ̺α,δ i ′ ̺α,δ ̺α,δ (uα,δ · ∇)βδ 1 − − (uα,δ · ∇)Hδ G′δ 1 − Gδ 1 − dx = dx = 0. b b b D D


20

Similarly to (3.4e), we have that −∇σ α,δ :: ∇G′δ (σ α,δ ) ≥ δ 2 k∇G′δ (σ α,δ )k2 ̺α,δ ̺α,δ ̺α,δ (5.13b) = −b∇ 1 − · ∇G′δ 1 − ∇̺α,δ · ∇G′δ 1 − b b b

2 ̺

α,δ ≥ b δ 2 ∇G′δ 1 −

b Combining (5.9)–(5.13a,b) yields the desired result (5.8). (5.13a)

a.e. in DT ,

a.e. in DT . tr(σ

)

α,δ ) I and Remark 5.1. We note if we multiply the advection term in (5.5c) by −G′δ (1 − b integrate over D, then we obtain, on noting uα,δ ∈ V and (5.12), that Z tr(σ α,δ ) ′ − (uα,δ · ∇)βδ (σ α,δ ) : Gδ 1 − Idx b D Z tr(σ α,δ ) tr(βδ (σ α,δ )) (uα,δ · ∇)G′δ 1 − = dx b D Z tr(σ α,δ ) tr(βδ (σ α,δ )) (uα,δ · ∇)G′δ 1 − dx = −b 1− b b D Z tr(σ α,δ ) tr(σ α,δ ) βδ 1 − 6= −b (uα,δ · ∇)G′δ 1 − dx = 0. b b D

Hence, the need for the new variable ̺α,δ in order to mimic the free energy structure of (Pα ). The following Corollary follows from (5.8) on noting the proof of Corollary 3.1. Corollary 5.1. Under the assumptions of Proposition 5.1 it follows that Z 1−ε (5.14) k∇uα,δ k2 dx dt sup Fδ (uα,δ (t, ·), σ α,δ (t, ·), ̺α,δ (t, ·)) + 2 t∈(0,T ) DT Z

̺α,δ αεδ 2

2

′ 2 ′ k∇Gδ (σ α,δ )k + b ∇Gδ 1 − +

dx dt 2Wi DT b Z ε 2 tr (A (σ , ̺ )) β (σ ) dx dt + δ α,δ α,δ δ α,δ 2Wi2 DT 1 + CP 2 0 0 ≤ 2 F (u , σ ) + kf kL2 (0,T ;H −1 (D)) . 2(1 − ε) 6. Finite element approximation of (Pα,δ ) 6.1. Finite element discretization. We now introduce a conforming finite element discretization of (Pα,δ ), (5.5a–h), which satisfies a discrete analogue of (5.8). As noted in Section 5, we cannot use S0h with the desirable property (4.6) to approximate σ α,δ , as we now have the added diffusion term. In the following, we choose Wh1 := Wh2 ⊂ W

(6.1a)

or Wh1,+ ⊂ W,

Q1h = {q ∈ C(D) : q |Kk ∈ P1

(6.1b)

S1h

(6.1d)

k = 1, . . . , NK } ⊂ Q,

: φ |Kk ∈ [P1 ]d×d k = 1, . . . , NK } ⊂ S = {φ ∈ S Z 1 1 1 Vh = v ∈ Wh : q div v dx = 0 ∀q ∈ Qh ;

(6.1c) and

[C(D)]d×d S D

where (4.4), (6.2)

Wh2

is defined as in (4.3a) and, on recalling the barycentric coordinate notation used in

Wh1,+ :=

  

"

v ∈ [C(D)]d ∩ W : v |Kk ∈ P1 ⊕ span

d Y

i=0

ηik

#d

k = 1, . . . , NK

  

.


21

The velocity-pressure choice, Wh2 × Q1h , is the lowest order Taylor-Hood element. It satisfies (4.1) with Wh0 and Q0h replaced by Wh2 and Q1h , respectively, provided, in addition to {Th }h>0 being a regular family of meshes, that each simplex has at least one vertex in D, see p177 in Girault and Raviart [16] in the case d = 2 and Boffi [10] in the case d = 3. Of course, this is a very mild restriction on {Th }h>0 . The velocity-pressure choice, Wh1,+ × Q1h , is called the mini-element. It satisfies (4.1) with Wh0 and Q0h replaced by Wh1,+ and Q1h , respectively; see Chapter II, Section 4.1 in Girault and Raviart [16] in the case d = 2 and Section 4.2.4 in Ern and Guermond [15] in the case d = 3. Hence for both choices of Wh1 , it follows that for all v ∈ V there exists a sequence {v h }h>0 , with v h ∈ Vh1 , such that (6.3)

lim kv − v h kH 1 (D) = 0 .

h→0+

We recall the well-known local inverse inequality for Q1h Z −1 kqkL∞ (Kk ) ≤ C |Kk | (6.4) |q| dx ∀q ∈ Q1h , Kk Z −1 ⇒ kχkL∞ (Kk ) ≤ C |Kk | kχk dx ∀χ ∈ S1h ,

k = 1, . . . , NK k = 1, . . . , NK .

Kk

We recall a similar well-known local inverse inequality for S1h and Vh1 (6.5a) (6.5b)

k∇φkL2 (Kk ) ≤ C h−1 k kφkL2 (Kk ) k∇vkL2 (Kk ) ≤ C

h−1 k kvkL2 (Kk )

∀φ ∈ S1h ,

∀v ∈

Vh1 ,

k = 1, . . . , NK , k = 1, . . . , NK .

We introduce the interpolation operator πh : C(D) → Q1h , and extended naturally to πh : [C(D)]d×d → S1h , such that for all η ∈ C(D) and φ ∈ [C(D)]d×d S S (6.6)

πh η(Pp ) = η(Pp )

and

πh φ(Pp ) = φ(Pp )

p = 1, . . . , NP ,

P where {Pp }N the vertices of Th . As φ ∈ S1h and q ∈ Q1h do not imply that G′δ (φ) ∈ p=1 are S1h and G′δ 1 − qb ∈ Q1h , we have toh test the nfinite i element approximation of (5.5c,d) with ̺α,δ,h ε ε ′ ′ n 1 ∈ Q1h , respectively, where σ nα,δ,h ∈ S1h − 2Wi πh [Gδ (σ α,δ,h )] ∈ Sh and 2Wi πh Gδ 1 − b

and ̺nα,δ,h ∈ Q1h are our finite element approximations to σ α,δ and ̺α,δ at time level tn . Therefore the finite element approximation of (5.5c,d) have to be constructed the h to mimic i results ̺n ε ε (5.9)–(5.13a,b), when tested with − 2Wi ∈ Q1h , πh [G′δ (σ nα,δ,h )] ∈ S1h and 2Wi πh G′δ 1 − α,δ,h b respectively. In order to mimic (5.10) and the (Pα,δ ) analogue of (3.20), we need to use numerical integration (vertex sampling). We note the following results. As the basis functions associated with Q1h and S1h are nonnegative and sum to unity everywhere, we have, on noting (1.5b), for k = 1, . . . , NK that (6.7a)

kπh [φ ψ]k ≤ πh [ kφk kψk ]

1

1

≤ [πh [ kφkr1 ] ] r1 [πh [ kψkr2 ] ] r2

(6.7b)

kπh φk2 ≤ πh [ kφk2 ]

on Kk , on Kk ,

∀φ, ψ ∈ [C(Kk )]d×d , S

, ∀φ ∈ [C(Kk )]d×d S

where r1 , r2 ∈ (1, ∞) satisfy r1−1 + r2−1 = 1. In addition, we have for k = 1, . . . , NK that Z Z Z 2 2 (6.8) kχk2 dx ∀χ ∈ S1h . πh [ kχk ] dx ≤ C kχk dx ≤ Kk

Kk

Kk

The first inequality in (6.8) follows immediately from (6.7b), and the second from applying (6.4) and a Cauchy–Schwarz inequality. Of course, scalar versions of (6.7a,b) and (6.8) hold with [C(Kk )]d×d and S1h replaced by C(Kk ) and Q1h , respectively. S Furthermore, for later use, we recall the following well-known results concerning the interpolant πh for k = 1, . . . , NK : (6.9a)

k(I − πh )qkW 1,∞ (Kk ) ≤ C h |q|W 2,∞ (Kk )

∀q ∈ W 2,∞ (Kk ),


22

(6.9b)

k(I − πh )[q1 q2 ]kL1 (Kk ) ≤ C h2k k∇q1 kL2 (Kk ) k∇q2 kL2 (Kk ) ≤ C hk kq1 kL2 (Kk ) k∇q2 kL2 (Kk )

∀q1 , q2 ∈ Q1h .

In order to mimic (5.11) and (5.12), we have to carefully construct our finite element approximation of the advective terms in (5.5c,d). Our construction is a non-trivial extension of an approach that has been used in the finite element approximation of fourth-order degenerate nonlinear parabolic equations, such as the thin film equation; see e.g. Gr¨ un and Rumpf [17] and Barrett and N¨ urnberg [2]. Let {ei }di=1 be the orthonormal vectors in Rd , such that the j th component of ei b be the standard open reference simplex in Rd with vertices {Pbi }d , is δij , i, j = 1, . . . , d. Let K i=0 where Pb0 is the origin and Pbi = ei , i = 1, . . . , d. Given a simplex Kk ∈ Th with vertices {Pik }di=0 , then there exists a non-singular matrix Bk such that the linear mapping b ∈ Rd b ∈ Rd 7→ P0k + Bk x Bk : x

(6.10)

b to Kk . For all η ∈ Q1 and Kk ∈ Th , maps vertex Pbi to vertex Pik , i = 0, . . . , d. Hence Bk maps K h we define b ⇒ ∇η(Bk (b b η (b b, (6.11) ηb(b x) := η(Bk (b x)) ∀b x∈K x)) = (BkT )−1 ∇b x) ∀b x∈K b b∈K where for all x (6.12)

b η (b [∇b x)]j =

∂ ηb(b x) = ηb(Pbj ) − ηb(Pb0 ) = η(Pjk ) − η(P0k ) ∂b xj

j = 1, . . . , d.

Such notation is easily extended to φ ∈ S1h . b k (b Given q ∈ Q1h and Kk ∈ Th , then first, for j = 1, . . . , d, we find the unique Λ δ,j q ) ∈ R, which is continuous on q, such that ∂ ∂ b k (b b Λ (6.13) π bh [G′δ (b q )] = π bh [Hδ (G′δ (b q ))] on K, δ,j q ) ∂b xj ∂b xj

b and η ∈ C(Kk ). We set b ) for all x b∈K where (b πh ηb)(b x) ≡ (πh η)(Bk x  ′ k ′ k   Hδ (Gδ (q(Pj ))) − Hδ (Gδ (q(P0 ))) if βδ (q(Pjk )) 6= βδ (q(P0k )), k b (b Λ (6.14) G′δ (q(Pjk )) − G′δ (q(P0k )) δ,j q ) :=   β (q(P k )) = β (q(P k )) if βδ (q(Pjk )) = βδ (q(P0k )), δ δ 0 j

where we have noted that G′δ (βδ (s)) = G′δ (s) for all s ∈ R and G′δ (·) is strictly decreasing on b k (b [δ, ∞). Clearly, Λ δ,j q ) ∈ R, j = 1, . . . , d, satisfies (6.13) and depends continuously on q |Kk . Next, we extend the construction (6.14) for a given φ ∈ S1h and Kk ∈ Th , to find for j = 1, . . . , d b ∈ Rd×d , which is continuous on φ, such that b k (φ) the unique Λ δ,j S (6.15)

b = ∂ π b b : ∂ π b k (φ) bh [G′δ (φ)] bh [tr(Hδ (G′δ (φ)))] Λ δ,j ∂b xj ∂b xj

b on K.

b k (φ) b satisfying (6.15), we note the following. We have from (5.7) and (1.9) that To construct Λ δ,j (6.16)

βδ (φ(Pjk )) : (G′δ (φ(Pjk )) − G′δ (φ(P0k ))) ≤ tr(Hδ (G′δ (φ(Pjk )) − Hδ (G′δ (φ(P0k )))

≤ βδ (φ(P0k )) : (G′δ (φ(Pjk )) − G′δ (φ(P0k ))).

Since G′δ (βδ (s)) = G′δ (s) for all s ∈ R, it follows from (3.4e) that (6.17)

− (βδ (φ(Pjk )) − βδ (φ(P0k ))) : (G′δ (φ(Pjk )) − G′δ (φ(P0k )))

= −(βδ (φ(Pjk )) − βδ (φ(P0k ))) : (G′δ (βδ (φ(Pjk ))) − G′δ (βδ (φ(P0k )))) ≥ δ 2 kG′δ (βδ (φ(Pjk ))) − G′δ (βδ (φ(P0k )))k2 .

Therefore the left-hand side of (6.17) is zero if and only if G′δ (βδ (φ(Pjk ))) = G′δ (βδ (φ(P0k ))), which is equivalent to βδ (φ(Pjk )) = βδ (φ(P0k )) as G′δ (·) is invertible on [δ, ∞), the range of βδ (·). Hence, on noting (6.12), (6.16), (6.17) and (1.3b), we have that (6.18a)

b := (1 − λk )βδ (φ(P k )) + λk βδ (φ(P k )) b k (φ) Λ δ,j 0 δ,j δ,j j


23

if (βδ (φ(Pjk )) − βδ (φ(P0k ))) : (G′δ (φ(Pjk )) − G′δ (φ(P0k ))) 6= 0 ,

(6.18b)

b := βδ (φ(P k )) = βδ (φ(P k )) b kδ,j (φ) Λ j 0

if (βδ (φ(Pjk )) − βδ (φ(P0k ))) : (G′δ (φ(Pjk )) − G′δ (φ(P0k ))) = 0

satisfies (6.15) for j = 1, . . . , d; where λkδ,j ∈ [0, 1] is defined as i h tr(Hδ (G′δ (φ(Pjk )) − Hδ (G′δ (φ(P0k ))) − βδ (φ(Pjk )) : (G′δ (φ(Pjk )) − G′δ (φ(P0k ))) λkδ,j := . (βδ (φ(P0k )) − βδ (φ(Pjk ))) : (G′δ (φ(Pjk )) − G′δ (φ(P0k ))) b ∈ Rd×d , j = 1, . . . , d, depends continuously on φ |K . b k (φ) Furthermore, Λ k δ,j S Therefore given q ∈ Q1h and φ ∈ S1h , we introduce, for m, p = 1, . . . , d,

(6.19a)

Λδ,m,p (q) =

d X j=1

(6.19b)

Λδ,m,p (φ) =

d X j=1

b kδ,j (b q ) [BkT ]jp ∈ R, [(BkT )−1 ]mj Λ

b [B T ]jp ∈ Rd×d b kδ,j (φ) [(BkT )−1 ]mj Λ k S

on Kk ,

k = 1, . . . , NK ,

on Kk ,

k = 1, . . . , NK .

