2004-Fez conference on Differential Equations and Mechanics Electronic Journal of Differential Equations, Conference 11, 2004, pp. 41–51. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

NUMERICAL ANALYSIS OF EULER-SUPG MODIFIED METHOD FOR TRANSIENT VISCOELASTIC FLOW

MOHAMMED BENSAADA, DRISS ESSELAOUI

Abstract. We study a new approximation scheme of transient viscoelastic fluid flow obeying an Oldroyd-B type constitutive law. The approximation stress, velocity, pressure are respectively P_{1}-continuous, P_{2}-continuous, P_{1}-continuous. We use the modified streamline upwinding Petrov-Galerkin method induced by the modified Euler method. We assume that the continuous problem admits a sufficiently smooth and sufficiently small solution. We show that the approximate problem has a solution and we give an error bound.

1. Introduction and Presentation of the problem

In the numerical simulation of the viscoelastic fluid flows, the hyperbolic character of the constitutive equation (when using differential models) has to be taken into account (see [8, 13]). This hyperbolic character implies that some upwinding is needed to avoid oscillations as in the method of characteristics [8, 2], the Lesaint-Raviart discontinuous finite element method [1, 5], the streamline-upwind method (SU) and the streamline-upwind-Petrov-Galerkin method (SUPG) [6, 13]. The numerical analysis of the steady case of the viscoelastics fluids flows is abundant. Although the list is not exhaustive, one may see for example [1, 6, 15]. Moreover the numerical analysis for transient viscoelastic flow remain quite few [11, 16]. For example, some difficulties appear, when we use continuous finite element approximation for (σ, u, p) and the standard SUPG method for the convection of the extra stress tensor. To give some response to this difficulties, we develop in this paper the study of continuous finite element(F.E) approximation of a transient viscoelastic fluid flow obeying an Oldroyd-B model. For the convective term of the constitutive equation we use some modified SUPG method linked to a variant of implicit Euler method (see [3]). Under this condition we are able to show that the approximate problem is stabilized and has a solution and we give an error bound.

The transient viscoelastic fluid flow obeying an Oldroyd-B type constitutive law is considered flowing in a bounded, connected open set Ω in IR^{2} with lipshitzian boundary Γ; n is the outward unit normal to Γ. The basic set of equations of the Oldroyd-B model with a single relaxation time is given by

2000 Mathematics Subject Classification. 65N30, 34K25, 74S05.

Key words and phrases. Stabilized method; finite elements; modified SUPG method;

transient viscoelastic flow.

\begin{matrix}c \\ ◯ \end{matrix}2004 Texas State University - San Marcos.

Published October 15, 2004.

Supported by CNRST - CNRS, Program STIC01/03, and by CNRST- GRICES Portugal.

λ(\frac{∂σ}{∂t} + (u.∇)σ + β(σ, ∇u)) + σ − 2αD(u) = 0 in Ω×]0, T [,

Re \frac{∂u}{∂t} − ∇.σ − 2(1 − α)∇.D(u) + ∇p = f in Ω×]0, T [,

∇.u = 0 in Ω×]0, T [,
u = 0 on Γ×]0, T [,
u = u_{0}, σ = σ_{0} in Ω t = 0.

(1.1)

Where λ > 0, Re and 0 < α < 1 are respectively the Weissenberg number, the Reynolds number and the viscosity ratio constant. f is a density of forces. D(u) = \frac{1}{2}(∇u + ∇u^{⊤}) the rate of strain tensor, and β(σ, ∇u) = −∇uσ − σ∇u^{⊤}.

Remark 1.1. The boundary condition u = 0 on Γ can be replaced by u = u_{d} on Γ. Regarding σ and the hyperbolic character of the constitutive equation, we have to impose σ = σ_{d} on Γ^{−} = {x ∈ Γ; u_{d}.n(x) < 0}.

Remark 1.2. The inertia term (u.∇)u is neglected in the momentum equation in order to make the analysis simpler.

Let us define the following spaces:

T = {τ = (τ_{ij})_{1≤i,j≤2} : τ_{ij} = τ_{ji}; τ_{ij} ∈ L^{2}(Ω); i, j = 1, 2}, X = (H_{0}^{1}(Ω))^{2}

Q = {q ∈ L^{2}(Ω)/_{∫Ω}q dx = 0}, V = {v ∈ X/(q, ∇.v) = 0; ∀q ∈ Q}.

The norm and scalar product in L^{2}(Ω) of functions, vectors and tensors are denoted respectively by |·| and (·, ·); (|·|_{Γ} and (·, ·)_{Γ} in L^{2}(Γ)); ⟨f, v⟩ will denote the duality between f ∈ (H^{−1}(Ω))^{2} and v ∈ X.

