Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers

DOI:10.1051/m2an/2011062 www.esaim-m2an.org

FINITE ELEMENT APPROXIMATION OF FINITELY EXTENSIBLE NONLINEAR ELASTIC DUMBBELL MODELS FOR DILUTE POLYMERS

John W. Barrett

and Endre S¨ uli

Abstract. We construct a Galerkin finite element method for the numerical approximation of weak solutions to a general class of coupled FENE-type finitely extensible nonlinear elastic dumbbell models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain Ω⊂R^d,d= 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. We perform a rigorous passage to the limit as first the spatial discretization parameter, and then the temporal discretization parameter tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier–Stokes–Fokker–Planck system. The passage to the limit is performed under minimal regularity assumptions on the data: a square-integrable and divergence-free initial velocity datumu_∼0for the Navier–Stokes equation and a nonnegative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M.

Mathematics Subject Classification. 35Q30, 35K65, 65M12, 65M60, 76A05, 82D60.

Received March 19, 2011. Revised August 3, 2011.

Published online February 13, 2012.

1. Introduction

This paper is concerned with the construction and convergence analysis of a finite element method for the numerical approximation of weak solutions to a system of nonlinear partial differential equations that arises from the kinetic theory of dilute polymer solutions. The mathematical model under consideration couples the incompressible Navier–Stokes equations, with the divergence of the elastic extra-stress tensor appearing on its right-hand side, to a high-dimensional Fokker–Planck equation with an unbounded drift term, whose transport coefficients depend on the fluid velocity and whose solution is a probability density function featuring in the

Keywords and phrases.Finite element method, convergence analysis, existence of weak solutions, kinetic polymer models, FENE dumbbell, Navier–Stokes equations, Fokker–Planck equations.

1 Department of Mathematics, Imperial College London, London SW7 2AZ, UK.j.barrett@ imperial.ac.uk

2 Mathematical Institute, University of Oxford, 24–29 St. Giles’, OX1 3LB Oxford, UK.endre.suli@ maths.ox.ac.uk

Article published by EDP Sciences c EDP Sciences, SMAI 2012

J.W. BARRETT AND E. S ¨ULI

definition of the elastic extra stress tensor. The existence and exponential equilibration of global weak solutions to the system was proved in two substantial recent papers, [7,9], for both finitely extensible nonlinear bead- spring chain type polymer models and Hookean type bead-spring chain models, using a delicate combination of entropy estimates and weak compactness arguments applied to sequences of approximations that were generated by a temporal semidiscretization of the system. Weak solutions to the Navier–Stokes–Fokker–Planck system, whose existence was thus shown, were also shown to satisfy the natural energy inequality originating from the formal balance law for the system. In this paper, we construct a fully-discrete numerical approximation of the model and show that, as the spatial mesh size tends to zero, a subsequence of the sequence of numerical solutions converges to a weak solution of the temporally semidiscrete scheme, a subsequence of whose solutions is, in turn, known from [7] to converge to a weak solution of the coupled Navier–Stokes–Fokker–Planck system as the time step tends to zero. Hence we deduce that a subsequence of the sequence of numerical approximations converges to a weak solution of the problem. To the best of our knowledge, this is the first example of a general family of fully-discrete, convergent numerical methods for the coupled Navier–Stokes–Fokker–Planck system. The key contributions of the paper are the following: (1) the finite element method constructed here reproduces at the discrete level the energy inequality satisfied by weak solutions of the Navier–Stokes–Fokker–Planck system; and (2) convergence to weak solutions is established under minimal regularity assumptions on the data: a square- integrable and divergence-free initial velocity datum for the Navier–Stokes equation and a nonnegative initial probability density function for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian of the model.

We shall now describe the Navier–Stokes–Fokker–Planck system under consideration, arising from the kinetic theory of dilute polymer solutions. The solvent is an incompressible, viscous, isothermal Newtonian fluid confined to a bounded open setΩ⊂R^d,d= 2 or 3, with a Lipschitz-continuous boundary∂Ω. For the sake of simplicity of presentation we shall suppose thatΩhas ‘solid boundary’∂Ω; the velocity fieldu_∼will then satisfy the no-slip boundary conditionu_∼= 0_∼on∂Ω. The polymer chains, which are suspended in the solvent, are assumed not to interact with each other. The conservation of momentum and mass equations for the solvent then have the form of the incompressible Navier–Stokes equations in which the elasticextra-stresstensorτ_≈(i.e., the polymeric part of the Cauchy stress tensor), appears as a source term:

GivenT ∈R>0, findu_∼ : (x_∼, t)∈Ω×[0, T]→u_∼(x_∼, t)∈R^dandp : (x_∼, t)∈Ω×(0, T]→p(x_∼, t)∈Rsuch that