It follows from (6.19a,b), (6.13), (6.15) and (6.11) that (6.20)

Λδ,m,p (q) ≈ βδ (q) δmp ,

and for m = 1, . . . , d (6.21a)

d X

Λδ,m,p (q)

p=1

(6.21b)

d X

Λδ,m,p (φ) :

p=1

Λδ,m,p (φ) ≈ βδ (φ) δmp

m, p = 1, . . . , d ;

∂ ∂ πh [G′δ (q)] = πh [Hδ (G′δ (q))] ∂xp ∂xm

on Kk ,

k = 1, . . . , NK ,

∂ ∂ πh [G′δ (φ)] = πh [tr(Hδ (G′δ (φ)))] ∂xp ∂xm

on Kk ,

k = 1, . . . , NK .

For a more precise version of (6.20), see Lemma 6.4 below. Of course, for (6.19a) and (6.21a) we can adopt the more compact notation on Kk , k = 1, . . . , NK , b k (b (6.22) Λδ (q) = (B T )−1 Λ q ) B T ≈ βδ (q) I ⇒ Λδ (q)∇πh [G′ (q)] = ∇πh [Hδ (G′ (q))], k

δ

k

(6.23)

Rd×d S

δ

δ

where ∈ is diagonal with = j = 1, . . . , d, so that Λδ (q) ∈ Rd×d with S [Λδ (q)]mp = Λδ,m,p (q), m, p = 1, . . . , d. Finally, as the partitioning Th consists of regular simplices, we have that Λkδ (b q)

[Λkδ (b q )]jj

Λkδ,j (b q ),

k(BkT )−1 k kBkT k ≤ C,

k = 1, . . . , NK .

Hence, it follows from (6.19a,b), (6.23) and (6.18a,b) that for k = 1, . . . , NK (6.24a) (6.24b)

kΛδ,m,p(q)kL∞ (Kk ) ≤ C kπh [βδ (q)]kL∞ (Kk )

kΛδ,m,p(φ)kL∞ (Kk ) ≤ C kπh [βδ (φ)]kL∞ (Kk )

∀q ∈ Q1h ,

∀φ ∈ S1h .

In order to mimic (5.13a,b), we shall assume from now on that the family of meshes, {Th }h>0 , for the polytope D consists of non-obtuse simplices only, i.e. all dihedral angles of any simplex in Th are less than or equal to π2 . Let Kk have vertices {Pjk }dj=0 , and let ηjk (x) be the basis functions on Kk associated with Q1h and S1h , i.e. ηjk |Kk ∈ P1 and ηjk (Pik ) = δij , i, j = 0, . . . , d. As Kk is non-obtuse it follows that (6.25)

∇ηik · ∇ηjk ≤ 0

on Kk ,

We then have the following result.

i, j = 0, . . . , d, with i 6= j.

Lemma 6.1. Let g ∈ C 0,1 (R) be monotonically increasing with Lipschitz constant gLip . As Th consists of only non-obtuse simplices, then we have for all q ∈ Q1h , φ ∈ S1h that (6.26)

gLip ∇πh [g(q)] · ∇q ≥ k∇πh [g(q)]k2

and

gLip ∇πh [g(φ)] :: ∇φ ≥ k∇πh [g(φ)]k2 on Kk ,

k = 1, . . . , NK .


24

Proof. See the proof of Lemma 5.1 in [1].

Of course, the construction of a non-obtuse mesh in the case d = 3 is not straightforward for a general polytope D. However, we stress that our numerical method (P∆t α,δ,h ), see (6.34a–c) below, does not require this constraint. It is only required to show that (P∆t α,δ,h ) mimics the free energy structure of (Pα,δ ). 6.2. A free energy preserving approximation, (P∆t α,δ,h ), of (Pα,δ ). In addition to the assumptions on the finite element discretization stated in subsection 6.1, and our definition of ∆t in subsection 4.1, we shall assume for the convergence analysis, see Section 7, that there exists a C ∈ R>0 such that

(6.27)

∆tn ≤ C ∆tn−1 ,

n = 2, . . . , N,

as

∆t → 0+ .

We note that this constraint is not required for the results in this section, in particular Theorem 6.1. With ∆t1 and C as above, let ∆t0 ∈ R>0 be such that ∆t1 ≤ C∆t0 . Given initial data satisfying (3.14), we choose u0h ∈ Vh1 and σ 0h ∈ S1h throughout the rest of this paper such that Z Z 0 (6.28a) uh · v + ∆t0 ∇u0h : ∇v dx = u0 · v dx ∀v ∈ Vh1 , D ZD Z 0 0 (6.28b) πh [σ h : χ] + ∆t0 ∇σ h :: ∇χ dx = σ 0 : χ dx ∀χ ∈ S1h . D

D

It follows from (6.28a,b), (6.8) and (3.14) that Z 0 2 kuh k + kσ0h k2 + ∆t0 k∇u0h k2 + k∇σ 0h k2 dx ≤ C. (6.29) D

In addition, we note the following result.

Lemma 6.2. For p = 1, . . . , NP we have that (6.30)

0 0 σmin kξk2 ≤ ξ T σ 0h (Pp ) ξ ≤ σmax kξk2

∀ξ ∈ Rd

and

tr(σ 0h (Pp )) ≤ b⋆ .

Proof. For the proof of the first result in (6.30), see the proof of Lemma 5.2 in [1]. We now prove the second result in (6.30). On choosing χ = Iη, with η ∈ Q1h , in (6.28b) yields that zh := tr(σ 0h ) − b⋆ ∈ Q1h satisfies Z Z (6.31) z η dx ∀η ∈ Qh1 , [πh [zh η] + ∆t0 ∇zh · ∇η] dx = 0

D ⋆

D

∞

where z := tr(σ ) − b ∈ L (D) and is non-positive on recalling (3.14). Choosing η = πh [zh ]+ ∈ Q1h , it follows, on noting the Q1h version of (6.8) and (6.26) with g(·) = [ · ]+ , that Z Z (6.32) πh [zh ]2+ + ∆t0 ∇zh · ∇πh [zh ]+ dx πh [zh ]+ ]2 + ∆t0 k∇πh [zh ]+ k2 dx ≤ D Zd z πh [zh ]+ dx ≤ 0. = D

Hence πh [zh ]+ ≡ 0 and so the second result in (6.30) holds.

Furthermore, it follows from (6.28a,b), (6.29), (3.14), (6.3) and (6.9a,b) that, as h, ∆t0 → 0+ ,

(6.33)

u0h → u0

weakly in [L2 (D)]d

and

σ 0h → σ 0

weakly in [L2 (D)]d×d .

Our approximation (P∆t α,δ,h ) of (Pα,δ ) is then: ∆t 0 (Pα,δ,h ) Setting (uα,δ,h , σ 0α,δ,h , ̺0α,δ,h ) = (u0h , σ0h , tr(σ 0h )) ∈ Vh1 × (S1h ∩ S>0,b ) × Q1h , with u0h and 0 σ h as defined in (6.28a,b), then for n = 1, . . . , NT find (unα,δ,h , σ nα,δ,h , ̺nα,δ,h ) ∈ Vh1 × S1h × Q1h such that for any test functions (v, φ, η) ∈ Vh1 × S1h × Q1h (6.34a) Z " Re D

n−1 unα,δ,h − uα,δ,h

∆tn

!

·v+

i Re h n−1 n−1 · ∇)v (uα,δ,h · ∇)unα,δ,h · v − unα,δ,h · (uα,δ,h 2


+ (1 −

ε)∇unα,δ,h

25

# ε n n n n n πh κδ (σ α,δ,h , ̺α,δ,h ) Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) : ∇v dx : ∇v + Wi = hf n , viH01 (D) ,

(6.34b) # ! " Z n−1 σ nα,δ,h − σ α,δ,h Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) : φ dx :φ+ πh ∆tn Wi D Z α∇σ nα,δ,h :: ∇φ − 2∇unα,δ,h : πh [κδ (σ nα,δ,h , ̺nα,δ,h ) φ βδ (σ nα,δ,h ) dx + D

− (6.34c) Z D

Z X d X d D m=1 p=1



πh 

+

Z

D

+b

n−1 [uα,δ,h ]m Λδ,m,p (σ nα,δ,h ) :

n−1 ̺nα,δ,h − ̺α,δ,h

∆tn

!

∂φ dx = 0, ∂xp

 tr Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) η+ η  dx Wi

α∇̺nα,δ,h · ∇η − 2∇unα,δ,h : πh [κδ (σ nα,δ,h , ̺nα,δ,h ) η βδ (σ nα,δ,h )] dx

̺nα,δ,h ∂η n−1 [uα,δ,h ]m Λδ,m,p 1 − dx = 0. b ∂x p D m=1 p=1

Z X d X d

In deriving (P∆t α,δ,h ), we have noted (4.14), (1.5a), (6.19a,b) and (6.20). We note that on replacn ing Aδ (σ α,δ,h , tr(̺nα,δ,h )) with I − G′δ (σ nα,δ,h ) and κδ (σ nα,δ,h , ̺nα,δ,h ) by 1 then (P∆t α,δ,h ), (6.34a,b), collapses to the corresponding finite element approximation of Oldroyd-B with stress diffusion studied in [1], see (5.34a,b) with no L cut-off there. Before proving existence of a solution to (P∆t α,δ,h ), we first derive a discrete analogue of the energy bound (5.8) for (Pα,δ ). 6.3. Energy bound. On setting Z Z i h η Re ε (6.35) Fδ,h (v, φ, η) := + tr (Gδ (φ) + I) dx kvk2 dx − πh b Gδ 1 − 2 D 2Wi D b ∀(v, φ, η) ∈ Vh1 × S1h × Q1h ,

we have the following discrete analogue of Proposition 5.1. Proposition 6.1. For n = 1, . . . , NT , a solution unα,δ,h , σ nα,δ,h , ̺nα,δ,h ∈ Vh1 ×S1h ×Q1h to (P∆t α,δ,h ), (6.34a–c), if it exists, satisfies (6.36)

n−1 n−1 n−1 Fδ,h (unα,δ,h , σ nα,δ,h , ̺nα,δ,h ) − Fδ,h (uα,δ,h , σ α,δ,h , ̺α,δ,h )

∆tn Z Z Re n−1 2 + kunα,δ,h − uα,δ,h k dx + (1 − ε) k∇unα,δ,h k2 dx 2∆tn D D Z h i 2 ε n n n tr A (σ , ̺ ) β (σ ) dx π + δ δ h α,δ,h α,δ,h α,δ,h 2Wi2 D "

# Z

̺nα,δ,h 2 αεδ 2 ′ ′ n 2

dx + k∇πh [Gδ (σ α,δ,h )]k + b

∇πh Gδ 1 − b 2Wi D Z 1 + CP 1 k∇unα,δ,h k2 dx + kf n k2H −1 (D) . ≤ hf n , unα,δ,h iH01 (D) ≤ (1 − ε) 2 2(1 − ε) D

Proof. The proof is similar to that of Proposition 4.1, we choose as test functions v = unα,δ,h ∈ Vh1 , h i ̺n ε ε ∈ Q1h in (6.34a–c), and obtain, on πh [G′δ (σ nα,δ,h )] ∈ S1h and η = 2Wi πh G′δ 1 − α,δ,h φ = − 2Wi b


26

noting (4.15), (3.4a,d), (3.6), (6.26) with g = −G′δ having Lipschitz constant δ −2 , (6.21a,b) and (6.35) that (6.37) hf n , unα,δ,h iH01 (D) ≥

n−1 n−1 n−1 , σ α,δ,h , ̺α,δ,h ) Fδ,h (unα,δ,h , σ nα,δ,h , ̺nα,δ,h ) − Fδ,h (uα,δ,h

∆tn Z Z Re n−1 2 n ku − uα,δ,h k dx + (1 − ε) k∇unα,δ,h k2 dx + 2∆tn D α,δ,h D Z h i 2 ε n n n + tr A (σ , ̺ ) β (σ ) dx π δ δ h α,δ,h α,δ,h α,δ,h 2Wi2 D

# Z "

̺nα,δ,h 2 αεδ 2 ′ ′ n 2

dx

k∇πh [Gδ (σ α,δ,h )]k + b ∇πh Gδ 1 − +

2Wi D b Z ̺nα,δ,h ε n−1 ′ ′ n + dx. u · ∇πh tr(Hδ (Gδ (σ α,δ,h ))) + bHδ Gδ 1 − 2Wi D α,δ,h b

The first desired inequality in (6.36) follows immediately from (6.37) on noting (6.1a,d), (1.12) and that πh : C(D) → Q1h . The second inequality in (6.36) follows immediately from (2.9) with ν 2 = (1 − ε)/(1 + CP ). 6.4. Existence of discrete solutions. R n−1 n−1 [tr(σ α,δ,h ) − ̺α,δ,h ] dx D n n n = 0 and for any time step ∆tn > 0, then there exists at least one solution uα,δ,h , σ α,δ,h , ̺α,δ,h ∈ R n n Vh1 × S1h × Q1h to (P∆t α,δ,h ), (6.34a–c), such that D [tr(σ α,δ,h ) − ̺α,δ,h ] dx = 0. n−1 n−1 n−1 Proposition 6.2. Given (uα,δ,h , σα,δ,h , ̺α,δ,h ) ∈ Vh1 × S1h × Q1h such that

Proof. The proof is similar to that of Proposition 4.2. We introduce the following inner product on the Hilbert space Vh1 × S1h × Q1h Z h ((w, ψ, ξ), (v, φ, η))D = [w · v + πh [ψ : φ + ξ η] ] dx ∀(w, ψ, ξ), (v, φ, η) ∈ Vh1 × S1h × Q1h . D

n−1 n−1 n−1 Given (uα,δ,h , σ α,δ,h , ̺α,δ,h )∈ 1 for any (w, ψ, ξ) ∈ Vh × S1h ×

Vh1 × S1h × Q1h , let F h : Vh1 × S1h × Q1h → Vh1 × S1h × Q1h be such that Q1h

(6.38)

h F h (w, ψ, ξ), (v, φ, η) D ! Z " n−1 w − uα,δ,h ε · v + (1 − ε)∇w : ∇v + πh [κδ (ψ, ξ) Aδ (ψ, ξ) βδ (ψ)] : ∇v := Re ∆t Wi n D i Re h n−1 n−1 (uα,δ,h · ∇)w · v − w · (uα,δ,h · ∇)v + 2 # + α [∇ψ :: ∇φ + ∇ξ · ∇η] − 2∇w : πh [ κδ (ψ, ξ) [φ + ηI] βδ (ψ)] dx

! # n−1 ξ − ̺α,δ,h Aδ (ψ, ξ) βδ (ψ) tr (Aδ (ψ, ξ) βδ (ψ)) :φ+ η+ :φ+ η dx πh + ∆tn Wi ∆tn Wi D Z X d X d ∂φ ξ ∂η n−1 − [uα,δ,h ]m Λδ,m,p (ψ) : dx − hf n , viH01 (D) − b Λδ,m,p 1 − ∂xp b ∂xp D m=1 p=1 Z

n−1 ψ − σ α,δ,h

!