Remark 1.3. Existence results for problem (1.1) are proved in [9]. In order to make some theoretical analysis of approximate problem of (1.1) we use the regularity imposed in [9].

2. Description of the approximation scheme

FE approximation. We suppose Ω polygonal and we consider a triangulation ℑ_{h} on Ω made of triangles K such that \overline{Ω} = \{⋃ K; K ∈ ℑ_{h}\} uniformly regular, ∃ν_{0}, ν_{1} : ν_{0}h ≤ h_{K} ≤ ν_{1}ϱ_{K} where ϱ_{K} is the diameter of the greatest ball included in K and h_{max} = max_{K∈ℑ_{h}} h_{K}.

We use the Taylor-Hood finite element method for approximations in space of (u, p): P_{2}-continuous in velocity, P_{1}-continuous in pressure and we consider P_{1}- continuous approximation of the stresses. The corresponding F E space are:

X_{h} = {v ∈ X ∩ C^{0}(Ω)^{2} : v_{|K} ∈ P_{2}(K)^{2}, ∀K ∈ ℑ_{h}}
Q_{h} = {q ∈ Q ∩ C^{0}(Ω) : q_{|K} ∈ P_{1}(K), ∀K ∈ ℑ_{h}}
V_{h} = {v ∈ X_{h} : (q, ∇.v) = 0, ∀q ∈ Q_{h}}.
T_{h} = {τ ∈ T ∩ C^{0}(Ω) : τ|_{K} ∈ P_{1}(K), ∀K ∈ ℑ_{h}} ,

where P_{m}(K) denotes the space of polynomials of degrees less or equal to m on K ∈ T_{h}. The term ((u.∇)σ, τ) is approximated by means of an operator B on X_{h} × T_{h} × T_{h} defined by

B(u_{h}, σ_{h}; τ_{h}) = ((u_{h}(t).∇)σ_{h}(t), τ_{h}) + δ_{0}(h, t)((u_{h}(t)∇)σ_{h}(t), (u_{h}(t).∇)τ)
+ (1/2)((∇.u_{h})(t)σ, τ).

For the steady case you can see [15].

Numerical method. We propose Euler-SUPG modified scheme, implicit in time, based on the scheme proposed for the transport equation (see [3]). We construct an approximation of the solution at each time step nk, n = 0, …, N in the following way. We start with u_{h}^{0} = ũ_{0}: elliptic projection of u_{0} into V_{h}, σ_{h}^{0} = σ̃_{0}: orthogonal projection of σ_{0} into T_{h}. Given u_{h}^{0}, …, u_{h}^{n}; σ_{h}^{0}, …, σ_{h}^{n}, because (X_{h}, Q_{h}) satisfies the inf sup condition, we look for the solution of the following problem, find (u_{h}^{n+1}, σ_{h}^{n+1}) ∈ V_{h} × T_{h}, such that

\begin{matrix}λ(\frac{σ_{hu_{h}^{n},δ}^{n+1} − σ_{hu_{h}^{n},δ}^{n}}{k} , τ_{u_{h}^{n},λ}) + (σ_{h}^{n+1}, τ_{u_{h}^{n},λ}) + B(λu_{h}^{n}, σ_{h}^{n+1}; τ) \\ − 2α(D(u_{h}^{n+1}), τ_{u_{h}^{n},λ}) + λ(β(σ_{h}^{n+}, ∇u_{h}^{n+1}), τ_{u_{h}^{n},λ}) = 0 ∀τ ∈ T_{h}; \end{matrix}  (2.1)

\begin{matrix}Re(\frac{u_{h}^{n+1} − u_{h}^{n}}{k} , v) + (σ_{h}^{n+1}, D(v)) + 2(1 − α))(D(u_{h}^{n+1}), D(v)) \\ = ⟨f(t_{n+1}, x), v⟩ ∀v ∈ V_{h}, \end{matrix}  (2.2)

where σ_{hu_{h}^{n},δ}^{i} = σ_{h}^{i} + δ(h, k)(u_{h}^{n}.∇)σ_{h}^{i} (i = n, n + 1); τ_{u_{h}^{n},λ} = τ + λδ_{0}(h, k)(u_{h}^{n}.∇)τ for all τ ∈ T_{h}, and δ(,) (resp. δ_{0}(,)) will be specified later. In order to show that equation (2.1) − (2.2) defined uniquely (u_{h}^{n+1}, σ_{h}^{n+1}), we multiply equation (2.2) by 2α and add the equation obtained to equation (2.1), we get