∂u∼

∂t + (u

∼· ∇

∼x)u

∼−ν Δ_xu

∼+∇

∼xp=f

∼

+∇

∼x ·τ

≈ inΩ×(0, T], (1.1a)

∇∼x ·u

∼= 0 inΩ×(0, T], (1.1b)

u∼= 0

∼ on∂Ω×(0, T], (1.1c)

u∼(x

∼,0) =u

∼0(x

∼) ∀x

∼∈Ω. (1.1d)

It is assumed that each of the equations above has been written in its nondimensional form; u_∼ denotes a nondimensional velocity, defined as the velocity field scaled by the characteristic flow speedU₀;ν∈R>0is the reciprocal of the Reynolds number,i.e.the ratio of the kinematic viscosity coefficient of the solvent andL₀U₀, whereL₀is a characteristic length-scale of the flow;pis the nondimensional pressure andfis the nondimensional density of body forces.

In abead-spring chain model, consisting ofK+ 1 beads coupled withKelastic springs to represent a polymer chain, the extra-stress tensorτ_≈is defined by theKramers expression as a weighted average ofψ, the probability density function of the (random) conformation vector q

∼:= (q

∼

T1, . . . , q

∼

TK)^T∈R^Kd of the chain (cf.(1.7) below, in the case ofK= 1), withq

∼i representing thed-component conformation/orientation vector of theith spring.

The Kolmogorov equation satisfied by ψ is a second-order parabolic equation, the Fokker–Planck equation, whose transport coefficients depend on the velocity fieldu_∼. The domain D of admissible conformation vectors D ⊂R^Kd is aK-fold Cartesian product D₁× · · · ×D_K of balanced convex open setsD_i ⊂R^d,i = 1, . . . , K;

the term balanced means that q

∼i ∈ D_i if, and only if, −q

∼i ∈ D_i. Hence, in particular, 0_∼ ∈D_i, i = 1, . . . , K.

Typically D_i is the whole of R^d or a bounded open d-dimensional ball centred at the origin 0_∼∈ R^d for each i= 1, . . . , K. WhenK= 1, the model is referred to as thedumbbell model.

Henceforth we shall confine ourselves to the case of K = 1 – the dumbbell model; then, D =D₁, and we shall simply write D instead of D₁. Concerning extensions to the case of K≥1, the reader is referred to the concluding remarks in Section5.

LetO ⊂[0,∞) denote the image ofDunder the mappingq

∼∈D→ ¹₂|q

∼|², and consider thespring potentialU∈ C²(O;R_≥0). Clearly, 0∈ O. We shall suppose thatU(0) = 0 and thatU is monotonic increasing and unbounded onO. The elastic spring-forceF_∼ : D⊆R^d→R^dof the spring is defined by

F∼(q

∼) =U(1 2|q

∼|²)q

∼. (1.2)

Remark 1.1. In the Hookean dumbbell model the spring force is defined by F_∼(q

∼) = q

∼, with q

∼ ∈ D = R^d, corresponding toU(s) =s,s∈ O= [0,∞). This model is physically unrealistic as it admits an arbitrarily large extension.

We shall therefore assume in what follows thatD is aboundedopen ball, centred at the origin 0_∼∈R^d, with boundary∂D. We shall further suppose that there exist constantsc_j>0,j= 1,2,3,4, andγ >1 such that the (normalized) MaxwellianM, defined by

M(q

∼) = 1

Ze^−U(12|q

∼|²)

, Z:=

D

e^−U(12|q

∼|²)

dq∼, (1.3)

and the associated spring potentialU satisfy c₁[dist(q

∼

, ∂D)]^γ≤M(q

∼

) ≤c₂[dist(q

∼

, ∂D)]^γ ∀q

∼∈D, (1.4a)

c₃≤[dist(q

∼

, ∂D)]U(1 2|q

∼|²)≤c₄ ∀q

∼∈D. (1.4b)

Observe that

M(q

∼)∇_∼q[M(q

∼)]⁻¹=−[M(q

∼)]⁻¹∇_∼qM(q

∼) =∇_∼qU(1

2|q_∼|²) =U(1 2|q_∼|²)q

∼. (1.5)

Since [U(¹₂|q

∼|²)]² = (−logM(q

∼) + Const.)², it follows from (1.4a), (1.4b) that (ifγ >1, as has been assumed

here,)

D

1 + [U(1 2|q

∼|²)]²+ [U(1 2|q

∼|²)]²

M(q

∼) dq

∼<∞. (1.6)

Remark 1.2. In the FENE (finitely extensible nonlinear elastic) dumbbell model the spring force is given by F_∼(q

∼) = (1− |q

∼|²/b)⁻¹q

∼, q

∼ ∈ D = B(0_∼, b¹²), corresponding to U(s) = −₂^blog 1−^2s_b

, s ∈ O = [0,₂^b).