∀(v, φ, η) ∈ Vh1 × S1h × Q1h .

A solution (unα,δ,h , σ nα,δ,h , ̺nα,δ,h ) to (4.13a–c), if it exists, corresponds to a zero of F h . On recalling (6.19a,b), (6.18a,b) and (6.14), it is easily deduced that the mapping F h is continuous. For any


27

h i ε ε (w, ψ, ξ) ∈ Vh1 × S1h × Q1h , on choosing (v, φ, η) = w, − 2Wi , we πh [G′δ (ψ)], 2Wi πh G′δ 1 − ξb obtain analogously to (6.36) that (6.39) h ξ ε ε πh [G′δ (ψ)], πh G′δ 1 − F h (w, ψ, ξ), w, − 2Wi 2Wi b D Z n−1 n−1 n−1 Fδ,h (w, ψ, ξ) − Fδ,h (uα,δ,h , σα,δ,h , ̺α,δ,h ) Re n−1 2 ≥ + kw − uα,δ,h k dx ∆tn 2∆tn D Z Z h i ε 1 + CP 1−ε 2 2 kf n kH −1 (D) + k∇wk2 dx − + πh tr (Aδ (ψ, ξ)) βδ (ψ) dx 2 2 2(1 − ε) 2Wi D D "

2 # Z 2

ξ αεδ ′ ′ 2

dx. k∇πh [Gδ (ψ)]k + b ∇πh Gδ 1 − +

2Wi D b Let

k(v, φ, η)khD

1 := ((v, φ, η), (v, φ, η))hD 2 =

Z

D

2

2

2

kvk + πh [ kφk + |η| ] dx

12

.

If for any γ ∈ R>0 , the continuous mapping F h has no zero (unα,δ,h , σ nα,δ,h , ̺nα,δ,h ), which lies in the ball n o Bγh := (v, φ, η) ∈ Vh1 × S1h × Q1h : k(v, φ, η)khD ≤ γ ; then for such γ, we can define the continuous mapping Gγh : Bγh → Bγh such that for all (v, φ, η) ∈ Bγh Gγh (v, φ, η) := −γ

F h (v, φ, η)

h

kF h (v, φ, η)kD

.

By the Brouwer fixed point theorem, Gγh has at least one fixed point (w γ , ψ γ , ξγ ) in Bγh . Hence it satisfies

(w γ , ψ γ , ξγ ) h = Gγh (w γ , ψ γ , ξγ ) h = γ. (6.40) D

D

h h In addition, (w γ , ψ γ , ξγ ), (v, φ, η) D = G h (wγ , ψ γ , ξγ ), (v, φ, η) D with (v, φ, η) = (0, I, −1) yields that Z Z n−1 n−1 (6.41) [tr(σ α,δ,h ) − ̺α,δ,h ] dx = 0. [tr(ψ γ ) − ξγ ] dx = D

D

On noting (6.4), we have that there exists a µh ∈ R>0 such that for all φ ∈ S1h , Z πh [ kφk2 ] dx, (6.42) kπh [ kφk ]k2L∞ (D) = kπh [ kφk2 ]kL∞ (D) ≤ µ2h D

and an equivalent result holding for all η ∈ and (6.40) that

Q1h .

It follows from (6.35), (6.41), (3.5), (3.7a), (6.42)

(6.43) Fδ,h (w γ , ψ γ , ξγ ) Z Z Re ε ξγ 2 = kwγ k dx + πh tr(ψ γ − Gδ (ψ γ ) − I) − b Gδ 1 − − ξγ dx 2 D 2Wi D b Z Z ε Re πh [ kψγ k + |ξγ | ] dx − (2d + 3b)|D| kwγ k2 dx + ≥ 2 D 4Wi D Z ε(2d + 3b)|D| Re kwγ k2 dx − ≥ 2 D 4Wi Z Z ε ∞ ∞ kπh [ kψ γ k ]kL (D) πh [ kψγ k] dx + kπh [ |ξγ | ]kL (D) πh [ |ξγ | ] dx + 4Wi µh γ D D Z ε(2d + 3b)|D| ε Re kwγ k2 + πh [ kψ γ k2 + |ξγ |2 ] dx − , ≥ min 2 4Wi µh γ 4Wi D


28

= min

ε Re , 2 4Wi µh γ

γ2 −

ε(2d + 3b)|D| . 4Wi

Hence for all γ sufficiently large, it follows from (6.39) and (6.43) that h ξγ ε ε πh [G′δ (ψ γ )], πh G′δ 1 − ≥ 0. (6.44) F h (w γ , ψγ , ξγ ), w γ , − 2Wi 2Wi b D

On the other hand as (wγ , ψ γ , ξγ ) is a fixed point of Gγh , we have that h ε ξγ ε h ′ ′ F (w γ , ψ γ , ξγ ), wγ , − πh [Gδ (ψ γ )], πh Gδ 1 − (6.45) 2Wi 2Wi b D

h

h Z

F (w γ , ψ γ , ξγ ) ε ξγ 2 ′ ′ D πh ψ γ : Gδ (ψ γ ) − ξγ Gδ 1 − kwγ k − dx. =− γ 2Wi b D

It follows from (6.41), (3.5), (3.7b), and similarly to (6.43), on noting (6.42) and (6.40), that Z ξγ ε ′ ′ 2 (6.46) dx πh ψ γ : Gδ (ψ γ ) − ξγ Gδ 1 − kwγ k − 2Wi b D Z ε ξγ 2 ′ ′ kwγ k + = −1 dx πh ψ γ : (I − Gδ (ψ γ )) + ξγ Gδ 1 − 2Wi b ZD h i ε kwγ k2 + ≥ πh [ kψγ k + |ξγ | ] − 2(d + b) dx 4Wi D ε(d + b)|D| ε γ2 − . ≥ min 1, 4Wi µh γ 2Wi

Therefore on combining (6.45) and (6.46), we have for all γ sufficiently large that h ξγ ε ε (6.47) πh [G′δ (ψ γ )], πh G′δ 1 − < 0, F h (w γ , ψ γ , ξγ ), w γ , − 2Wi 2Wi b D

which obviously contradicts (6.44). Hence the mapping F h has a zero, (unα,δ,h , σ nα,δ,h , ̺nα,δ,h ) ∈ Bγh for γ sufficiently large. Finally, similarly to (6.41), it follows, on choosing (v, φ, η) = (0, I, −1) h R R n−1 in F h (unα,δ,h , σ nα,δ,h , ̺nα,δ,h ), (v, φ, η) )− = 0, that D [tr(σ nα,δ,h ) − ̺nα,δ,h ] dx = D [tr(σ α,δ,h n−1 ̺α,δ,h ] dx = 0.

D

We now have the analogue of Theorem 4.1. Theorem 6.1. For any δ ∈ (0, min{ 21 , b}], NT ≥ 1 and any partitioning of [0, T ] into NT time T ∈ [V1 ×S1h ×Q1h ]NT to (P∆t steps, there exists a solution {(unα,δ,h , σ nα,δ,h , ̺nα,δ,h )}N α,δ,h ), (6.34a–c). R n=1 n h n In addition, it follows for n = 1, . . . , NT that D [tr(σ α,δ,h ) − ̺α,δ,h ] dx = 0 and n Z h i 1 X m−1 2 m 2 (6.48) Fδ,h (unα,δ,h , σ nα,δ,h , ̺nα,δ,h ) + dx Rekum − u k + (1 − ε)∆t k∇u k m α,δ,h α,δ,h α,δ,h 2 m=1 D Z n h i 2 ε X m + ∆t βδ (σ m dx πh tr Aδ (σ m m α,δ,h , ̺α,δ,h ) α,δ,h ) 2 2Wi m=1 D

2 # Z " n

̺m αεδ 2 X α,δ,h ′ ′ m 2

dx k∇πh [Gδ (σ α,δ,h )]k + b ∇πh Gδ 1 − + ∆tm

2Wi m=1 b D ≤ Fδ,h (u0h , σ 0h , tr(σ 0h )) +

n 1 + CP X ∆tm kf m k2H −1 (D) ≤ C, 2(1 − ε) m=1

which yields that (6.49) Z n kuα,δ,h k2 + kσnα,δ,h k + |̺nα,δ,h | + δ −1 πh k[σnα,δ,h ]− k + [b − ̺nα,δ,h ]− dx ≤ C; max n=0,...,NT

D


29

where C is independent of α, as well as δ, h and ∆t. Proof. Existence and the stability result (6.48) follow immediately from Propositions 6.2 and 6.1, respectively, on noting (6.35), (3.1), (6.29), (6.30), (4.11a) and (3.14). The bounds (6.49) follow immediately from (6.48), on noting (3.6), (3.4b), (6.35), (3.5) and that Z ̺nα,δ,h ̺nα,δ,h n n (6.50) − Gδ 1 − + tr(σ α,δ,h − Gδ (σ α,δ,h )) dx πh b 1 − b b D Z ̺nα,δ,h n = πh b 1 − Gδ 1 − − tr(Gδ (σ α,δ,h )) dx ≤ C. b D Remark 6.1. We recall that we have used S1h for the approximation of σ α,δ in (P∆t α,δ,h ), (6.34a–c), due to the presence of the diffusion term in (5.5c). Secondly, due to the advective term in (5.5c), one has to introduce the variable ̺nα,δ,h and its equation (6.34c) in (P∆t α,δ,h ) in order to obtain the entropy bound (6.36). However, we now have a bound on πh [[b − ̺nα,δ,h ]− ] in (6.49), as opposed to [b − tr(σ nδ,h )]− in (4.37). Now, it does not seem possible to pass to the limit δ → 0 in (P∆t α,δ,h ) ∆t to prove well-posedness of the corresponding direct approximation of (Pα ), i.e. (Pα,h ) without the regularization δ, as we did for (P∆t δ,h ) in subsection 4.5. Finally, we note the following Lemmas for later purposes. Lemma 6.3. For all Kk ∈ Th and φ ∈ [C(Kk )]d×d S , we have, for r ∈ [1, ∞), that Z Z r r (6.51) kπh φkr dx. [πh [kφk ] + |πh [kφk]| ] dx ≤ C Kk

Kk

Proof. It follows immediately from (6.4) that Z Z r [πh [kφkr ] + |πh [kφk]| ] dx ≤ 2 |Kk | kπh φkrL∞ (Kk ) ≤ C Kk

Kk

kπh φkr dx,

and hence the desired result(6.51).

Lemma 6.4. Let g ∈ C 0,1 (R) with Lipschitz constant gLip . For all Kk ∈ Th , and for all q ∈ Q1h , φ ∈ S1h we have that Z Z kπh [βδ (φ)] − βδ (φ)k2 dx + max (6.52a) kΛδ,m,p(φ) − βδ (φ) δmp k2 dx m,p=1,...,d K Kk k Z ≤ C h2 k∇φk2 dx, Kk Z Z 2 2 2 kπh [g(q)] − g(q)k dx ≤ C gLip h (6.52b) k∇qk2 dx Kk Kk Z Z 2 2 2 kπh [g(φ)] − g(φ)k dx ≤ C gLip h and k∇φk2 dx. Kk

Kk

Q1h ,

In addition, if g is monotonic then, for all Kk ∈ Th and for all q ∈ we have that Z Z kπh [g(q)] − g(q)k2 dx ≤ C h2 (6.53) k∇πh [g(q)]k2 dx. Kk

Kk

Proof. The results (6.52a) are proved in Lemma 5.3 of [1] for the case when βδ , Λδ,m,p and S1h are replaced by β, Λm,p and S1h ∩ S>0 . The proofs given there are trivially adapted to the present case. In fact, the proof of the first result in (6.52a) in [1] is easily adapted to any function g ∈ C 0,1 (R). Hence, we have the results (6.52b). The result (6.53) is a simple variation of (6.52b) and follows on noting that (6.54)

kπh [g(q)] − g(q)kL∞ (Kk ) ≤

where {Pjk }dj=0 are the vertices of Kk .

max

i, j=0,...,Pdk

|g(q(Pik )) − g(q(Pjk ))|,


30

7. Convergence of (P∆t α,δ,h ) to (Pα ) in the case d = 2 7.1. Stability. Before proving our stability results, we introduce some further notation. We require the L2 projector Rh : V → Vh1 defined by Z (7.1) (v − Rh v) · w dx = 0 ∀w ∈ Vh1 . D

In addition, we require Ph : [L (D)]d×d → S1h defined by S Z Z (7.2) πh [Ph χ : φ] dx = χ : φ dx 2

D

∀φ ∈ S1h .