λ(\frac{σ_{hu_{h}^{n},δ}^{n+1} − σ_{hu_{h}^{n},δ}^{n}}{k} , τ_{u_{h}^{n},λ}) + 2αRe(\frac{u_{h}^{n+1} − u_{h}^{n}}{k} , v_{h}) + B(λu_{h}^{n}, σ_{h}^{n+1}; τ_{h})

+ A(u_{h}^{n}; (σ_{h}^{n+1}, u_{h}^{n+1}), (τ, v)) + λ(β(σ_{h}^{n+1}, ∇u_{h}^{n+1}), τ_{u_{h}^{n},λ})

= 2α⟨f(t_{n+1}), v_{h}⟩ , ∀(τ_{h}, v_{h}) ∈ T_{h} × V_{h}.

where A(., .) is a bilinear form on T_{h} × V_{h} defined as

A(w; (σ, u), (τ, v)) = (σ, τ_{w,λ})+2α(σ, D(v))−2α(D(u), τ_{w,λ})+4α(1−α)(D(u), D(v))

From this, we can establish some error bound and then the following existence result

Theorem 2.1. There exists M_{0}, and h_{0} such that if problem (1.1) admits a solution (σ, u, p) with,

M = max {∥σ∥_{C^{1}([t_{n},t_{n+1}];H^{2})}, ∥u∥_{C^{1}([t_{n},t_{n+1}];H^{3})}, ∥p∥_{C^{0}([t_{n},t_{n+1}];H^{2})},
∥σ∥_{C^{2}([t_{n},t_{n+1}];L^{2})}, ∥u∥_{C^{2}([t_{n},t_{n+1}],T;L^{2})}} < M_{0},

then if δ(,) satisfies δ(,) = λδ_{0}(,) and h ≤ h_{0} there exists a unique solution in T_{h} × V_{h} of problem (2.1)- (2.2).

Proof. For this purpose we define a mapping Φ : T_{h} × V_{h} ⟼ T_{h} × V_{h}, which to (σ_{1}, u_{1}) associates (σ_{2}, u_{2}) = Φ(σ_{1}, u_{1}), where (σ_{2}, u_{2}) ∈ T_{h} × V_{h} satisfies

λ(\frac{σ_{2u_{h}^{n},δ} − σ_{hu_{h}^{n},δ}^{n}}{k} , τ_{u_{h}^{n},λ}) + 2αRe(\frac{u_{2} − u_{h}^{n}}{k} , v_{h}) + B(λu_{h}^{n}, σ_{h}^{n+1}; τ_{h})

+ A(u_{h}^{n}; (σ_{2}, u_{2}), (τ, v))
= −λ(β(σ_{1}, ∇u_{1}), τ_{u_{h}^{n},λ}) + 2α⟨f(t_{n+1}), v_{h}⟩ , ∀(τ_{h}, v_{h}) ∈ T_{h} × V_{h}.

We define a ball B_{h}^{(n+1)} as follows: let C^{∗} be given. Then we define

B_{h}^{(n+1)} = \{(τ_{h}, v_{h}) ∈ T_{h} × V_{h} : [\frac{αRe}{k}∥v_{h} − u(t_{n+1})∥_{0,Ω}^{2}

+ \frac{λ}{2k} ∥(τ_{h} − σ(t_{n+1}))_{u_{h}^{n},λ}∥_{0,Ω}^{2}]^{1/2} ≤ C^{∗}(k + δ + h\sqrt{δ_{0}} + \frac{h^{2}}{\sqrt{δ_{0}}} + h^{\frac{3}{2}} ),

[\frac{1}{4}\{4α(1 − α)∥D(v_{h} − u(t_{n+1})∥_{0,Ω}^{2} + ∥τ_{h} − σ(t_{n+1})∥_{0,Ω}^{2}]^{1/2}

≤ C^{∗}(k + δ + h\sqrt{δ_{0}} + \frac{h^{2}}{\sqrt{δ_{0}}} + h^{\frac{3}{2}} )\}.