Here B(0_∼, b¹²) is a bounded open ball inR^d centred at the origin 0_∼∈ R^d and of fixed radius b¹², with b >0.

Direct calculations show that the Maxwellian M and the elastic potential U of the FENE model satisfy the conditions (1.4a), (1.4b) withγ:=₂^b provided thatb >2. Thus, (1.6) also holds forb >2.

The governing equations of the general FENE-type dumbbell model with centre-of-mass diffusion are (1.1a)–

(1.1d), where the extra-stress tensor τ_≈, depending on the probability density function ψ, is defined by the Kramers expression:

τ≈(ψ) =k C≈(ψ)

−k ρ(ψ)I_≈. (1.7)

Here the dimensionless constantk >0 is a constant multiple of the product of the Boltzmann constantk_B and the absolute temperatureT,I_≈is the unitd×dtensor, and

C≈(ψ)(x_∼, t) =

D

ψ(x_∼, q

∼, t)q

∼q

∼

TU 1

2|q

∼|²

dq∼, and ρ(ψ)(x_∼, t) =

D

ψ(x_∼, q

∼, t) dq

∼, (1.8)

J.W. BARRETT AND E. S ¨ULI

whereρ(ψ)(x_∼, t) is the density of polymer chains located atx_∼ at timet. The probability density functionψis a solution of the Fokker–Planck equation

∂ψ

∂t + (u_∼· ∇_∼x)ψ+∇_∼q · σ_≈(u_∼)q

∼ψ

=ε Δ_xψ+ 1 2λ∇_∼q ·

M∇_∼q

ψ M

inΩ×D×(0, T], (1.9) with σ_≈(v_∼) ≡ ∇_≈xv_∼, where (∇_≈xv_∼)(x_∼, t) ∈ R^d×d and {∇_≈xv_∼}ij = _∂x^∂vi

j. In (1.9), ε > 0 is the centre-of-mass diffusion coefficient defined asε:= (₀/L₀)²/(8λ) with₀:=

k_BT/Hsignifying the characteristic microscopic length-scale and λ:= (ζ/4H)(U0/L₀), whereζ >0 is a friction coefficient and H>0 is a spring-constant. The dimensionless parameter λ∈R_>0, called the Weissenberg number (and usually denoted byWi), characterizes the elastic relaxation property of the fluid.

We impose the following boundary and initial conditions onψ:

M 1

2λ∇

∼q

ψ M

−σ

≈(u

∼)q

∼

ψ M

· q

|q∼

∼| = 0 onΩ×∂D×(0, T], (1.10a) ε∇

∼xψ · n

∼= 0 on∂Ω×D×(0, T], (1.10b)

ψ(·,·,0) =ψ₀(·,·)≥0 onΩ×D, (1.10c)

whereq

∼is normal to∂D, asD is a bounded ball centred at the origin inR^d, andn_∼is normal to∂Ω. The initial conditionψ₀ is nonnegative, defined onΩ×D, with

Dψ₀(x_∼, q

∼) dq

∼ = 1 for a.e. x_∼ ∈Ω, and assumed to have finite relative entropy with respect to the MaxwellianM;i.e.

Ω×Dψ₀(x_∼, q

∼) log(ψ₀(x_∼, q

∼)/M(q

∼)) dq

∼dx_∼<∞. The boundary and initial conditions forψhave been chosen so as to ensure that

D

ψ(x_∼, q

∼, t) dq

∼=

D

ψ(x_∼, q

∼,0) dq

∼= 1 ∀(x_∼, t)∈Ω×(0, T]. (1.11) Remark 1.3. The collection of equations and structural hypotheses ((1.1a)–(1.1d))–((1.10a)–(1.10c)) will be referred to throughout the paper as model (P), or as thegeneral FENE-type dumbbell model with centre-of-mass diffusion.

A noteworthy feature of equation (1.9) in the model (P) compared to classical Fokker–Planck equations for dumbbell models in the literature is the presence of thex_∼-dissipative centre-of-mass diffusion termε Δ_xψon the right-hand side of the Fokker–Planck equation (1.9). We refer to Barrett and S¨uli [3] for the derivation of (1.9);