D

It is easily deduced for p = 1, . . . , NP and i, j = 1, . . . , d that Z 1 (7.3) [Ph χ]ij ηp dx, [Ph χ]ij (Pp ) = R η D p D

where ηp ∈ Q1h is such that ηp (Pr ) = δpr for p, r = 1, . . . , NP . It follows from (7.2) and (6.7b) with φ = Ph χ, in both cases, that Z Z Z (7.4) kχk2 dx ∀χ ∈ [L2 (D)]d×d . πh [ kPh χk2 ] dx ≤ kPh χk2 dx ≤ S D

D

D

We shall assume from now on that D is convex and that the family {Th }h>0 is quasi-uniform, i.e. hk ≥ C h, k = 1, . . . , NK . It then follows that (7.5)

kRh vkH 1 (D) ≤ CkvkH 1 (D)

∀v ∈ V,

see Lemma 4.3 in Heywood and Rannacher [18]. Similarly, it is easily established that (7.6)

kPh χkH 1 (D) ≤ CkχkH 1 (D)

∀χ ∈ [H 1 (D)]d×d . S

We also require the scalar analogue of Ph , where d = 1 and S1h is replaced by Q1h , satisfying the analogues of (7.2)–(7.4) and (7.6). Let ([H 1 (D)]d×d )′ be the topological dual of [H 1 (D)]d×d with [L2 (D)]d×d being the pivot space. S S S d×d ′ d×d 1 1 Let E : ([H (D)]S ) → [H (D)]S be such that Eχ is the unique solution of the Helmholtz problem Z (7.7) [∇(Eχ) :: ∇φ + (Eχ) : φ] dx = hχ, φiH 1 (D) ∀φ ∈ [H 1 (D)]d×d , S D

where h·, ·iH 1 (D) denotes the duality pairing between ([H 1 (D)]d×d )′ and [H 1 (D)]d×d . We note that S S (7.8)

hχ, EχiH 1 (D) = kEχk2H 1 (D)

∀χ ∈ ([H 1 (D)]d×d )′ , S

and kE · kH 1 (D) is a norm on ([H 1 (D)]d×d )′ . We also employ this operator in the scalar case, d = 1, S 1 ′ 1 i.e. E : H (D) → H (D). Let V′ be the topological dual of V with the space of weakly divergent free functions in [L2 (D)]d being the pivot space. Let S : V′ → V be such that Sw is the unique solution to the HelmholtzStokes problem Z (7.9) [∇(Sw) : ∇v + (Sw) · v] dx = hw, viV ∀v ∈ V, D

where h·, ·iV denotes the duality pairing between V′ and V. We note that (7.10)

hw, SwiV = kSwk2H 1 (D)

∀w ∈ V′ ,

and kS · kH 1 (D) is a norm on the reflexive space V′ . We recall the following well-known Gagliardo-Nirenberg inequality. Let r ∈ [2, ∞) if d = 2, and r ∈ [2, 6] if d = 3 and θ = d( 12 − 1r ). Then, there exists a positive constant C(D, r, d) such that (7.11)

θ kηkLr (D) ≤ C(D, r, d)kηk1−θ L2 (D) kηkH 1 (D)

∀η ∈ H 1 (D).


31

We recall also the following compactness results. Let Y0 , Y and Y1 be real Banach spaces, Yi , i = 0, 1, reflexive, with a compact embedding Y0 ֒→ Y and a continuous embedding Y ֒→ Y1 . Then, for µi > 1, i = 0, 1, the following embedding is compact: { η ∈ Lµ0 (0, T ; Y0 ) :

(7.12)

∂η ∈ Lµ1 (0, T ; Y1 ) } ֒→ Lµ0 (0, T ; Y); ∂t

see Theorem 2.1 on p184 in Temam [25]. Let X0 , X and X1 be real Hilbert spaces with a compact embedding X0 ֒→ X and a continuous embedding X ֒→ X1 . Then, for γ > 0, the following embedding is compact: { η ∈ L2 (0, T ; X0 ) : Dtγ η ∈ L2 (0, T ; X1 ) } ֒→ L2 (0, T ; X ),

(7.13)

where Dtγ η is the time derivative of order γ of η, which can be defined in terms of the Fourier transform of η; see Theorem 2.2 on p186 in Temam [25]. Finally, we recall the discrete Gronwall inequality: (r0 )2 + (s0 )2 ≤ (q 0 )2 ,

(7.14)

(rm )2 + (sm )2 ≤ ⇒

m 2

m 2

m−1 X

(η n )2 (rn )2 +

n=0

m−1 X

(r ) + (s ) ≤ exp(

(q n )2

n=0 m X n 2 n 2

(η ) )

n=0

m X

(q )

n=0

m≥1 m ≥ 1.

Theorem 7.1. Under the assumptions of Theorem 6.1, there exists a solution {(unα,δ,h , σ nα,δ,h , 1 NT 1 1 T ̺nα,δ,h )}N of (P∆t n=1 ∈ [Vh × Sh × Qh ] α,δ,h ), (6.34a–c), such that the bounds (6.48) and (6.49) hold. Moreover, if d = 2, (6.27) holds and ∆t ≤ C⋆ (ζ −1 ) α1+ζ h2 , for a ζ > 0 and a C⋆ (ζ −1 ) ∈ R>0 sufficiently small, then the following bounds hold: (7.15a) max

n=0,...,NT

(7.15b) max

n=0,...,NT

Z

πh [ kσ nα,δ,h k2 ] dx +

Z

πh [ |̺nα,δ,h |2 ] dx +

D

D

NT Z h i X n−1 2 ∆tn αk∇σ nα,δ,h k2 + πh [ kσ nα,δ,h − σ α,δ,h k ] dx ≤ C, D

n=1

NT Z h i X n−1 2 ∆tn αk∇̺nα,δ,h k2 + πh [ |̺nα,δ,h − ̺α,δ,h | ] dx ≤ C,

n=1

D

(7.15c) # "

NT 2 X 2 X X

4 ̺nα,δ,h 4 n

Λδ,m,p (σ α,δ,h ) 4 dx ≤ C. + ∆tn

4

Λδ,m,p 1 − b L (D) L (D) n=1 m=1 p=1

Proof. Existence and the bounds (6.48) and (6.49) were proved in Theorem 6.1. On choosing φ = σ nα,δ,h in (6.34b), it follows from (3.2), and (4.15), on applying a Young’s inequality, for any ζ > 0 that (7.16) Z Z 1 n−1 2 πh [ kσnα,δ,h k2 + kσnα,δ,h − σ α,δ,h k ] dx + ∆tn α k∇σ nα,δ,h k2 dx 2 D D Z ̺nα,δ,h ∆tn n 2 ′ dx tr (βδ (σ α,δ,h )) + πh Gδ 1 − Wi D b Z Z 1 n−1 2 ≤ πh [ kσ α,δ,h k ] dx + 2∆tn ∇unα,δ,h : πh [κδ (σ nα,δ,h , ̺nα,δ,h ) σ nα,δ,h βδ (σ nα,δ,h )] dx 2 D D Z ̺nα,δ,h ∆tn n n n ′ 1 − dx tr β (σ ) (β (σ ) − σ ) G π + δ δ h α,δ,h α,δ,h α,δ,h δ Wi D b


32

Z X d X d ∂σ nα,δ,h ∆tn n−1 n 1 + dx kσ α,δ,h kL (D) + ∆tn [uα,δ,h ]m Λδ,m,p (σ nα,δ,h ) : Wi ∂xp D m=1 p=1 Z ∆tn 1 n−1 2 kσ nα,δ,h kL1 (D) πh [ kσ α,δ,h k ] dx + ≤ 2 D Wi Z + C δ −1 ∆tn πh tr βδ (σ nα,δ,h ) βδ (σ nα,δ,h ) − σ nα,δ,h dx D

+ 2 ∆tn k∇unα,δ,h kL2 (D) kπh [κδ (σ nα,δ,h , ̺nα,δ,h ) σ nα,δ,h βδ (σ nα,δ,h )]kL2 (D) n−1 + C ∆tn kuα,δ,h k

L

2(2+ζ) ζ

(D)

kΛδ,m,p (σ nα,δ,h )kL2+ζ (D) k∇σ nα,δ,h kL2 (D) .

We deduce from (3.3), (1.8), (6.7b), (1.5b), (5.2), (5.3), (6.51) and (7.11), as d = 2, that (7.17a) tr βδ (σ n ) βδ (σ n ) − σ n ≤ δ tr βδ (σ n ) − σ n ≤ C δ (δ + kσ n α,δ,h α,δ,h α,δ,h kπh [κδ (σ nα,δ,h , ̺nα,δ,h ) σ nα,δ,h βδ (σ nα,δ,h )]k2L2 (D)

(7.17b)

α,δ,h

α,δ,h k),

α,δ,h

Z

πh kκδ (σ nα,δ,h , ̺nα,δ,h ) σ nα,δ,h βδ (σ nα,δ,h )k2 dx D Z Z n 3 2 n 3 n ≤Cb πh kσα,δ,h k kβδ (σ α,δ,h )k dx ≤ C δ + πh [ kσα,δ,h k ] dx D D ≤ C δ 3 + kσ nα,δ,h k3L3 (D) ≤ C δ 3 + kσnα,δ,h k2L2 (D) kσ nα,δ,h kH 1 (D) . ≤

Similarly, as d = 2, it follows from (6.24b), (3.3), (6.4) and (7.11) that for all ζ > 0 (7.18) kΛδ,m,p (σ nα,δ,h )k2+ζ ≤ L2+ζ (D)

NK X

k=1

|Kk | kΛδ,m,p(σ nα,δ,h )k2+ζ L∞ (Kk ) ≤ C

NK X

k=1

|Kk | kπh [βδ (σ nα,δ,h )]k2+ζ L∞ (Kk )

i h ≤ C + C(ζ) kσ nα,δ,h k2L2 (D) kσnα,δ,h kζH 1 (D) . ≤ C δ 2+ζ + kσnα,δ,h k2+ζ L2+ζ (D)

In addition, as d = 2, we note from (7.11), (1.13) and (6.49) that for all ζ > 0 n−1 kuα,δ,h k

(7.19)

ζ

2(2+ζ) L ζ

(D)

2

2

n−1 2+ζ n−1 2+ζ n−1 2+ζ kH 1 (D) ≤ C(ζ −1 ) k∇uα,δ,h kL2 (D) . ≤ C(ζ −1 ) kuα,δ,h kL2 (D) kuα,δ,h

Combining (7.16)–(7.19), yields, on applying a Young’s inequality, that for all ζ > 0 (7.20) Z

D

πh [ kσnα,δ,h k2

≤

Z

D

+

kσ nα,δ,h

−

n−1 2 σ α,δ,h k ] dx

+ ∆tn α

Z

D

k∇σ nα,δ,h k2 dx

n−1 2 πh [ kσα,δ,h k ] dx

i h n−1 2 kL2 (D) 1 + kσ nα,δ,h k2L2 (D) . + C(ζ −1 ) ∆tn α−(1+ζ) 1 + k∇unα,δ,h k2L2 (D) + k∇uα,δ,h

Hence, summing (7.20) from n = 1, . . . , m for m = 1, . . . , NT yields, on noting (6.7b), that for any ζ >0 Z Z m m Z X X n−1 2 2 n 2 (7.21) k ] dx + α ∆t πh [ kσnα,δ,h − σ α,δ,h k ] dx πh [ kσ m k∇σ k dx + n α,δ,h α,δ,h D

≤

Z

n=1

D

n=1

D

πh [ kσ0h k2 ] dx + C(ζ −1 ) α−(1+ζ) #Z " n m X X k 2 −1 −(1+ζ) k∇uα,δ,h kL2 (D) πh [ kσ nα,δ,h k2 ] dx. + C(ζ ) α ∆tn 1 + D

n=1

k=n−1

D

Applying the discrete Gronwall inequality (7.14) to (7.21), and noting (6.27), (6.8), (6.29), (6.5b), (6.48), (6.49), and that ∆t ≤ C⋆ (ζ −1 ) α1+ζ h2 , for a ζ > 0 where C⋆ (ζ −1 ) is sufficiently small, yields the bounds (7.15a).


33

Similarly to (7.16) on choosing η = ̺nα,δ,h in (6.34c), it follows from (4.15), (1.3b), (1.4b) and (1.8), on applying a Young’s inequality, for any ζ > 0 that (7.22) Z Z 1 n−1 2 πh [ |̺nα,δ,h |2 + |̺nα,δ,h − ̺α,δ,h | ] dx + ∆tn α k∇̺nα,δ,h k2 dx 2 D D Z Z 1 n−1 2 n πh [ |̺α,δ,h | ] dx + 2∆tn ∇uα,δ,h : πh [κδ (σ nα,δ,h , ̺nα,δ,h ) ̺nα,δ,h βδ (σ nα,δ,h )] dx ≤ 2 D D Z ∆tn − πh tr Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) ̺nα,δ,h dx Wi D Z X d X d ̺nα,δ,h ∂̺nα,δ,h n−1 − ∆tn b [uα,δ,h ]m Λδ,m,p 1 − dx b ∂xp D m=1 p=1 Z 1 n−1 2 ≤ πh [ |̺α,δ,h | ] dx + 2 ∆tn k∇unα,δ,h kL2 (D) kπh [κδ (σ nα,δ,h , ̺nα,δ,h ) ̺nα,δ,h βδ (σ nα,δ,h )]kL2 (D) 2 D

̺n n−1

Λδ,m,p 1 − α,δ,h k∇̺nα,δ,h kL2 (D) + C ∆tn kuα,δ,h k 2(2+ζ)

2+ζ

b (D) L ζ L (D) Z i h ∆tn 1 2 + πh tr Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) + d 2 (̺nα,δ,h )2 kβδ (σ nα,δ,h )k dx. 2Wi D

Similarly to (7.17b), as d = 2, we deduce from (6.7b), (5.2), (5.3), (3.3), (6.4), (7.11) and (7.15a) that (7.23) Z πh (̺nα,δ,h )2 kβδ (σ nα,δ,h )k dx kπh [κδ (σ nα,δ,h , ̺nα,δ,h ) ̺nα,δ,h βδ (σ nα,δ,h )]k2L2 (D)≤ C b D ≤ C 1 + kσ nα,δ,h kL3 (D) k̺nα,δ,h k2L3 (D) 2 4 1 3 3 k̺nα,δ,h kL3 2 (D) k̺nα,δ,h kH ≤ C 1 + kσnα,δ,h kH 1 (D) 1 (D) .

Similarly to (7.18), as d = 2, it follows from (6.24a), (3.3), (6.4) and (7.11) that for all ζ > 0

2+ζ

̺n

Λδ,m,p 1 − α,δ,h (7.24) ≤ C + C(ζ) k̺nα,δ,h k2L2 (D) k̺nα,δ,h kζH 1 (D) .