The proof is decomposed in five parts:

(a) Φ is well defined for h ≤ h_{1} = h_{1}(h_{max}, α) and bounded on bounded sets,

(b) let C_{0} be a positive constant independent of h, k and λ, α, Re. If h ≤ h_{0} = min \{h_{1}, \sqrt{k}(\frac{C^{∗}}{C_{0}M\sqrt{αRe}})^{2/3}, \frac{1}{C_{0}M} \sqrt{k/λ}\} and \sqrt{δ/k} ≤ \frac{1}{C_{0}M}, we have B_{h}^{(n+1)} is non-

empty. On the other hand, if M_{0} = C^{∗} \frac{γ}{λ+γ}, γ = \sqrt{α(1 − α)}, we can prove that

Φ(\overbrace{B}^{∘}{(n+1) \atop h}) ⊂ \overbrace{B}^{∘}{(n+1) \atop h},

(c) Φ is continuous on T_{h} × V_{h}. Φ(B_{h}^{(n+1)}) ⊂ B_{h}^{(n+1)}, Brouwer’s theorem then gives the existence of fixed point (σ_{h}^{n+1}, u_{h}^{n+1}) of Φ solution of problem (1) − (2).

(d) Furthermore, if λM and λMγ^{−1} is sufficiently small, Φ is a contraction mapping on B_{h}^{(n+1)}.

Then a result of existence and uniqueness follows from the fixed theorem. □

Remark 2.2. When we use only the classical Euler-scheme in time and SUPG method, we can’t point out result of (a) in the above proof.

3. Main result and error estimates

Suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution, we can show, the following result.

Theorem 3.1. There exists M_{0} and h_{0} such that if problem (1.1) admits a solution (σ, u, p) with σ ∈ C^{1}([0, T], (H^{2})^{4}) ∩ C^{2}([0, T], (L^{2})^{4}); u ∈ C^{1}([0, T], (H^{3})^{2}) ∩ C^{2}([0, T], (L^{2})^{2}); p ∈ L^{2}([0, T], (H^{2}) ∩ L_{0}^{2}) ∩ C^{0}([0, T], H^{2}), satisfying

max{∥σ∥_{C^{1}(0,T;H^{2})}, ∥u∥_{C^{1}(0,T;H^{3})}, ∥p∥_{C^{0}(0,T;H^{2})}, ∥σ∥_{C^{2}(0,T;L^{2})}, ∥u∥_{C^{2}(0,T;L^{2})}} ≤ M_{0}

then for kh^{−(1+ε)}(0 < ε ≤ 1/2) bounded, there exists a constant C independent of h and k such that

max_{0≤n≤N} |(σ_{h}^{n} −σ(t_{n}))_{u_{h}^{n−1}}|+(∑_{n=0}^{N} k|σ_{h}^{n} −σ(t_{n})|)^{2})^{1/2} ≤ C(k +δ +h\sqrt{δ_{0}} + \frac{h^{2}}{\sqrt{δ_{0}}} +h^{1+ε})

and

max_{0≤n≤N} |u_{h}^{n} − u(t_{n})| + (∑_{n=0}^{N} k|D(u_{h}^{n} − u(t_{n}))|^{2})^{1/2} ≤ C(k + δ + h\sqrt{δ_{0}} + \frac{h^{2}}{\sqrt{δ_{0}}} + h^{1+ε})

where N ∈ N^{∗} : Nk = T, u_{h}^{−1} = u_{h}^{0} and (σ_{h}^{i}, u_{h}^{i})_{1≤i≤N} are solutions of ((2.1) − (2.2))_{1≤i≤N}.

Remark 3.2. The discontinuous stresses approach case with Euler semi-implicit method in time was treated by Baranger and all. [16]. In the proof of the present result we can fund some similar technical like in [16].

Proof of Theorem 3.1. For 0 ≤ n < N, let u(t_{n}), σ(t_{n}), p(t_{n}) be respectively in H^{3}(Ω), ^{2}(Ω), H^{2}(Ω) and so, there exists (ũ(t_{n}), p̃(t_{n})) ∈ V_{h} × Q_{h} such that,

|(u − ũ)(t_{n})| + h∥(u − ũ)(t_{n})∥_{1,2} ≤ C_{1}h^{3}∥u(t_{n})∥_{3,2},  (3.1)
|(p − p̃)(t_{n})| ≤ C_{2}h^{2}∥p(t_{n})∥_{2,2},  (3.2)

(see [10]) and there exists σ̃(t_{n}) ∈ T_{h} such that

|(σ − σ̃)(t_{n})| ≤ C_{3}h^{2}∥σ(t_{n})∥_{2,2}  (3.3)

(see [10]). We remark that we can define ũ(.) by the elliptic projection of u(.) on V_{h} such that a((u − ũ)(.), v_{h}) = 0, ∀v_{h} ∈ V_{h}, where a(u, v) = (d(u), d(v)); then the following properties are also satisfied

dũ/dt = (du/dt)^{∼}

and

\begin{aligned}|(du/dt)(s) − (du/dt)^{∼}(s)| & ≤ C_{4}h∥(du/dt)(s) − (du/dt)^{∼}(s)∥_{1,2} \\ & ≤ C_{5}h^{3}∥(du/dt)(s)∥_{3,2} \end{aligned}  (3.4)