see also the article by Schieber [28] concerning generalized dumbbell models with centre-of-mass diffusion, and the recent paper of Degond and Liu [14] for a careful justification of the presence of the centre-of-mass diffusion term through asymptotic analysis. In standard derivations of bead-spring models the centre-of-mass diffusion term is routinely omitted on the grounds that it is several orders of magnitude smaller than the other terms in the equation. Indeed, when the characteristic macroscopic length-scaleL₀≈1, (for example,L₀= diam(Ω)), Bhave et al.[10] estimate the ratio²₀/L²₀ to be in the range of about 10⁻⁹to 10⁻⁷. However, the omission of the term ε Δ_xψfrom (1.9) in the case of a heterogeneous solvent velocityu_∼(x_∼, t) is a mathematically counterproductive model reduction. Whenε Δ_xψis absent, (1.9) becomes a degenerate parabolic equation exhibiting hyperbolic behaviour with respect to (x_∼, t). Since the study of weak solutions to the coupled problem requires one to work with velocity fieldsu_∼that have very limited Sobolev regularity (typicallyu_∼∈L^∞(0, T;L_∼²(Ω))∩L²(0, T;H_∼¹₀(Ω))), one is then forced into the technically unpleasant framework of hyperbolically degenerate parabolic equations with rough transport coefficients (cf. Ambrosio [1] and DiPerna and Lions [15]). The resulting difficulties are further exacerbated by the fact that a typical spring forceF_∼(q

∼) for a finitely extensible model (such as FENE) explodes asq

∼approaches∂D; see Remark1.2above. For these reasons, here we shall retain the centre-of-mass diffusion term in (1.9).

Lions and Masmoudi [24] proved the global existence of weak solutions for the simplifiedcorotational FENE dumbbell model, i.e. with σ(u_∼) = ∇_≈xu_∼ replaced by its skew-symmetric part ¹₂(∇_≈xu_∼−(∇_≈xu)^T), and with

ε = 0; see also the work of Masmoudi [25]. Under very general assumptions on the finite-dimensional spaces used for the purpose of spatial discretization, including, in particular, classical conforming finite element spaces and spectral Galerkin subspaces, Barrett and S¨uli [5] showed the convergence of a (sub)sequence of numerical approximations to a weak solution of the coupled Navier–Stokes–Fokker–Planck system (P), for a large class of unbounded spring potentials, including the FENE potential, in the case of the simplified corotational model.

Recently, Masmoudi [26] has extended the analysis of Lions and Masmoudi [24] to the noncorotational case. For a fuller literature survey on the mathematical analysis of FENE-type dumbbell models we refer the reader to Barrett and S¨uli [7]. In the rest of this section we concentrate on those references that are relevant to the finite element approximation developed and analyzed in this paper.

In Barrett and S¨uli [4] we showed the existence of global-in-time weak solutions to the general class of noncorotational FENE type dumbbell models (including the standard FENE dumbbell model) with centre-of- mass diffusion and microscopic cut-off (cf.(1.13) and (1.14) below) in the drag term

∇∼q ·(σ_≈(u_∼)q

∼ψ) =∇_∼q ·

σ≈(u_∼)q

∼M ψ

M

· (1.12)

We observe that ifψ/M is bounded above then, forL∈R>0sufficiently large, the drag term (1.12) is equal to

∇∼q ·

σ≈(u_∼)q

∼M β^L ψ

M

, (1.13)

whereβ^L∈C(R) is a cut-off function defined as

β^L(s) := min(s, L). (1.14)

It then follows that, forL1, any solutionψof (1.9), such thatψ/M is bounded above, also satisfies

∂ψ

∂t + (u

∼· ∇

∼x)ψ+∇

∼q ·

σ

≈(u

∼)q

∼

M β^L ψ

M

=ε Δ_xψ+ 1 2λ∇

∼q ·

M∇

∼q

ψ M

inΩ×D×(0, T], (1.15) and the following boundary and initial conditions:

M 1

2λ∇

∼q

ψ M

−σ

≈(u

∼)q

∼

β^L ψ

M

· q

|q∼

∼| = 0 onΩ×∂D×(0, T], (1.16a) ε∇

∼xψ · n

∼= 0 on∂Ω×D×(0, T], (1.16b)

ψ(·,·,0) =M(·)β^L(ψ₀(·,·)/M(·))≥0 onΩ×D. (1.16c) Clearly, if there existsL >0 such that 0≤ψ₀≤L M, thenM β^L(ψ₀/M) =ψ₀. HenceforthL >1 is assumed.

Remark 1.4. The coupled problem (1.1a)–(1.1d), (1.7), (1.8), (1.15), (1.16a)–(1.16c) will be referred to as model (P_L), or as thegeneral FENE-type dumbbell model with centre-of-mass diffusion and microscopic cut-off, with cut-off parameterL >1.

In order to highlight the dependence on L, in subsequent sections the solution to (1.15), (1.10a)–(1.10c) will be labelled ψ_L, and we work with the variable ψ_L := ψ_L/M. Due to the coupling of (1.15) to (1.1a)–

(1.1d) through (1.7), the velocity and the pressure will also depend on Land we shall therefore denote them in subsequent sections by u_∼L and p_L. As has been already emphasized earlier, the centre-of-mass diffusion coefficientε >0 is a physical parameter and is regarded as being fixed, so we do not highlight its presence in the model through our notation.