2+ζ b L (D)

Combining (7.22)–(7.24) and (7.19), yields, on applying a Young’s inequality, that for all ζ > 0 Z Z n−1 2 (7.25) πh [ |̺nα,δ,h k2 + |̺nα,δ,h − ̺α,δ,h | ] dx + ∆tn α k∇̺nα,δ,h k2 dx D D Z n−1 2 ≤ πh [ |̺α,δ,h | ] dx + 2 ∆tn k∇unα,δ,h k2L2 (D) D Z i h 2 ∆tn πh tr Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) dx + Wi D i h n−1 2 kL2 (D) 1 + k̺nα,δ,hk2L2 (D) . + C(ζ −1 ) ∆tn α−(1+ζ) 1 + kσnα,δ,h k2H 1 (D) + k∇uα,δ,h

Hence, summing (7.25) from n = 1, . . . , m for m = 1, . . . , NT yields, on noting (6.7b) and (6.48), that for any ζ > 0 Z Z m m Z X X n−1 2 m 2 n 2 πh [ |̺α,δ,h | ] dx + α ∆tn (7.26) πh [ |̺nα,δ,h − ̺α,δ,h k ] dx k∇̺α,δ,h k dx + D

≤

Z

n=1

D

n=1

D

πh [ (tr(σ 0h ))2 ] dx + C(ζ −1 ) α−(1+ζ) D

+ C(ζ

−1

−(1+ζ)

)α

m X

n=1

h iZ n−1 2 n 2 ∆tn 1 + kσα,δ,h kH 1 (D) + k∇uα,δ,h kL2 (D) πh [ |̺nα,δ,h |2 ] dx. D


34

Applying the discrete Gronwall inequality (7.14) to (7.26), and noting (6.27), (6.30), (6.5a,b), (6.48), (6.49), (6.8), (7.15a), and that ∆t ≤ C⋆ (ζ −1 ) α1+ζ h2 , for a ζ > 0 where C⋆ (ζ −1 ) is sufficiently small, yields the bounds (7.15b). Finally, the desired result (7.15c) follows immediately from (7.18) and (7.24) with ζ = 2, on noting (7.15a,b) and (6.8). Remark 7.1. Our final convergence result will be restricted to d = 2 for the same reason as why our result for Oldroyd-B in [1] was restricted to d = 2. For example, the control of the term (7.17b) necessitates the restriction to d = 2. This could be overcome for FENE-P by replacing the regularization βδ in (Pα,δ ), (5.5a–h), by βδb and using test functions based on Gbδ in place of Gδ , where (Gbδ )′ (s) = βδb (s) for all s ∈ R. On making a similar change to our numerical approximation h,∆t (Pα,δ ), (6.34a–c), it is then possible to prove the analogues of Theorem 6.1 and Theorem 7.1 for d = 2 and 3 with no restriction on ∆t. One can then establish analogues of Lemmas 7.1, 7.2 and Theorem 7.2, below, but now the limits involve the cut-off b, i.e. ubα , σ bα and ̺bα . Moreover, it does not seem possible to establish that ̺bα = tr(σ bα ), as we now have Λbδ,m,p in place of Λδ,m,p in the analogue of (7.60).Hence,in taking the limit δ, h, ∆t → 0+ , the last term in the analogue of ̺b

(7.61) would have b β b 1 − bα + tr(β b (σ bα )) instead of −(̺bα − tr(σ bα )), where β b (s) = min{s, b}. Therefore, the extension of the convergence analysis in this paper to d = 3 and with a weaker restriction on ∆t will be a topic of further research. Lemma 7.1. Under all of the assumptions of Theorem 7.1, the solution {(unα,δ,h , σ nα,δ,h , ∆t T ̺nα,δ,h )}N n=1 of (Pα,δ,h ), (6.34a–c), satisfies the following bounds:

∆tn S

n=1

NT

X

∆tn E

! ϑ4 n−1

unα,δ,h − uα,δ,h

≤ C,

1 ∆tn H (D)

! 2 NT n−1

X σ nα,δ,h − σ α,δ,h

+ ∆tn E

1

∆tn

NT X

(7.27a)

(7.27b)

n=1 NT X

(7.27c)

n=1 NT X

(7.27d)

n=1

! 2 n−1

̺nα,δ,h − ̺α,δ,h

∆tn

n=1

H (D)

H 1 (D)

≤ C,

2 ∆tn πh κδ (σ nα,δ,h , ̺nα,δ,h ) Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) L2 (D) ≤ C,

8 ∆tn πh Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) 5 8

L 5 (D)

≤ C,

where ϑ ∈ (2, 4] and C in (7.27a,c) is independent of α, as well as δ, h and ∆t. n uα,δ,h −un−1 α,δ,h Proof. On choosing v = Rh S ∈ Vh1 in (6.34a) yields, on noting (7.1), (7.10), ∆tn (7.5) and Sobolev embedding, that (7.28)

Re S

ε = Wi

! 2 n−1

unα,δ,h − uα,δ,h

∆tn

= Re

H 1 (D)

Z

D

Z


∆tn

D

"

· Rh S


∆tn

" " n n n n n πh κδ (σ α,δ,h , ̺α,δ,h ) Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) : ∇ Rh S − (1 − ε) Re − 2

Z

D

∇unα,δ,h

"

"

: ∇ Rh S


" Z n−1 (uα,δ,h · ∇)unα,δ,h · Rh S D

∆tn

!##


∆tn

dx


dx

!#

!#

dx

∆tn

!##

dx


Re + 2 * +

Z

unα,δ,h

D

"

n

n−1 (uα,δ,h

·


· ∇) Rh S


f , Rh S

"

"

∆tn

!#+

∆tn

!##!

35

dx

H01 (D)

2 ≤ C πh κδ (σ nα,δ,h , ̺nα,δ,h ) Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) L2 (D) + k∇unα,δ,h k2L2 (D)

n−1 n−1 k k∇unα,δ,h k k2L1+θ (D) + kf n k2H −1 (D) , + k kuα,δ,h k kunα,δ,h k k2L2 (D) + k kuα,δ,h

where θ > 0 as d = 2. It follows from (6.7b) and (5.4) that

πh κδ (σ nα,δ,h , ̺nα,δ,h ) Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) 2 2 (7.29) L (D) Z h

2 i πh κδ (σ nα,δ,h , ̺nα,δ,h ) Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) dx ≤ D Z h i 2 ≤b πh tr Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) dx. D

Applying the Cauchy–Schwarz and the algebraic-geometric mean inequalities, in conjunction with (7.11), for d = 2, and the Poincaré inequality (1.13) yields that

(7.30)

n−1 2 n−1 kL4 (D) kunα,δ,h k2L4 (D) ≤ k kuα,δ,h k kunα,δ,h k k2L2 (D) ≤ kuα,δ,h

n X

1 2

m=n−1

4 kum α,δ,h kL4 (D)

n i h X m 2 2 kum ≤C α,δ,h kL2 (D) k∇uα,δ,h kL2 (D) . m=n−1

Similarly, we have for any θ ∈ (0, 1), as d = 2, but now using a Young’s inequality (7.31)

n−1 n−1 2 k kuα,δ,h k k∇unα,δ,h k k2L1+θ (D) ≤ kuα,δ,h k 2(1+θ) L

1−θ

(D)

k∇unα,δ,h k2L2 (D)

2(1−θ)

4θ

n−1 n−1 1+θ n 2 ≤ C kuα,δ,h kL21+θ (D) k∇uα,δ,h kL2 (D) k∇uα,δ,h kL2 (D) 2(1−θ)

n−1 ≤ C kuα,δ,h kL21+θ (D) 2 ϑ

n X

m=n−1

2(1+3θ)

1+θ k∇um α,δ,h kL2 (D) .

On taking the power of both sides of (7.28), multiplying by ∆tn , summing from n = 1, . . . , NT ϑ−2 ⇔ ϑ = 2(1+3θ) and noting (7.29), (7.30), (7.31) with θ = 6−ϑ (1+θ) ∈ (2, 4), (6.27), (4.11a), (6.48), (6.49) and (6.29) yields that

! ϑ4 NT n−1

X unα,δ,h − uα,δ,h

(7.32) ∆tn S

1

∆t n n=1 H (D)

≤C

"

NT X

∆tn

D

n=1

+C

Z

"

NT X

n=1

+C 1+ ≤ C,

# ϑ2 h i 2 πh tr Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) dx #2 i ϑ h ∆tn k∇unα,δ,h k2L2 (D) + kf n k2H −1 (D) max

n=0,...,NT

# "N T X n 2 n 2 ∆tn k∇uα,δ,h kL2 (D) kuα,δ,h kL2 (D)

and hence the bound (7.27a) with C independent of α.

n=0


36

n σ α,δ,h −σ n−1 α,δ,h Choosing φ = Ph E ∈ S1h in (6.34b) yields, on noting (7.2) and (7.8), that ∆tn

(7.33) E

! 2 n−1

σ nα,δ,h − σ α,δ,h

1 ∆tn H (D) " " ! !## Z n−1 n−1 σ nα,δ,h − σ α,δ,h σ nα,δ,h − σ α,δ,h πh = : Ph E dx ∆tn ∆tn D " " !## Z n−1 σ nα,δ,h − σ α,δ,h 1 n n n =− dx πh Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) : Ph E Wi D ∆tn " " !## Z n−1 σ nα,δ,h − σ α,δ,h n ∇σ α,δ,h :: ∇ Ph E −α dx ∆tn D !# # " " Z n−1 σ nα,δ,h − σ α,δ,h n n n n βδ (σ α,δ,h ) dx ∇uα,δ,h : πh κδ (σ α,δ,h , ̺α,δ,h ) Ph E +2 ∆tn D " " !## Z X n−1 d X d n σ − σ ∂ α,δ,h α,δ,h n−1 Ph E dx. + [uα,δ,h ]m Λδ,m,p (σ nα,δ,h ) : ∂xp ∆tn D m=1 p=1

It follows from (1.5a,b), (1.3b), (1.4b) and (1.8) that for any ζ ∈ R>0

(7.34) Z " " !## n−1 σ nα,δ,h − σ α,δ,h n n n dx πh Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) : Ph E D ∆tn

" # !# " Z n−1 n

σ − σ 1 1

α,δ,h α,δ,h πh kAδ (σ nα,δ,h , ̺nα,δ,h ) [βδ (σ nα,δ,h )] 2 k Ph E ≤

k[βδ (σ nα,δ,h )] 2 k dx

∆tn D Z ≤ ζ −1 πh tr (Aδ (σ nα,δ,h , ̺nα,δ,h ))2 βδ (σ nα,δ,h ) dx D  "  !# 2 Z n−1

σ nα,δ,h − σ α,δ,h 1

+ ζ d2 πh  Ph E

kβδ (σ nα,δ,h )k dx.

∆tn D Similarly to (7.17b), it follows from (6.7b), (1.5b), (5.2), (5.3) and (1.8) that

" !# # 2 " n−1


n n n βδ (σ α,δ,h ) (7.35)

πh κδ (σ α,δ,h , ̺α,δ,h ) Ph E

2

∆tn L (D)  "  !# 2 Z n−1


≤C πh  Ph E

kβδ (σ nα,δ,h )k dx.

∆tn D

In addition, (3.3), (6.4), (6.8) and (7.15a) imply that for all φ ∈ S1h Z Z NK h i X n 2 n ∞ kσ α,δ,h kL (Kk ) + δ (7.36) πh kφk kβδ (σ α,δ,h )k dx ≤ D

k=1

Kk

2

kφk dx

≤ C kσ nα,δ,h kL2 (D) + δ kφk2L4 (D) ≤ C kφk2H 1 (D) .

Combining (7.33)–(7.36), yields, on noting (7.6), that (7.37)

E

! 2 n−1


∆tn

H 1 (D)


≤C

Z

37

πh tr (Aδ (σ nα,δ,h , ̺nα,δ,h ))2 βδ (σ nα,δ,h ) dx + αk∇σ nα,δ,h k2L2 (D)

D

n−1 2 + k∇unα,δ,h k2L2 (D) + kuα,δ,h kL4 (D)

Z

2 X 2 X

Λδ,m,p (σ nα,δ,h ) 2 4 dx . L (D)

D m=1 p=1

n ̺α,δ,h −̺n−1 α,δ,h Similarly to (7.33)–(7.37), choosing η = Ph E ∈ Q1h in (6.34c) yields that ∆tn

(7.38) E

! 2 n−1

̺nα,δ,h − ̺α,δ,h

1 ∆tn H (D) Z ≤C πh tr (Aδ (σ nα,δ,h , ̺nα,δ,h ))2 βδ (σ nα,δ,h ) dx + αk∇̺nα,δ,h k2L2 (D) D

n−1 2 + k∇unα,δ,h k2L2 (D) + kuα,δ,h kL4 (D)

2 Z X 2 X 2

̺n

Λδ,m,p 1 − α,δ,h dx .

4 b D m=1 p=1 L (D)

Multiplying (7.37) and (7.38) by ∆tn , summing from n = 1, ..., NT and noting (6.48), (6.49), (7.30), (6.27), (6.29) and (7.15a–c) yields the bounds (7.27b). Multiplying (7.29) by ∆tn , summing from n = 1, ..., NT and noting (6.48) yields the result (7.27c) with C independent of α. Finally, it follows from (6.7a) that

8

πh Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) 5 8 (7.39) L 5 (D)

Z

54

45 1 2 1 2

≤ πh Aδ (σ nα,δ,h , ̺nα,δ,h ) [βδ (σ nα,δ,h )] 2 πh [βδ (σ nα,δ,h )] 2 dx D

45

1 2

n n n

≤ πh Aδ (σ α,δ,h , ̺α,δ,h ) [βδ (σ α,δ,h )] 2

1

L (D)

45

1 2 n

πh

[βδ (σ α,δ,h )] 2

.

L4 (D)

Multiplying (7.39) by ∆tn , summing from n = 1, ..., NT and noting (1.3b), (1.4b), (6.48), (1.8), (3.3), (6.51), (7.11) with d = 2, (6.8) and (7.15a) yields that (7.40)

NT X

n=1

8 ∆tn πh Aδ (σ nα,δ,h , ̺nα,δ,h ) βδ (σ nα,δ,h ) 5 8

≤

L 5 (D)

NT X

n=1

1 2

n n n

2 (σ , ̺ ) [β (σ )] π ∆tn

A δ δ h α,δ,h α,δ,h α,δ,h

L1 (D)

×

≤C ≤C

NT X

n=1

NT X

n=1

4 ∆tn πh kβδ (σ nα,δ,h )k L4 (D)

4 ∆tn πh βδ (σ nα,δ,h ) L4 (D)

δ4 +

NT X

n=1

! 15

≤C

! 51

δ +

2

2 ∆tn σ nα,δ,h L2 (D) σ nα,δ,h H 1 (D)

Hence, we have the desired result (7.27d).