(see([14])), u being in C^{1}([0, T], H^{3}); same properties are satisfied for σ:

dσ̃/dt = (dσ/dt)^{∼}

and

|\frac{dσ}{dt} (s) − ˜ \frac{dσ}{dt} (s)| ≤ C_{6}h∥\frac{dσ}{dt} (s) − ˜ \frac{dσ}{dt} (s)∥_{1,2} ≤ C_{7}h^{2}∥\frac{dσ}{dt} (s)∥_{2,2},  (3.5)

(see([14])). □

In the sequel we shall use the following inverse inequalities (see [4]).

Lemma 3.3. Let k ≥ 0 be an integer and W_{h} = {v, v_{|_{K}} ∈ P_{k}(K) ∀K ∈ ℑ_{h}}. Let r and p be reals with 1 ≤ r, p ≤ ∞ and let l ≥ 0 and m ≥ 0 be integers such that l ≤ m. Then there exists a constant C = C(ν_{0}, ν_{1}, l, r, m, p, k) such that ∀v ∈ W_{h} ∩ W^{l,r}(Ω) ∩ W^{m,p}(Ω), |v|_{m,p} ≤ Ch^{l−m−2 max{0,1/r−1/p}}|v|_{l,r}.

We shall also use the following Sobolev’s imbedding theorems.

Lemma 3.4. Let m ≥ 0 be an integer. The following embedding hold algebraically and topologically:

W^{m+1,2}(Ω) ⊂ W^{m,q}(Ω) ∀q ∈ [1, ∞[, and W^{m,p}(Ω) ⊂ C^{0}(Ω̄).

Now, let us denote e_{h}^{n} = u_{h}^{n} − ũ(t_{n}), ε_{h}^{n} = σ_{h}^{n} − σ̃(t_{n}). From equations (2.1)-(2.2) we have for (τ, v) ∈ T_{h} × V_{h}

\begin{matrix}2αRe(\frac{e_{h}^{n+1} − e_{h}^{n})}{k}, v) + λ(\frac{ε_{hu_{h}^{n},δ}^{n+1} − ε_{hu_{h}^{n},δ}^{n}}{k} , τ_{u_{h}^{n},λ}) + B(λu_{h}^{n}, ε_{h}^{n+1}; τ) \\ + A(u_{h}^{n}; (ε_{h}^{n+1}, e_{h}^{n+1}), (τ, v)) \\ = −λ(\frac{σ̃_{hu_{h}^{n},δ}(t_{n+1}) − σ̃_{hu_{h}^{n},δ}(t_{n})}{k}, τ_{u_{h}^{n},λ}) − 2αe(\frac{ũ(t_{n+1}) − ũ(t_{n})}{k}, v) \\ − B(λu_{h}^{n}, σ̃(t_{n+1}); τ) + A(u_{h}^{n}; (−σ̃(t_{n+1}), −ũ(t_{n+1})), (τ, v)) \\ − λ(β(σ_{h}^{n}, ∇u_{h}^{n}), τ_{u_{h}^{n},λ}) + 2α⟨f(t_{n+1}, x), v⟩, ∀(τ, v) ∈ T_{h} × V_{h}. \end{matrix}  (3.6)

But (σ, u, p) being the exact solution of problem (1.1), it satisfies the following consistency equation

2αRe(\frac{du}{dt}(t), v) + λ(\frac{dσ}{dt} (t), τ_{u_{h}^{n},λ}) + B(λu(t), λu_{h}^{n}, σ(t); τ)

+ A(u_{h}^{n}; (σ(t), u(t)), (τ, v))

= 2α(p(t), ∇.v) + 2α⟨f(t), v⟩ − λ(β(σ(t), ∇u(t)), τ_{u_{h}^{n},λ}) ∀(τ, v) ∈ T_{h} × V_{h}

Inserting the value of ⟨f(t_{n+1}), .⟩ in equation (3.6) we obtain

Re(\frac{e_{h}^{n+1} − e_{h}^{n})}{k}, v) + λ(\frac{ε_{hu_{h}^{n},δ}^{n+1} − ε_{hu_{h}^{n},δ}^{n}}{k} , τ_{u_{h}^{n},λ}) + B(λu_{h}^{n}, ε_{h}^{n+1}; τ)