J.W. BARRETT AND E. S ¨ULI

Barrett and S¨uli [8] constructed a Galerkin finite element approximation, and proved (sub)sequence conver- gence, to a weak solution of a system similar to (P_L), where ψ_L in the convective term, in addition to the drag term, is replaced byβ^L(ψ_L). Finally, Barrett and S¨uli proved in [7,9] the existence and exponential equili- bration of global-in-time weak solutions to Navier–Stokes–Fokker–Planck systems for general classes of finitely extensible nonlinear elastic and Hookean-type bead-spring chains models.

After this brief literature survey, we now turn our attention to the description of the formal energy bounds on which the constructions of the temporally semidiscrete and fully-discrete approximations are based.

2. Formal energy bounds for (P) and (P

)

In this section we identify formally the energy structure for (P), and related regularized models. Before doing so, we note that the notation| · |will be used to signify one of the following. When applied to a real numberx,

|x|will denote the absolute value of the numberx; when applied to a vectorv_∼,|v_∼|will stand for the Euclidean norm of the vectorv_∼; and, when applied to a square matrixA,|A|will signify the Frobenius norm, [tr(A^TA)]¹², of the matrixA, where, for a square matrixB,tr(B) denotes the trace ofB.

Multiplying (1.1a) byu_∼, integrating overΩ, and noting (1.1b), (1.1c) yields that 1

2 d dt

Ω

|u∼|² dx

∼

+ν

Ω

|∇≈xu

∼|²dx

∼−

Ω

f

∼·u

∼dx

∼=−

Ω

τ≈(Mψ) : ∇

≈xu

∼ dx

∼

=−k

Ω

C≈(Mψ) : ∇

≈xu

∼ dx

∼, (2.1)

where ψ := ψ/M. Let F(s) := (lns−1)s+ 1 for s > 0, with F(0) := 1. Multiplying the Fokker–Planck equation (1.9) by F(ψ) ≡lnψ, on assuming that ψ > 0, integrating overΩ×D and noting (1.10a), (1.10b) and (1.1b), (1.1c) yields that

d dt

Ω×DMF(ψ) dq

∼

dx∼

+ 1

2λ

Ω×DM∇

∼qψ· ∇

∼q[F(ψ)] dq

∼

dx∼

+ε

Ω×DM∇

∼xψ· ∇

∼x[F(ψ)] dq

∼

dx∼=

Ω×DMψ[(∇

≈xu

∼)q

∼

]· ∇

∼q[F(ψ)] dq

∼

dx∼. (2.2) It follows, on noting that F(s) =s⁻¹ >0 for s > 0 and hence that ψ∇_∼q[F(ψ)] = ∇_∼qψ, (1.5), (1.1b) and M = 0 on∂D that

Ω×D

Mψ[(∇

≈xu

∼)q

∼

]· ∇

∼q[F(ψ)] dq

∼

dx∼=

Ω×D

M[(∇

≈xu

∼)q

∼

]· ∇

∼qψdq

∼

dx∼

=

Ω×DM U(1 2|q

∼|²)q

∼·[(∇

≈xu

∼)q

∼

]ψdq

∼

dx∼

=

Ω

C≈(Mψ) : ∇

≈xu

∼dx

∼, (2.3)

on recalling (1.8). Combining (2.1)–(2.3), we obtain the following energy law for (P):

d dt

1 2

Ω

|u∼|² dx

∼+k

Ω×DMF(ψ) dq

∼

dx∼

+ν

Ω

|∇≈xu

∼|² dx

∼+k ε

Ω×DMψ|∇

∼x[F(ψ)] |²dq

∼

dx∼

+ k 2λ

Ω×DMψ|∇

∼q[F(ψ)]| ²dq

∼

dx∼=

Ω

f

∼·u

∼dx

∼. (2.4)

To make the above rigorous, and for computational purposes, we replace the convex function F ∈C(R_≥0)∩ C^∞(R_>0) by its convex regularization F_δ^L∈C^2,1(R) defined, for anyδ∈(0,1) andL >1, as follows:

F_δ^L(s) :=

⎧⎪

⎨

⎪⎩

s²−δ²

2δ + (lnδ−1)s+ 1 s≤δ, F(s)≡(lns−1)s+ 1 δ≤s≤L,

s²−L²

2L + (lnL−1)s+ 1 L≤s.

(2.5)

Hence, we have that [F_δ^L](s) =

⎧⎨

⎩

s

δ + lnδ−1 s≤δ,

lns δ≤s≤L,

s

L+ lnL−1 L≤s,

and [F_δ^L](s) =

⎧⎨

⎩

δ⁻¹ s≤δ, s⁻¹ δ≤s≤L, L⁻¹ L≤s.

(2.6)

In addition, we introduce

β_δ^L(s) := [[F_δ^L](s)]⁻¹=

⎧⎨

⎩

δ s≤δ, s δ≤s≤L, L L≤s.