4

NT X

n=1

! 15

! 45

4 ∆tn σ nα,δ,h L4 (D)

! 15

≤ C.

Unfortunately, the bound (7.27a) is not useful for obtaining compactness via (7.12), see the discussion in the proof of Theorem 7.2 below. Instead one has to exploit the compactness result (7.13). This we now do, by following the proof of Lemma 5.6 on p237 in Temam [25]. Here the introduction of κδ (σ nα,δ,h , ̺nα,δ,h ) in the extra stress term, as discussed above (5.2), is crucial, as this yields an L2 (DT ) bound on this stress term via the bound (5.4). For this purpose, we introduce


38

∆t,± 1 ∞ 1 the following notation in line with (4.10). Let u∆t α,δ,h ∈ C([0, T ]; Vh ) and uα,δ,h ∈ L (0, T ; Vh ) be such that for n = 1, . . . , NT

t − tn−1 n tn − t n−1 uα,δ,h (·) + u (·) t ∈ [tn−1 , tn ], ∆tn ∆tn α,δ,h

(7.41a)

u∆t α,δ,h (t, ·) :=

(7.41b)

n u∆t,+ α,δ,h (t, ·) := uα,δ,h (·),

(7.41c)

and

n−1 u∆t,− α,δ,h (t, ·) := uα,δ,h (·)

∆(t) := ∆tn

t ∈ [tn−1 , tn ),

t ∈ [tn−1 , tn ).

We note that ∆t,± n u∆t α,δ,h − uα,δ,h = (t − t± )

(7.42)

∂u∆t α,δ,h ∂t

t ∈ (tn−1 , tn ), ∆t(,±)

where tn+ := tn and tn− := tn−1 . We shall adopt uα,δ,h ∆t(,±)

n = 1, . . . , NT ,

∆t,± as a collective symbol for u∆t α,δ,h , uα,δ,h .

∆t(,±)

We also define σ α,δ,h and ̺α,δ,h similarly to (7.41a,b). Using the notation (7.41a,b), (P∆t α,δ,h ), i.e. (6.34a–c) multiplied by ∆tn and summed for n = 1, . . . , NT , can be restated as: # Z " ∂u∆t α,δ,h ∆t,+ Re (7.43a) · v + (1 − ε)∇uα,δ,h : ∇v dx dt ∂t DT Z hh i h i i Re ∆t,+ ∆t,− ∆t,+ (u∆t,− + α,δ,h · ∇)uα,δ,h · v − (uα,δ,h · ∇)v · uα,δ,h dx dt 2 DT Z h i ε ∆t,+ ∆t,+ ∆t,+ ∆t,+ πh κδ (σ ∆t,+ + α,δ,h , ̺α,δ,h ) Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) : ∇v dx dt Wi DT Z T hf + , viH01 (D) dt ∀v ∈ L2 (0, T ; Vh1 ), = 0 " # Z ∆t,+ ∆t,+ Aδ (σ ∆t,+ ∂σ∆t α,δ,h α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) πh (7.43b) :φ+ : φ dx dt ∂t Wi DT Z Z ∆t,+ ∆t,+ ∆t,+ ∇u∆t,+ ∇σ ∆t,+ :: ∇φ dx dt − 2 +α α,δ,h : πh [κδ (σ α,δ,h , ̺α,δ,h ) φ βδ (σ α,δ,h )] dx dt α,δ,h DT

DT

Z

d X

d X

∂φ dx dt = 0 ∀φ ∈ L2 (0, T ; S1h ), ∂x p DT m=1 p=1   Z tr Aδ (σ ∆t,+ , ̺∆t,+ ) βδ (σ ∆t,+ ) ∂̺∆t α,δ,h α,δ,h α,δ,h α,δ,h πh  η+ η  dx dt (7.43c) ∂t Wi DT Z Z ∆t,+ ∆t,+ ∆t,+ ∆t,+ ∇u∆t,+ ∇̺α,δ,h · ∇η dx dt − 2 +α α,δ,h : πh [κδ (σ α,δ,h , ̺α,δ,h ) η βδ (σ α,δ,h )] dx dt −

∆t,+ [u∆t,− α,δ,h ]m Λδ,m,p (σ α,δ,h ) :

DT

DT

+b

Z

d d X X

DT m=1 p=1

∆t,− [uα,δ,h ]m

Λδ,m,p 1 −

̺∆t,+ α,δ,h b

!

∂η dx dt = 0 ∂xp

∀η ∈ L2 (0, T ; Q1h)

0 ∆t 0 ∆t 0 subject to the initial conditions u∆t α,δ,h (0) = uh , σ α,δ,h (0) = σ h and ̺α,δ,h (0) = tr(σ h ). ∆t ∆t Lemma 7.2. Under all of the assumptions of Theorem 7.1, the solution (u∆t α,δ,h , σ α,δ,h , ̺α,δ,h ) of (P∆t α,δ,h ), (7.43a–c), satisfies the following bound:

(7.44)

Z

0

T

2 kDtγ u∆t α,δ,h kL2 (D) dt ≤ C,

where γ ∈ (0, 14 ) and C is independent of α, as well as δ, h and ∆t.


Proof. Equation (7.43a) can be reinterpreted as Z Z d (7.45) ∇g ∆t,+ (t) : ∇v dx u∆t (t) · v dx = Re dt D α,δ,h D

∀v ∈ Vh1 ,

39

t ∈ (0, T ),

where g ∆t,+ (t) ∈ Vh1 is defined by Z Z (7.46) ∇u∆t,+ ∇g ∆t,+ (t) : ∇v dx = hf + (t), viH01 (D) − (1 − ε) α,δ,h (t) : ∇v dx D D Z hh i h i i Re ∆t,+ ∆t,− ∆t,+ − (u∆t,− α,δ,h (t) · ∇)uα,δ,h (t) · v − (uα,δ,h (t) · ∇)v · uα,δ,h (t) dx 2 D Z h i ε ∆t,+ ∆t,+ ∆t,+ ∆t,+ − πh κδ (σ ∆t,+ α,δ,h (t), ̺α,δ,h (t)) Aδ (σ α,δ,h (t), ̺α,δ,h (t)) βδ (σ α,δ,h (t)) : ∇v dx. Wi D Similarly to (7.28) and (7.29) with θ ∈ (0, 1), it follows from (7.46) that

(7.47)

k∇g ∆t,+ (t)kL2 (D) ≤ C kf + (t)kH −1 (D) + k∇u∆t,+ α,δ,h (t)kL2 (D)

∆t,+ ∆t,− ∆t,+ + k ku∆t,− α,δ,h (t)k kuα,δ,h (t)k kL2 (D) + k kuα,δ,h (t)k k∇uα,δ,h (t)k kL1+θ (D) Z 12 2 ∆t,+ ∆t,+ . + tr Aδ (σ ∆t,+ (t), ̺ (t)) β (σ (t)) dx δ α,δ,h α,δ,h α,δ,h D

On noting (7.30), (7.31) and the bound on unα,δ,h in (6.49), we deduce from (7.47) that 1+3θ 1+3θ ∆t,+ 1+θ 1+θ ∆t,+ (7.48) k∇g (t)kL2 (D) ≤ C 1 + kf + (t)kH −1 (D) + k∇u∆t,− α,δ,h (t)kL2 (D) + k∇uα,δ,h (t)kL2 (D) +

Z

D

12 2 ∆t,+ ∆t,+ ∆t,+ tr Aδ (σ α,δ,h (t), ̺α,δ,h (t)) βδ (σ α,δ,h (t)) dx .

Similarly to (7.32), on recalling (6.29), (4.11a) and (6.49), we deduce from (7.48) that Z T 4 (7.49) k∇g ∆t,+ (t)kLϑ2 (D) dt ≤ C, 0

where ϑ ∈ (2, 4] and C is independent of α, as well as δ, h and ∆t. The rest of the proof follows as on p1825–6 in [1], which is based on the proof of Lemma 5.6 on p237 in [25]. 7.2. Convergence. It follows from (6.48), (6.49), (7.15a–c), (6.29), (6.8), (7.27a–d), (7.44) and (7.41a–c) that # ∆t,− 2 Z T" ku∆t,+ α,δ,h − uα,δ,h kL2 (D) ∆t(,±) 2 ∆t(,±) 2 k∇uα,δ,h kL2 (D) + sup kuα,δ,h kL2 (D) + (7.50a) dt ≤ C, ∆(t) t∈(0,T ) 0 # ∆t,− 2 Z T" kσ∆t,+ α,δ,h − σ α,δ,h kL2 (D) ∆t(,±) 2 ∆t(,±) 2 αk∇σ α,δ,h kL2 (D) + (7.50b) sup kσα,δ,h kL2 (D) + dt ≤ C, ∆(t) t∈(0,T ) 0 # ∆t,− 2 Z T" k̺∆t,+ α,δ,h − ̺α,δ,h kL2 (D) ∆t(,±) 2 ∆t(,±) 2 (7.50c) dt sup k̺α,δ,h kL2 (D) + αk∇̺α,δ,h kL2 (D) + ∆(t) t∈(0,T ) 0 " !# 2 Z T

̺∆t,+

α,δ,h ′ 2 +δ dt ≤ C,

∇πh Gδ 1 −

2 b 0 L (D) Z h i ∆t,+ 2 ∆t,+ (7.50d) , ̺ ) β (σ ) dx dt ≤ C, πh tr Aδ (σ ∆t,+ δ α,δ,h α,δ,h α,δ,h DT Z h i ∆t(,±) ∆t(,±) (7.50e) sup πh k[σα,δ,h ]− k + |[b − ̺α,δ,h ]− | dx ≤ C δ, t∈(0,T )

D


40

(7.50f)

(7.50g) (7.50h) (7.50i)

 

4

∂u∆t ϑ α,δ,h 2   + kDtγ u∆t

S α,δ,h kL2 (D) dt ≤ C,

∂t 1 0 H (D)  

Z T

∂̺∆t 2

∂σ ∆t 2

α,δ,h α,δ,h  dt ≤ C,  + E

E

∂t 1 ∂t 1 0 H (D) H (D)

h i

∆t,+ ∆t,+ ∆t,+ ∆t,+ ∆t,+ ≤ C,

πh κδ (σ α,δ,h , ̺α,δ,h ) Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) 2 L (DT )

h i

∆t,+ ∆t,+ ∆t,+ ≤ C,

πh Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) 8 Z

T

L 5 (DT )

where ϑ ∈ (2, 4], γ ∈ (0, 41 ) and C in (7.50a,d,e,f,h) is independent of α, as well as δ, h and ∆t. We are now in a position to prove the following convergence result concerning (P∆t α,δ,h ). Theorem 7.2. Under all of the assumptions of Theorem 7.1, there exists a subsequence of ∆t ∆t {(u∆t α,δ,h , σ α,δ,h , ̺α,δ,h )}δ>0,h>0,∆t>0 , and functions 4

uα ∈ L∞ (0, T ; H) ∩ L2 (0, T ; V) ∩ W 1, ϑ (0, T ; V′ ) with uα (0) = u0 ,

(7.51a)

2×2 2×2 ′ 2 1 1 1 σ α ∈ L∞ (0, T ; [L2 (D)]2×2 )) S ) ∩ L (0, T ; [H (D)]S ) ∩ H (0, T ; ([H (D)]S

(7.51b)

with σ α non-negative definite a.e. in DT and σ α (0) = σ 0 ,

̺α ∈ L∞ (0, T ; L2(D)) ∩ L2 (0, T ; H 1(D)) ∩ H 1 (0, T ; (H 1 (D))′ )

(7.51c)

with ̺α ≤ b a.e. in DT and ̺α (0) = tr(σ 0 ),

such that, as δ, h, ∆t → 0+ ,

∆t(,±)

weak* in L∞ (0, T ; [L2 (D)]2 ),

∆t(,±)

weakly in L2 (0, T ; [H 1 (D)]2 ),

(7.52a)

uα,δ,h → uα

(7.52b)

uα,δ,h → uα

(7.52c)

S

(7.52d)

∂u∆t ∂uα α,δ,h →S ∂t ∂t ∆t(,±) uα,δ,h → uα

weak* in L∞ (0, T ; [L2 (D)]2×2 ),

∆t(,±)

weakly in L2 (0, T ; [H 1 (D)]2×2 ),

σ α,δ,h → σ α

(7.53b)


E

strongly in L2 (0, T ; [Lr (D)]2 ),

∆t(,±)

(7.53a)

(7.53c)

4

weakly in L ϑ (0, T ; V),

∂σ ∆t α,δ,h ∂t

→E

∂σ α ∂t

∆t(,±)

(7.53d)


(7.53e)

πh [βδ (σ α,δ,h )] → σ α

(7.53f)

Λδ,m,p (σ α,δ,h ) → σ α δmp

∆t(,±)

∆t(,±)

weakly in L2 (0, T ; [H 1 (D)]2×2 S ), strongly in L2 (0, T ; [Lr (D)]2×2 ), strongly in L2 (0, T ; [L2(D)]2×2 ), strongly in L2 (0, T ; [L2(D)]2×2 ),

m, p = 1, 2,

and ∆t(,±)

weak* in L∞ (0, T ; L2 (D)),

∆t(,±)

weakly in L2 (0, T ; H 1 (D)),

(7.54a)

̺α,δ,h → ̺α

(7.54b)

̺α,δ,h → ̺α

∂̺∆t ∂̺α α,δ,h →E ∂t ∂t ∆t(,±) ̺α,δ,h → ̺α !# ∆t(,±) ̺α,δ,h ̺α 1− → 1− b b E

(7.54c) (7.54d) "

(7.54e) πh βδ

weakly in L2 (0, T ; H 1 (D)), strongly in L2 (0, T ; Lr (D)), strongly in L2 (0, T ; L2 (D)),

FINITE ELEMENT APPROXIMATION OF THE FENE-P MODEL ∆t(,±)

(7.54f)

Λδ,m,p 1 −

̺α,δ,h b

!