+ A(u_{h}^{n}; (ε_{h}^{n+1}, e_{h}^{n+1}), (τ, v))

= λ(g_{a}(σ(t_{n+1}), ∇u(t_{n+1})) − g_{a}(σ_{h}^{n}, ∇u_{h}^{n}), τ)

+ 2αRe(\frac{du}{dt}(t_{n+1}) − \frac{ũ(t_{n+1}) − ũ(t_{n})}{k}, v)

+ λ(\frac{dσ}{dt} (t_{n+1}) − \frac{σ̃_{u_{h}^{n}}(t_{n+1}) − σ̃_{u_{h}^{n}}(t_{n})}{k}, τ_{u_{h}^{n},λ})

+ A(u_{h}^{n}, ((σ − σ̃)(t_{n+1}), (u − ũ)(t_{n+1})), (τ, v))

+ λ[B(λu(t_{n+1}), λu_{h}^{n}, σ(t_{n+1}); τ) − B(λu_{h}^{n}, σ̃(t_{n+1}); τ)] + 2α(p(t_{n+1}), ∇.v).

(3.7)

Taking v = e_{h}^{n+1} and τ = ε_{h}^{n+1} in equation (3.7) and using the identity (a − b, a) = \frac{1}{2}(|a|^{2} − |b|^{2} + |a − b|^{2}), and coercivity, we obtain

\begin{matrix}\frac{αRe}{k}\{|e_{h}^{n+1}|^{2} − |e_{h}^{n}|^{2} + |e_{h}^{n+1} − e_{h}^{n}|^{2}\} + |ε_{h}^{n+1}|^{2} \\ + \frac{λ}{2k} \{|ε_{hu_{h}^{n},λ}^{n+1}|^{2} − |ε_{hu_{h}^{n},λ}^{n}|^{2} + |ε_{hu_{h}^{n},λ}^{n+1} − ε_{hu_{h}^{n},λ}^{n}|^{2}\} \\ + 2α(1 − α)|D(e_{h}^{n+1})|^{2} + (1/2)|ε_{h}^{n+1}|^{2} + (δ_{0}/4)|λu_{h}^{n}.∇ε_{h}^{n+1}|^{2} \end{matrix}  (3.8)

which is less than ro equal to the right-hand side of (3.7). To bound each term of the second member of inequality (3.8), let us define for C_{0} > 0, the ball B_{h,k}^{m} for 0 ≤ m ≤ N, by

B_{h,k}^{m}

= {(τ_{i}, v_{i})_{i=0,..,m} ∈ (T_{h} × V_{h})^{m+1} : max_{0≤i≤m}{|(τ_{i} − σ(t_{i}))_{v_{i−1}}|^{2} + |v_{i} − u(t_{i})|^{2}}^{1/2}

≤ C_{0}(k + δ + h\sqrt{δ_{0}} + \frac{h^{2}}{\sqrt{δ_{0}}} + h^{1+ε}) and

[∑_{n=0}^{m} k\{|τ_{i} − σ(t_{i})|^{2} + |D(v_{i} − u(t_{i}))|^{2}\}]^{1/2} ≤ C_{0}(k + δ + h\sqrt{δ_{0}} + \frac{h^{2}}{\sqrt{δ_{0}}} + h^{1+ε})\}.

Our aim is to prove that we can choose M_{0}, h_{0}, C_{0} such that for M ≤ M_{0}, h ≤ h_{0} if (σ_{h}^{n}, u_{h}^{n})_{0≤n≤m−1} ∈ B_{h,k}^{m−1} for a C_{0} = C_{0}(M_{0}, h_{0}, C_{i}) then (σ_{h}^{n}, u_{h}^{n})_{0≤n≤m} ∈ B_{h,k}^{m} for the same C_{0}, thus for all m such mk ≤ T. Firstly, by equations (3.1) to (3.5) we have

|(σ_{h}^{0} − σ(0))_{u_{h}^{0}}| + |u_{h}^{0} − u(0)| ≤ Mh^{2}{C_{3}(1 + λMδ_{0}h^{−1}) + C_{1}}

and

[k\{|σ_{h}^{0} − σ(0)|^{2} + |D(u_{h}^{0} − u(0)|^{2}\}]^{1/2} ≤ \sqrt{2k}(C_{1} + C_{3})Mh^{2}.