(2.7)

It follows from (2.7), for any sufficiently smoothϕ, that β_δ^L(ϕ)∇

∼x([F_δ^L](ϕ) ) =∇

∼xϕ and β_δ^L(ϕ)∇

∼q([F_δ^L](ϕ) ) = ∇

∼qϕ. (2.8)

Let {u_∼L,δ,ψ_L,δ} solve problem (P_L,δ), which is a regularization of the problem (P), similar to (P_L), where β^L(·) in (1.15) and (1.16a) is replaced byβ^L_δ(·). Multiplying the Fokker–Planck equation in (P_L,δ) by [F_δ^L](ψ_L,δ), integrating overΩ×D, noting the boundary conditions and (2.8) yields, similarly to (2.2) and (2.3), that

d dt

Ω×DMF_δ^L(ψ_L,δ) dq

∼

dx∼

+ 1

2λ

Ω×DM∇

∼qψ_L,δ· ∇

∼q

[F_δ^L](ψ_L,δ)

dq

∼

dx∼

+ε

Ω×DM∇

∼xψ_L,δ· ∇

∼x

[F_δ^L](ψ_L,δ)

dq

∼

dx∼=

Ω

C

≈(Mψ_L,δ) :∇

≈xu

∼L,δdx

∼. (2.9) Combining (2.9) and the (P_L,δ) version of (2.1), we obtain the following energy law for (P_L,δ), a regularized analogue of (2.4):

d dt

1 2

Ω

|u∼L,δ|²dx

∼+k

Ω×DMF_δ^L(ψ_L,δ) dq

∼

dx∼

+ν

Ω

|∇≈xu

∼L,δ|² dx

∼

+k ε

Ω×DM β_δ^L(ψ_L,δ)|∇

∼x

[F_δ^L](ψ_L,δ) |²dq

∼

dx∼

+ k 2λ

Ω×DM β_δ^L(ψ_L,δ)|∇

∼q

[F_δ^L](ψ_L,δ) |²dq

∼

dx∼=

Ω

f

∼·u

∼L,δ dx

∼. (2.10)

On noting that [F_δ^L]≥L⁻¹, and

min{F_δ^L(s), s[F_δ^L](s)} ≥ s²

2δ ifs≤0,

s²

4L−C(L) ifs≥0, (2.11)

one deduces from (2.10) that sup

t∈(0,T)

Ω

|u

∼L,δ|²dx

∼

+ν

ΩT

|∇≈xu

∼L,δ|²dx

∼dt+δ⁻¹ sup

t∈(0,T)

Ω×D

M|[ψ_L,δ]₋|²dq

∼

dx∼

≤C(L). (2.12)

J.W. BARRETT AND E. S ¨ULI

In addition, one can show that sup

t∈(0,T)

Ω×DM|ψ_L,δ|²dq

∼

dx∼

+1

λ _T

0

Ω×DM∇

∼qψ_L,δ²dq

∼

dx∼dt +ε

_T

0

Ω×D

M∇

∼xψ_L,δ² dq

∼

dx∼dt+ sup

t∈(0,T)

Ω

|C

≈(Mψ_L,δ)|²dx

∼≤C(L, T). (2.13) The above formal bounds can be made rigorous and the existence of a global-in-time weak solution{u_∼L,δ,ψ_L,δ} to (P_L,δ) can be established, see [4]. Moreover, one can take the limit δ→ 0₊ in problem (P_L,δ) to establish the existence of a global-in-time weak solution{u_∼L,ψ_L} to problem (P_L) with ψ_L ≥0 a.e. in Ω×D×(0, T).

Once again, see [4].

The aim of this paper is to construct a finite element approximation, (P^Δt,h_L,δ ), of problem (P_L,δ), which mimics the energy law (2.10) at a discrete level, and to show that a (sub)sequence of this approximation converges to a weak solution of (P^Δt_L ), as the spatial discretization parameterh, as well as the regularization parameterδ, tend to zero. Here (P^Δt_L ) is a time discretization of (P_L). Barrett and S¨uli [7] showed that for a specific time discretization (P^Δt_L ) a (sub)sequence of this approximation converges to a weak solution of (P), as the cut-off parameterLtends to infinity with the time discretization parameterΔt= o(L⁻¹).