̺α → 1− δmp b

strongly in L2 (0, T ; L2 (D)),

41

m, p = 1, 2,

where ϑ ∈ (2, 4] and r ∈ [1, ∞). Proof. The results (7.52a–c)S follow immediately from the bounds (7.50a,f) on noting the notation (7.41a–c). The denseness of h>0 Q1h in L2 (D) and (6.1d) yield that uα ∈ L2 (0, T ; V). Hence, the result (7.51a) holds on noting (7.10) and (6.33), where uα : [0, T ] → H is weakly continuous. The strong convergence result (7.52d) for u∆t α,δ,h and r = 2 follows immediately from (7.50a) and the second bound in (7.50f) and (7.13) with X0 = [H 1 (D)]2 and X = X1 = [L2 (D)]2 . Here we note that H 1 (D) is compactly embedded in L2 (D). We note here also that one cannot appeal to (7.12) for this strong convergence result with µ0 = 2, µ1 = 4/ϑ, Y0 = [H 1 (D)]2 , Y1 = V′ with norm kS · kH 1 (D) and Y = [L2 (D)]2 for the stated values of ϑ, since [L2 (D)]2 is not continuously embedded in V′ as V is not dense in [L2 (D)]2 . ∆t The result (7.52d) for u∆t,± α,δ,h and r = 2 follows immediately from this result for uα,δ,h and the the bound on the last term on the left-hand side of (7.50a), which yields (7.55)

∆t,± 2 ku∆t α,δ,h − uα,δ,h kL2 (0,T ;L2 (D)) ≤ C ∆t.

Finally, we note from (7.11), for d = 2, that, for all η ∈ L2 (0, T ; H 1(D)), (7.56)

θ kηkL2 (0,T ;Lr (D)) ≤ C kηk1−θ L2 (0,T ;L2 (D)) kηkL2 (0,T ;H 1 (D))

for all r ∈ [2, ∞) with θ = 1 −

2 r

∆t(,±) uα,δ,h

∆t(,±)

∈ (0, 1]. Hence, combining (7.56) and (7.52b,d) for uα,δ,h

with

r = 2 yields (7.52d) for for the stated values of r. Similarly, the results (7.53a–c) follow immediately from (7.50b,g). The strong convergence result (7.53d) for σ ∆t α,δ,h follows immediately from (7.53b,h), (7.8) and (7.12) with µ0 = µ1 = 2, Y0 = [H 1 (D)]d×d , Y1 = ([H 1 (D)]d×d )′ and Y = [Lr (D)]d×d for the stated values of r. Here we note that H 1 (D) is compactly embedded in Lr (D) for the stated values of r, and Lr (D) is continuously embedded in (H 1 (D))′ . Similarly to (7.55) and (7.56), the last bound in (7.50b) then yields that ∆t(,±) (7.53d) holds for σ α,δ,h . The results (7.54a–d) follow analogously from noting (7.50c,g). Hence, on noting (7.8) and (6.33), the results (7.51b,c) hold, where σ α : [0, T ] → [L2 (D)]d×d and ̺α : [0, T ] → L2 (D) are S weakly continuous, apart from the claims on the non-negative definiteness of σ α and the bound on ̺α . It remains to prove these, (7.53e,f) and (7.54e,f). It follows from (1.11), (6.52b) and (7.50b,e) that

(7.57)

k[σ α ]− kL2 (0,T ;L1 (D))

∆t(,±) ≤ [σ α ]− − [σ α,δ,h ]−

∆t(,±) ≤ σ α − σ α,δ,h

i h

∆t(,±) ∆t(,±) + [σ α,δ,h ]− − πh [σ α,δ,h ]− L2 (0,T ;L1 (D)) L2 (0,T ;L1 (D))

h i

∆t(,±) + πh [σ α,δ,h ]− 2 1

L2 (0,T ;L1 (D))

L (0,T ;L (D))

+ C [h + δ] .

The desired non-negative definiteness result on σ α in (7.51b) then follows from (7.53d). The desired bound on ̺α in (7.51c) follows similarly from (1.11), (6.52b), (7.50c,e) and (7.54d). The results (7.53e,f) follow immediately from (6.52a), (7.50b), (1.11), (7.53d), (3.3) and the nonnegative definiteness result on σ α in (7.51b). The results (7.54e,f) follow similarly from the scalar version of (6.52a), (7.50c), (1.11), (7.54d), (3.3) and the bound on ̺α in (7.51c). Lemma 7.3. Under all of the assumptions of Theorem 7.1, the subsequence of {(σ ∆t α,δ,h , ̺∆t )} of Theorem 7.2 and the limiting functions σ and ̺ , satisfying (7.51b,c), α α α,δ,h δ>0,h>0,∆t>0 are such that, as δ, h, ∆t → 0+ , h i ∆t(,±) (7.58) strongly in L2 (0, T ; L2(D)). πh βδb (̺α,δ,h ) → ̺α = tr(σ α )


42

In addition, we have with h = o(δ), as δ → 0+ , that

(7.59)

σ α is positive definite and tr(σ α ) < b

a.e. in DT .

Proof. Choosing φ = η I in (7.43b) and subtracting from (7.43c) yields that Z Z ∂ ∆t,+ ∆t ∆t πh (7.60) ∇ ̺∆t,+ ̺α,δ,h − tr(σ α,δ,h ) η dx dt + α α,δ,h − tr(σ α,δ,h ) · ∇η dx dt ∂t DT DT " ! # Z ∆t,+ d X d ∂η X ̺α,δ,h ∆t,+ ∆t,− + + tr Λδ,m,p (σ α,δ,h ) dx dt = 0 [uα,δ,h ]m b Λδ,m,p 1 − b ∂xp DT m=1 p=1 ∀η ∈ L2 (0, T ; Q1h).

It follows from (7.52d), (7.53b–d,f), (7.54b–e), (7.7), (7.51a–c) and (6.9a,b) that we may pass to the limit δ, h, ∆t → 0+ in (7.60) with η = πh χ to obtain Z Z T ∂ 1 (7.61) ∇ (̺α − tr(σ α )) · ∇χ dx dt h (̺α − tr(σ α )) , χiH (D) dt + α ∂t DT 0 Z (̺α − tr(σ α )) uα · ∇χ dx dt = 0 ∀χ ∈ C0∞ (0, T ; C ∞ (D)), − DT

where [̺α − tr(σ α )](0) = 0. For example, in order to pass to the limit on the first term in (7.60), we note that (7.62) Z ∂ ∆t ̺∆t − tr(σ ) π χ dx dt πh h α,δ,h ∂t α,δ,h DT Z ∂χ ∂ ∆t ∆t ∆t π χ + (I − π ) dx dt. ̺∆t − tr(σ ) ̺ − tr(σ ) π = h h h α,δ,h α,δ,h α,δ,h α,δ,h ∂t ∂t DT

Hence the desired first term in (7.61) follows from noting (7.53b,c), (7.54b,c), (7.7) and (6.9a,b). As C0∞ (0, T ; C ∞ (D)) is dense in L2 (0, T ; H 1 (D)), we have, on noting (7.51a–c), that (7.61) holds for all χ ∈ L2 (0, T ; H 1 (D)). It then follows from (7.51a–c) that we can choose χ = ̺α − tr(σ α ) in (7.61) to yield that ̺α = tr(σ α ) as [̺α − tr(σ α )](0) = 0. Recalling (7.51b,c), we have that ̺α ∈ [0, b] a.e. in DT . The desired convergence result (7.58) then follows from noting this, (5.3), (3.3), (6.52b), (7.50c) and (7.54d). We now improve on (7.51c) by establishing that ̺α = tr(σ α ) < b a.e. in DT . Assuming that tr(σ α ) = b a.e. in DTb ⊂ DT , we have Z (7.63) b |DTb | = tr(σ α ) dx dt b DT

=

Z

b DT

Z h i ∆t,+ πh tr(βδ (σ α,δ,h )) dx dt +

b DT

h i tr σ α − πh βδ (σ ∆t,+ dx dt =: T1 + T2 . α,δ,h )

We deduce from (3.6) and (3.2) that " Z ∆t,+ ∆t,+ (7.64) , ̺ ) β (σ ) + 2 βδ T1 = πh tr Aδ (σ ∆t,+ δ α,δ,h α,δ,h α,δ,h b DT

1−

̺∆t,+ α,δ,h b

!#

dx dt.

It follows from (1.3b) and (1.4b) that

(7.65) h i 21 ∆t,+ ∆t,+ ∆t,+ ∆t,+ 2 ∆t,+ ∆t,+ tr Aδ (σ ∆t,+ , ̺ ) β (σ ) ≤ tr (A (σ , ̺ )) β (σ ) tr β (σ ) . δ δ δ δ α,δ,h α,δ,h α,δ,h α,δ,h α,δ,h α,δ,h α,δ,h

Combining (7.64) and (7.65) yields, on noting a scalar version of (6.7a) over DTb , (7.50d), (1.8) and (3.3) that  21   !#2  " Z ∆t,+ h i ̺ α,δ,h  dx dt (7.66) T1 ≤ C  βδ 1 − πh  tr βδ (σ ∆t,+ α,δ,h ) + 1 b b DT


Z ≤C 1+

DT

i

h

4 dx dt πh kσ ∆t,+ α,δ,h k

81

 

Z

"

1−

πh βδ

b DT

̺∆t,+ α,δ,h b

43

!# 83

It follows from (6.51), (7.11), for d = 2, and (7.50b) that (7.67) Z

DT

Z i h ∆t,+ 4 4 ≤ C dx dt ≤ C kσ k k πh kσ ∆t,+ 4 α,δ,h L (DT ) α,δ,h

T

0

 38

dx dt .

∆t,+ 2 2 kσ ∆t,+ α,δ,h kL2 (D) kσ α,δ,h kH 1 (D) dt ≤ C.

Similarly to (6.51), it follows from (6.4) with Kk replaced by (Kk × (tn−1 , tn ))∩DTb , k = 1, . . . , NK and n = 1, . . . , NT , that (7.68)

Z

"

1−

πh βδ

b DT

where z := πh βδ 1 −

̺∆t,+ α,δ,h b

∆t,+ ̺α,δ,h b

!# 83

"

dx dt ≤ C πh βδ

− 1−

̺α b

1−

̺∆t,+ α,δ,h b

!# 38

8

L3

8

b ) (DT

≤ C kzk 3 8

b ) L 3 (DT

,

. Here we have noted that ̺α = tr(σ α ) = b a.e. in DTb .

Combining (7.66)–(7.68) yields, on noting (7.11), for d = 2, that

(7.69) T1 ≤ C kzk

8

L 3 (DT )

≤C

Z

T

kzk2L2(D) kzkH 1 (D) dt

0 1

1 2

2 3

! 83

3

1

≤ C kzkL4 3 (0,T ;L2 (D)) kzkL4 2(0,T ;H 1 (D))

1

≤ C kzkL2 (0,T ;L2 (D)) kzkL4 ∞(0,T ;L2 (D)) kzkL4 2(0,T ;H 1 (D)) . It follows from (7.69), (7.51c), (6.26), (3.3), (6.8), (7.50c) and (7.54e) that T1 = 0. In addition, it follows immediately from (7.53e) that T2 = 0. Hence, we conclude from (7.63) that |DTb | = 0, and so ρα = tr(σ α ) < b a.e. in DT ; that is, the second desired result in (7.59). We now establish the other result in (7.59) that σ α is symmetric positive definite a.e. in DT , which improves on (7.51b). This result requires the further assumption that h = o(δ), as δ → 0+ . Assume that σ α is not symmetric positive definite a.e. in DT0 ⊂ DT . Let v ∈ L∞ (0, T ; [L∞(D)]2 ) be such that σ α v = 0 with kvk = 1 a.e. in DT0 and v = 0 a.e. in DT \ DT0 . It then follows from (3.2) and (3.6) that Z h Z i

∆t,+ ∆t,+ 0 (7.70) kvk dx dt = |DT | =

πh G′δ (σ α,δ,h ) βδ (σ α,δ,h ) v dx dt DT ZDT h i

∆t,+ ∆t,+ ∆t,+ ≤

πh Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) v dx dt DT " ! # Z ∆t,+

̺

α,δ,h ∆t,+ + βδ (σ α,δ,h ) v dx dt =: T3 + T4 .