To ensure that (σ_{h}^{0}, u_{h}^{0}) = (σ̃_{0}, ũ_{0}) ∈ B_{h,k}^{0}, it suffices to impose, for h < h_{1},

Mh^{1/2}{C_{3}(1 + λMδ_{0}h^{−1}) + C_{1}} ≤ C_{0}

and for h ≤ h_{2},

(C_{1} + C_{3})\sqrt{2C̄}Mh^{\frac{3−ε}{2}} ≤ C_{0}.

So, if we take h_{0} ≤ min{h_{1}, h_{2}} we have for h ≤ h_{0}: B_{h}^{0} ≠ ∅. Now, let us suppose that (σ_{h}^{n}, u_{h}^{n})_{0≤n≤m−1} ∈ B_{h,k}^{(m−1)}. We multiply inequality (3.8) by k and sum it for n = 0 to n = m − 1,

αRe ∑_{n=0}^{m−1}[|e_{h}^{n+1}|^{2} − |e_{h}^{n}|^{2} + |e_{h}^{n+1} − e_{h}^{n}|^{2}] + (1/2) ∑_{n=0}^{m−1} k|ε_{h}^{n+1}|^{2}

+ \frac{λ}{2} ∑_{n=0}^{m−1}[|ε_{hu_{h}^{n},λ}^{n+1}|^{2} − |ε_{hu_{h}^{n},λ}^{n}| + |ε_{hu_{h}^{n},λ}^{n+1} − ε_{hu_{h}^{n},λ}^{n}|^{2}] + 2α(1 − α) ∑_{n=0}^{m−1} k|D(e_{h}^{n+1})|^{2}

+ (δ_{0}/4) ∑_{n=0}^{m−1} k|λu_{h}^{n}.∇ε_{h}^{n+1}|^{2}

≤ λ ∑_{n=0}^{m−1} k(g_{a}(σ(t_{n+1}), ∇u(t_{n+1})) − g_{a}(σ_{h}^{n+1}, ∇u_{h}^{n+1}), ε_{hu_{h}^{n},λ}^{n+1})  (3.9)

+ λ ∑_{n=0}^{m−1} k(\frac{dσ}{dt} (t_{n+1}) − \frac{σ̃_{u_{h}^{n},δ}(t_{n+1}) − σ̃_{u_{h}^{n},δ}(t_{n})}{k}, ε_{hu_{h}^{n},λ}^{n+1})  (3.10)

+ 2αRe ∑_{n=0}^{m−1} k(\frac{du}{dt}(t_{n+1}) − \frac{ũ(t_{n+1}) − ũ(t_{n})}{k}, e_{h}^{n+1})  (3.11)

+ ∑_{n=0}^{m−1} kA(u_{h}^{n}; (σ − σ̃)(t_{n+1}), (u − ũ)(t_{n+1}), (ε_{h}^{n+1}, e_{h}^{n+1}))  (3.12)

+ λ ∑_{n=0}^{m−1} k{B(λu(t_{n+1}), λu_{h}^{n}, σ(t_{n+1}); ε_{h}^{n+1}) − B(λu_{h}^{n}, σ̃(t_{n+1}); ε_{h}^{n+1})} (3.13)

+ 2α ∑_{n=0}^{m−1} k(p(t_{n+1}), ∇.e_{h}^{n+1}).  (3.14)

We use for the estimate of terms (3.11)–(3.14), the results (3.1)–(3.5). However, for the estimation of the term (3.9) we choose the fixed point (σ_{h}^{m}, u_{h}^{m}) of the mapping Φ defined in section 2. This choice is possible because we have, B_{h}^{(0)} ∩ B_{h,Δt}^{0} ≠ ∅ and by construction:

(σ_{h}^{l}, u_{h}^{l})_{l=0;..;m−1} ∈ B_{h,Δt}^{m−1} ⇒ (σ_{h}^{m−1}, u_{h}^{m−1}) ∈ B_{h}^{(m−1)}.

On the other hand for estimate of the term (3.10) we prepare the following lemma:

Lemma 3.5. Let (3.1)–(3.5) hold. Then

|(\frac{dσ}{dt} (t_{n+1}) − \frac{σ̃_{u_{h}^{n},δ}(t_{n+1}) − σ̃_{u_{h}^{n},δ}(t_{n})}{k}, ε_{hu_{h}^{n},λ}^{n+1})|

≤ λM[C_{7}h^{2} + k + δMC_{10}(M + hM + C_{1}Mh^{\frac{3}{2}} )]( ∑_{n=0}^{m−1} k)^{1/2} max_{0≤n≤m−1} |ε_{hu_{h}^{n},λ}^{n+1})|