The outline of this paper is as follows. In the next section, we introduce the necessary function spaces. In addition, we introduce the particular time discretization, (P^Δt_L ), of (P_L) and state the relevant convergence results from Barrett and S¨uli [7]. In Section 4, we introduce our finite element approximation, (P^Δt,h_L,δ ), of problem (P_L,δ) and show that a (sub)sequence of this approximation converges to a weak solution of (P^Δt_L ), as the spatial discretization parameterh, as well as the regularization parameterδ, tend to zero. Hence combining this with the convergence result in Section 3, we obtain the desired result that a (sub)sequence of our finite element approximation (P^Δt,h_L,δ ) converges to a weak solution of (P) as firsth, δ →0₊ and thenL→ ∞, with Δt= o(L⁻¹). In Section 5 we discuss possible extensions of our results. The paper closes with an Appendix, containing the proofs of some technical bounds required in the convergence analysis of the initialization of the scheme.

3. The discrete-in-time approximation (P

)

Let

H∼ :={w_∼ ∈L_∼²(Ω) :∇_∼x ·w_∼ = 0} and V_∼ :={w_∼ ∈H_∼ ¹₀(Ω) :∇_∼x ·w_∼ = 0}, (3.1) where the divergence operator∇_∼x· is to be understood in the sense of distributions onΩ. LetV_∼ be the dual ofV_∼. More generally, letV_∼σdenote the closure of the set of all divergence-freeC_∼^∞₀ (Ω) functions in the norm of H∼ 1

0(Ω)∩H_∼^σ(Ω),σ≥1, equipped with the Hilbert space norm, denoted by · Vσ, inherited fromH_∼ ^σ(Ω), and letV_∼σsignify the dual space ofV_∼σ, with duality pairing·,·Vσ. AsΩis a bounded Lipschitz domain, we have thatV_∼1=V_∼ (cf.Temam [30], Chap. 1, Thm. 1.6). Similarly,·,·_H¹_0(Ω)will denote the duality pairing between (H_∼¹₀(Ω)) andH_∼ ¹₀(Ω). The norm on (H_∼¹₀(Ω)) will be that induced from taking∇_≈x · L²(Ω) to be the norm on H∼ 1

0(Ω).

For later purposes, we recall the following well-known Gagliardo–Nirenberg inequality. Letr∈[2,∞) ifd= 2, andr∈[2,6] ifd= 3 andθ=d₁

2−¹_r

. Then, there is a constantC=C(Ω, r, d), such that, for allη∈H¹(Ω):

η_L^r_(Ω)≤Cη^1−θ_L2(Ω)η^θ_H1(Ω). (3.2) Let F ∈C(R>0) be defined by F(s) :=s(logs−1) + 1, s > 0. As lim_s→0+F(s) = 1, the function F can be considered to be defined and continuous on [0,∞), where it is a nonnegative, strictly convex function with F(1) = 0.

We recall our assumptions on the data:

(A1) ∂Ω ∈C^0,1; D=B(0

∼, r_D) withr_D>0 andU satisfying (1.4a), (1.4b) withγ >1;

u∼0∈H

∼ ; ψ₀:= ψ₀

M ≥0 a.e.onΩ×D with F(ψ₀)∈L¹_M(Ω×D) and

D

M(q

∼

)ψ₀(x

∼, q

∼

) dq

∼

= 1 for a.e.x

∼∈Ω; and f

∼∈L²(0, T; (H

∼

10(Ω))). (3.3)

Here,L^p_M(Ω×D), forp∈[1,∞), denotes the Maxwellian-weightedL^pspace overΩ×D with norm ϕ _L^p_M(Ω×D):=

Ω×DM|ϕ|^pdq

∼dx_∼ ¹_p

.

Similarly, we introduceL^p_M(D), the Maxwellian-weightedL^pspace overD. Letting ϕ _HM¹_(Ω×D):=

Ω×DM

|ϕ|²+|∇_∼xϕ|²+|∇_∼qϕ|² dq∼dx_∼

¹₂

, (3.4)

we then set

X≡H_M¹ (Ω×D) :=

ϕ∈L¹_loc(Ω×D) :ϕ _HM¹ _(Ω×D)<∞

. (3.5)

It is shown in Appendix C of [6] (the extended version of Barrett and S¨uli [7]) that

C^∞(Ω×D) is dense in X. (3.6)

In addition, we note that the embeddings

H_M¹ (D)→L²_M(D), (3.7a)

H_M¹ (Ω×D)≡L²(Ω;H_M¹ (D))∩H¹(Ω;L²_M(D))→L²_M(Ω×D)≡L²(Ω;L²_M(D)) (3.7b) are compact if γ ≥1 in (1.4a), (1.4b); see Appendix D in [6]. Throughout we will assume that (3.3) hold, so that (1.6) and (3.7a), (3.7b) hold. We note for future reference that (1.8) and (1.6) yield that, forϕ∈L²_M(Ω×D),

Ω

|C≈(Mϕ)| ²dx

∼=

Ω

D

Mϕ U q

∼

q

∼

Tdq

∼

² dx

∼≤C

Ω×DM|ϕ|²dq

∼

dx∼

, (3.8)

whereC is a positive constant.