πh G′δ 1 −

b DT A simple variation of (6.7a), (1.3b), (1.4a,b) and (7.50d) yield that (7.71) T3 ≤

Z

DT

≤C

2 21 h i 21

∆t,+ ∆t,+ ∆t,+ 12 T β (σ ) :: (v v ) π dx dt , ̺ ) [β (σ )] πh Aδ (σ ∆t,+

δ h δ α,δ,h α,δ,h α,δ,h α,δ,h

Z

DT

12 h i T ) :: (v v ) dx dt πh βδ (σ ∆t,+ . α,δ,h

Then (7.53e) and the definition of v yield that, as δ, h, ∆t → 0+ , Z Z h i T (7.72) σ α :: (v v T ) dx dt = 0, ) :: (v v ) dx dt → πh βδ (σ ∆t,+ α,δ,h DT

DT


44

∆t,+ ̺α,δ,h b

h i πh βδ (σ ∆t,+ α,δ,h ) ,

so we have that T3 = 0. Similarly to (7.71), on setting

χ∆t,+ δ,h

we have from (6.7a) that Z 12 Z

∆t,+ 12 2 (7.73) T4 ≤ πh [χδ,h ] dx dt

12 h i ∆t,+ T :: (v v ) dx dt πh χδ,h

DT

DT

≤C

Z

DT

12 h i ∆t,+ T :: (v v ) dx dt πh χδ,h ,

=

G′δ

1−

where we have noted from (1.3b), (1.4b), (1.8), (6.51), (3.6) and (7.50i) that Z

i

h

∆t,+ 21 2 (7.74) πh [χδ,h ] dx dt ≤ C πh χ∆t,+ δ,h 1 L (DT ) DT

h

h i i

∆t,+ χ ≤ C πh χ∆t,+ ≤ C

8

π h δ,h δ,h L1 (DT ) L 5 (DT )

h i

∆t,+ ∆t,+ ∆t,+ ≤ C. ≤ C + C πh Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) 8 L 5 (DT )

We

will

now

show, that

̺∆t α,δ,h )}δ>0,h>0,∆t>0 , (7.75)

πh

"

G′δ

1−

̺∆t,+ α,δ,h b

on

!

possibly #

βδ (σ ∆t,+ α,δ,h )

extracting

a

ρα −1 → 1− σα b

further

subsequence

8

of

{(σ ∆t α,δ,h ,

8

weakly in L 5 (0, T ; [L 5 (D)]2×2 ).

as δ, h, ∆t → 0+ with h = o(δ). It follows immediately from (7.74) and our definition of χ∆t,+ δ,h h i 8 8 ∆t,+ 2×2 converges weakly in L 5 (0, T ; [L 5 (D)] ) to some limit for a subsequence. We that πh χδ,h just need to show it is the limit stated in (7.75). We have from (7.53e), (7.54d) and (3.1) that −1 χ∆t,+ → 1 − ρbα σ α a.e. on DT , for a subsequence, as we have already established that ρα = δ,h h i tr(σ α ) < b a.e. on DT . So it remains to establish that (I − πh ) χ∆t,+ converges to zero a.e. on δ,h

DT . As G′δ ∈ C 0,1 (R) is monotonic, it follows from (6.9b), (6.53), (6.4), (3.2) and (7.50b,c) that (7.76)

h i

∆t,+

(I − πh ) χδ,h

L1 (DT )

# !# " "

h i ̺∆t,+

α,δ,h ∆t,+ ′ πh βδ (σ α,δ,h ) ≤ (I − πh ) πh Gδ 1 −

1

b L (DT )

!# " ∆t,+

h i ̺α,δ,h

∆t,+ + (I − πh ) G′δ 1 −

πh βδ (σ α,δ,h ) 2

2

b L (DT ) L (DT )

!# "

h

i ̺∆t,+

α,δ,h ∆t,+ ≤ C h ∇πh G′δ 1 −

πh βδ (σ α,δ,h ) 2

2

b L (DT ) L (DT )

≤ C δ −1 h.

h i Hence, we have for a subsequence that (I − πh ) χ∆t,+ converges to zero a.e. on DT as δ, h, ∆t → δ,h

0+ with h = o(δ). Therefore we have established (7.75). Similarly to (7.72), we have that (7.73), (7.75) and our definitions of χ∆t,+ and v yield that δ,h T4 = 0. Hence it follows from (7.70) with T3 = T4 = 0 that |DT0 | = 0, and so σ α is positive definite a.e. in DT ; that is, the first desired result in (7.59). Lemma 7.4. Under all of the assumptions of Lemma 7.3, a further subsequence of the subsequence ∆t of {(σ ∆t α,δ,h , ̺α,δ,h )}δ>0,h>0,∆t>0 of Lemma 7.3 and the limiting function σ α , satisfying (7.51b) and (7.59), are such that, as δ, h, ∆t → 0+ , with h = o(δ), h i ∆t,+ ∆t,+ (7.77a) , ̺ ) β (σ ) → σα strongly in L2 (0, T ; [L2 (D)]2×2 ), πh κδ (σ ∆t,+ δ α,δ,h α,δ,h α,δ,h


(7.77b) (7.77c)

45

h i 8 8 ∆t,+ ∆t,+ , ̺ ) β (σ ) → A(σ α ) σ α weakly in L 5 (0, T ; [L 5 (D)]2×2 ), πh Aδ (σ ∆t,+ δ α,δ,h α,δ,h α,δ,h h i ∆t,+ ∆t,+ ∆t,+ ∆t,+ πh κδ (σ ∆t,+ α,δ,h , ̺α,δ,h ) Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) → A(σ α ) σ α weakly in L2 (0, T ; [L2(D)]2×2 ).

Proof. It follows from (6.7a), (1.3b), (1.4b), (5.2), (1.8), (3.3), (6.8) and (7.50b) that

h h i 2 i

∆t,+ ∆t,+ ∆t,+ ∆t,+ (7.78)

πh κδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) − πh βδ (σ α,δ,h ) 2 L (DT ) Z h i h i ∆t,+ ∆t,+ ∆t,+ 2 πh (κδ (σ α,δ,h , ̺α,δ,h ) − 1) tr(βδ (σ α,δ,h )) πh tr(βδ (σ ∆t,+ ≤ α,δ,h )) dx dt DT Z i h i h ∆t,+ πh kσ∆t,+ πh βδb (̺∆t,+ ≤C α,δ,h k + δ dx dt α,δ,h ) − tr(βδ (σ α,δ,h )

DT h i

∆t,+ ) − tr(β (σ ) ≤ C πh βδb (̺∆t,+ . δ α,δ,h α,δ,h 2 L (DT )

The desired result (7.77a) then follows immediately from (7.78), (7.53e) and (7.58). The desired result (7.77b) follows immediately from (3.6), (3.2), (7.75) and (1.2) as ρα = tr(σ α ). Similarly to (7.78), it follows from (6.7a), (1.3b), (1.4b), (5.2), (6.8), and (7.50d) that (7.79)

h h i i

∆t,+ ∆t,+ ∆t,+ ∆t,+ ∆t,+ ∆t,+ ∆t,+ ∆t,+

πh κδ (σ α,δ,h , ̺α,δ,h ) Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) − πh Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) 1 L (DT ) Z h i 12 ∆t,+ ≤C πh βδb (̺∆t,+ α,δ,h ) − tr(βδ (σ α,δ,h ) DT

×

" #! 21 h i1

2 ∆t,+ ∆t,+ ∆t,+ 2

πh Aδ (σ α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) dx dt

i 1

h

∆t,+ 2 ) − tr(β (σ ) ≤ C πh βδb (̺∆t,+ δ α,δ,h α,δ,h 1

L (DT )

h i 1

∆t,+ 2 ) − tr(β (σ ) ≤ C πh βδb (̺∆t,+ δ α,δ,h α,δ,h 2

L (DT )

.

It follows immediately from (7.79), (7.53e), (7.58) and (7.50h) that the weak limits in (7.77b,c) are the same. Hence, the desired result (7.77c). Theorem 7.3. Under all of the assumptions of Lemma 7.3, the limiting functions (uα , σ α ) satisfying (7.51a,b) and (7.59) solve the following problem: 4 (Pα ) Find uα ∈ L∞ (0, T ; H) ∩ L2 (0, T ; V) ∩ W 1, ϑ (0, T ; V′ ) and σ α ∈ L∞ (0, T ; [L2 (D)]2×2 S,>0,b ) ∩ 2×2 ′ 2×2 1 1 2 2 L2 (0, T ; [H 1(D)]2×2 ) ∩ H (0, T ; ([H (D)] ) ), with A(σ ) σ ∈ L (0, T ; ([L (D)] )), such that α α S S S uα (0) = u0 , σ α (0) = σ 0 and Z T Z ∂uα (7.80a) Re [(1 − ε) ∇uα : ∇v + Re [(uα · ∇)uα ] · v] dx dt ,v dt + ∂t 0 DT V Z Z T 4 ε A(σ α ) σ α : ∇v dx dt ∀v ∈ L 4−ϑ (0, T ; V), = hf , viH01 (D) dt − Wi DT 0 Z Z T ∂σ α ,φ (7.80b) [(uα · ∇)σ α : φ + α ∇σ α :: ∇φ] dx dt dt + ∂t DT 0 H 1 (D) Z 1 2 (∇uα ) σ α − ∀φ ∈ L2 (0, T ; [H 1 (D)]2×2 = A(σ α ) σ α : φ dx dt S ), Wi DT where ϑ ∈ (2, 4).

Proof. The function spaces and the initial conditions for (uα , σ α ) follow immediately from (7.51a,b), (7.59) and (7.77c). It remains to prove that (uα , σ α ) satisfy (7.80a,b). It follows from (6.3), (7.52b–d), (7.77c), (4.11b), (7.9) and (4.14) that we may pass to the limit, δ, h, ∆t → 0+ , with h = o(δ), in (7.43a) to obtain that (uα , σ α ) satisfy (7.80a).


46

It follows from (7.53b–f), (7.52b,d), (7.77a,b), (7.7), (6.9a,b), (1.5a) and as uα ∈ L2 (0, T ; V) that we may pass to the limit δ, h, ∆t → 0+ , with h = o(δ), in (7.43b) with χ = πh φ to obtain (7.80b) for any φ ∈ C0∞ (0, T ; [C ∞ (D)]2×2 S ). For example, similarly to (7.62), in order to pass to the limit on the first term in (7.43b), we note that (7.81) Z

DT

"

! # ∆t,+ ∆t,+ Aδ (σ ∆t,+ ∂σ∆t α,δ,h α,δ,h , ̺α,δ,h ) βδ (σ α,δ,h ) πh : πh φ dx dt + ∂t Wi h i  ∆t,+ ∆t,+ ∆t,+ Z ∆t A (σ , ̺ ) β (σ ) π δ δ h ∂σ α,δ,h α,δ,h α,δ,h  : πh φ dx dt  α,δ,h + = ∂t Wi DT h i   π A (σ ∆t,+ , ̺∆t,+ ) β (σ ∆t,+ ) Z h δ δ α,δ,h α,δ,h α,δ,h ∂φ − : πh φ dx dt. (I − πh ) σ ∆t + α,δ,h : πh ∂t Wi DT

2 The desired result (7.80b) then follows from noting that C0∞ (0, T ; [C ∞ (D)]2×2 S ) is dense in L (0, T ; 2×2 1 [H (D)]S ). Of course passing to the limit δ, h, ∆t → 0+ , with h = o(δ), in (7.43c), using in addition (7.54b–d,f), yields the weak formulation for tr(σ α ) consistent with (7.80b).

Remark 7.2. Choosing 21 (φ + φT ) as a test function in (7.80b) for any φ ∈ L2 (0, T ; [H 1 (D)]2×2 ) yields, on noting the symmetry of σ α , (7.80b) with the term 2 (∇uα )σ α replaced by (∇uα )σ α + σ α (∇uα )T , which is consistent with (5.1c). Finally, it follows from (7.50a,f,h), (7.52a–c) and (7.77c) that #

4 Z T"

∂uα ϑ 2 2 2

k∇uα kL2 (D) + (7.82) sup kuα kL2 (D) + + kA(σ α ) σ α kL2 (D) dt ≤ C,

S ∂t 1 t∈(0,T ) 0 H (D)

where ϑ ∈ (2, 4) and C is independent of the stress diffusion coefficient α. Of course, in addition, it follows from (7.59) and (1.8) that kσα kL∞ (DT ) < b. References

[1] J. W. Barrett and S. Boyaval, Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837. [2] J. W. Barrett and R. N¨ urnberg, Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces, IMA J. Numer. Anal. 24 (2004) 323–363. [3] J. W. Barrett and E. S¨ uli, Existence of global weak solutions to some regularized kinetic models of dilute polymers, Multiscale Model. Simul. 6 (2007) 506–546. [4] J. W. Barrett and E. S¨ uli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: Finitely extensible nonlinear bead-spring chains, Math. Models Methods Appl. Sci. 21 (2011) 1211–1289. [5] J. W. Barrett and E. S¨ uli, Existence of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers with variable density and viscosity, Imperial College London and University of Oxford, 2011. Available from: http://arxiv.org/abs/1112.4781 . [6] J. W. Barrett and E. S¨ uli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers II: Hookean-type bead-spring chains, Math. Models Methods Appl. Sci. 22 (2012) 1150024 (84 pages). [7] J. W. Barrett and E. S¨ uli, Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers, M2AN Math. Model. Numer. Anal. 46 (2012) 949–978. [8] J. W. Barrett and E. S¨ uli, Existence of global weak solutions to the kinetic Hookean dumbbell model for incompressible dilute polymeric fluids, Imperial College London and University of Oxford, 2017. Available from: http://arxiv.org/abs/1702.06502 . [9] R. Bird, C. Curtiss, R. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol 2 : Kinetic Theory (John Wiley and Sons, New York, 1987). [10] D. Boffi, Three-dimensional finite element methods for the Stokes problem, SIAM J. Numer. Anal. 34 (1997) 664–670. [11] S. Boyaval, T. Leli` evre and C. Mangoubi, Free-energy-dissipative schemes for the Oldroyd-B model, ESAIM Math. Model. Numer. Anal. 43 (2009) 523–561. [12] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods (Springer-Verlag, New York, 1992).


47

[13] P. Degond and H. Liu, Kinetic models for polymers with inertial effects, Networks and Heterogeneous Media 4 (2009) 625–647. [14] A. W. El-Kareh and L. G. Leal, Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion, J. Non-Newtonian Fluid Mech. 33 (1989) 257–287. [15] A. Ern and J. L. Guermond, Theory and Practice of Finite Elements (Springer Verlag, New York, 2004). [16] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, volume 5 of Springer Ser. Comp. Math. (Springer-Verlag, 1986). [17] G. Gr¨ un and M. Rumpf, Nonnegativity preserving numerical schemes for the thin film equation, Numer. Math. 87 (2000) 113–152. [18] J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem I: Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982) 275–311. [19] D. Hu and T. Leli` evre, New entropy estimates for the Oldroyd-B model, and related models, Comm. Math. Sci. 5 (2007) 906–916. [20] N. Masmoudi, Global existence of weak solutions to macroscopic models of polymeric flows, J. Math. Pures Appl. 96 (2011) 502–520. [21] R. G. Owens and T. N. Phillips, Computational Rheology (Imperial College Press, 2002). [22] M. Renardy, Mathematical Analysis of Viscoelastic Flows, volume 73 of CBMS-NSF Conference Series in Applied Mathematics (SIAM, 2000). [23] J. D. Schieber, Generalized Brownian configuration field for Fokker–Planck equations including center-of-mass diffusion, J. Non-Newtonian Fluid Mech. 135 (2006) 179–181. [24] R. Sureshkumar and A. Beris, Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows, J. Non-Newtonian Fluid Mech. 60 (1995) 53–80. [25] R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, volume 2 of Studies in Mathematics and its Applications (North-Holland, Amsterdam, 1984). [26] P. Wapperom and M. Hulsen, Thermodynamics of viscoelastic fluids: The temperature equation, J. NonNewtonian Fluid Mech. 42 (1998) 999–1019. Department of Mathematics, Imperial College London, London SW7 2AZ, UK, [email protected] Laboratoire d’hydraulique Saint-Venant, Universit´ e Paris-Est (Ecole des Ponts ParisTech) EDF R&D, 6 quai Watier, 78401 Chatou Cedex, France, [email protected]

Finite Element Approximation of the FENE-P Model

Finite Element Approximation of the FENE-P Model

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