We now formulate our discrete-in-time approximation of problem (P_L) for a fixed parameterL >1. For any T >0 andN ≥1, letN Δt=T andt_n=n Δt,n= 0, . . . , N. To prove existence of a solution under minimal smoothness requirements on the initial datum u_∼0 (recall (3.3)), we introduce u_∼⁰ = u_∼⁰(Δt) ∈ V_∼, the unique solution of

Ω

u∼

0·v

∼+Δt∇

≈xu

∼

0:∇

≈xv

∼

dx∼=

Ω

u∼0·v

∼dx

∼ ∀v

∼∈V

∼; (3.9)

and so

Ω

[|u_∼⁰|²+Δt|∇_≈xu_∼⁰|²] dx_∼≤

Ω

|u_∼0|²dx_∼≤C. (3.10)

In addition, we have thatu_∼⁰ converges tou_∼0 weakly inH_∼ in the limit ofΔt→0₊.

J.W. BARRETT AND E. S ¨ULI

Analogously to defining u_∼⁰ for a given initial velocity field u_∼0, we shall assign a certain ‘smoothed’ initial datum, ψ⁰=ψ⁰(L, Δt)∈H_M¹ (Ω×D), to the initial datumψ₀ such that

Ω×DM

ψ⁰ϕ+Δt ∇

∼xψ⁰· ∇

∼xϕ+∇

∼qψ⁰· ∇

∼qϕ

dq

∼

dx∼=

Ω×DM β^L(ψ₀)ϕdq

∼

dx∼ ∀ϕ∈H_M¹ (Ω×D).

(3.11) Forp∈[1,∞), let

Z_p:=

ϕ∈L^p_M(Ω×D) :ϕ≥0 a.e. onΩ×D and

D

M(q

∼

)ϕ(x

∼, q

∼

) dq

∼≤1 for a.e.x

∼∈Ω

. (3.12) It is proved in the appendix that there exists a uniqueψ⁰∈H_M¹ (Ω×D) satisfying (3.11); furthermore,

ψ⁰∈Z₁;

Ω×DMF(ψ⁰) dq

∼dx_∼+ 4Δt

Ω×DM∇_∼x

ψ⁰²+∇_∼q

ψ⁰²

dq∼dx_∼≤

Ω×DMF(ψ₀) dq

∼dx_∼; (3.13a) and

ψ⁰≡β^L(ψ⁰)→ψ₀ weakly inL¹_M(Ω×D) as L→ ∞ and Δt→0₊. (3.13b) It follows from (3.13a), (3.13b) and (1.14) thatψ⁰∈Z₂; in fact,ψ⁰∈L^∞(Ω×D)∩H_M¹ (Ω×D).

Our discrete-in-time approximation of (P_L) is then defined as follows.

(P^Δt_L )Let u_∼⁰_L :=u_∼⁰ ∈V_∼ and ψ⁰_L := ψ⁰ ∈ Z₂. Then, forn = 1, . . . , N, given (u_∼ⁿ⁻¹_L ,ψ_Lⁿ⁻¹)∈ V_∼ ×Z₂, find (u_∼ⁿ_L,ψⁿ_L)∈V_∼ ×(X∩Z₂) such that

Ω

u

∼

nL−u

∼

n−1L

Δt + (u

∼

n−1L · ∇

∼x)u

∼

nL

· w

∼dx

∼+ν

Ω

∇≈xu

∼

nL:∇

≈xw

∼dx

∼

=f

∼

n, w

∼_H0¹_(Ω)−k

Ω

C≈(Mψⁿ_L) :∇

≈xw

∼ dx

∼ ∀w

∼ ∈V

∼, (3.14a)

Ω×DM ψⁿ_L−ψⁿ⁻¹_L Δt ϕdq

∼

dx∼+

Ω×DM 1

2λ∇

∼qψ_Lⁿ−[σ

≈(u

∼

nL)q

∼

]β^L(ψⁿ_L)

· ∇∼qϕdq

∼

dx∼

+

Ω×DM

ε∇

∼xψ_Lⁿ−u

∼

n−1 L ψⁿ_L

· ∇

∼xϕdq

∼

dx∼= 0 ∀ϕ∈X; (3.14b) where, fort∈[t_n−1, t_n), andn= 1, . . . , N,

f

∼

Δt,+(·, t) =f

∼

n(·) := 1 Δt

_tn

tn−1

f

∼(·, t) dt∈(H

∼

10(Ω))⊂V

∼

. (3.15)

It follows from (3.3) and (3.15) that N

n=1

Δtf

∼

n^r_H−1(Ω)≤ _T

0 f

∼^r_H−1(Ω)dt≤C for anyr∈[1,2], (3.16a) f

∼

Δt,+→f

∼

strongly inL²(0, T; (H

∼

10(Ω))) as Δt→0₊. (3.16b)