+ {δM[C_{7}C_{8}kh( ∑_{n=0}^{m−1} k|∇(u_{h}^{n} − u(t_{n}))|^{2})^{1/2}

+ (C_{7}λ^{−1}h^{2} + \frac{C7}{C_{6}}C_{9}δ_{0}(M + max_{0≤n≤m−1} |u(t_{n}) − u_{h}^{n}|))]( ∑_{n=0}^{m−1} k)^{1/2}

+ λ(1 + \sqrt{δ_{0}})δh^{−1}( ∑_{n=0}^{m−1} k|∇(u_{h}^{n} − u(t_{n}))|^{2})^{1/2}\}

× ( ∑_{n=0}^{m−1} k{|ε_{hu_{h}^{n},λ}^{n+1}|^{2} + δ_{0}|λ(u_{h}^{n}.∇)ε_{h}^{n+1}|^{2}})^{1/2}.

Proof. To proof the inequality in this lemma, we write each term as follows:

\frac{dσ}{dt} (t_{n+1}) − \frac{σ̃_{u_{h}^{n},δ}(t_{n+1}) − σ̃_{u_{h}^{n},δ}(t_{n})}{k}

= (\frac{dσ}{dt} (t_{n+1}) − \frac{σ_{u_{h}^{n},δ}(t_{n+1}) − σ_{u_{h}^{n},δ}(t_{n})}{k})

+ (\frac{σ_{u_{h}^{n},δ}(t_{n+1}) − σ_{u_{h}^{n},δ}(t_{n})}{k} − \frac{σ̃_{u_{h}^{n},δ}(t_{n+1}) − σ̃_{u_{h}^{n},δ}(t_{n})}{k})

we can write the second term in the form,

\frac{1}{k} (∫_{t_{n}}^{t_{n+1}}(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s) ds, ε_{hu_{h}^{n},λ}^{n+1}) + \frac{δ}{k} (∫_{t_{n}}^{t_{n+1}} u_{h}^{n}.∇(\frac{dσ}{dt} − ( ˜\frac{dσ}{dt}))(s) ds, ε_{hu_{h}^{n},λ}^{n+1})

so,

|k(\frac{σ_{u_{h}^{n},δ}(t_{n+1}) − σ_{u_{h}^{n},δ}(t_{n+1})}{k} − \frac{σ̃_{u_{h}^{n},δ}(t_{n+1}) − σ̃_{u_{h}^{n},δ}(t_{n+1})}{k}, ε_{hu_{h}^{n},λ}^{n+1})|

≤ |(∫_{t_{n}}^{t_{n+1}}(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s) ds, ε_{hu_{h}^{n},λ}^{n+1})| + δ|(∫_{t_{n}}^{t_{n+1}} u_{h}^{n}.∇(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s) ds, ε_{hu_{h}^{n},λ}^{n+1})|

we estimate the first term in this inequality using (3.5):

|(∫_{t_{n}}^{t_{n+1}}(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s) ds, ε_{hu_{h}^{n},λ}^{n+1})| ≤ k max_{t_{n}≤s≤t_{n+1}} |(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s)||ε_{hu_{h}^{n},λ}^{n+1}|
≤ C_{7}Mkh^{2}|ε_{hu_{h}^{n},λ}^{n+1}|.

To study the second term, we write

(∫_{t_{n}}^{t_{n+1}} u_{h}^{n}.∇(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s) ds, ε_{hu_{h}^{n},λ}^{n+1})

= −(∫_{t_{n}}^{t_{n+1}}(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s) ds, (u_{h}^{n}.∇)ε_{h}^{n+1})

+ ((∇.u_{h}^{n}) ∫_{t_{n}}^{t_{n+1}}(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s) ds, ε_{h}^{n+1})

+ δ_{0}(∫_{t_{n}}^{t_{n+1}} u_{h}^{n}.∇(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s) ds, λ(u_{h}^{n}.∇)ε_{hu_{h}^{n},λ}^{n+1})

and

δ|(∫_{t_{n}}^{t_{n+1}} u_{h}^{n}.∇(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s) ds, ε_{hu_{h}^{n},λ}^{n+1})|

≤ δλ^{−1}k max_{t_{n}≤s≤t_{n+1}} |(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s)||λ(u_{h}^{n}.∇)ε_{h}^{n+1}|

+ δk max_{t_{n}≤s≤t_{n+1}} |(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s)||∇(u_{h}^{n} − u(t_{n}))||ε_{h}^{n+1}|_{0,∞}

+ kδ_{0}δ|u_{h}^{n}|_{0,∞} max_{t_{n}≤s≤t_{n+1}} |(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s)|_{1,2}|λ(u_{h}^{n}.∇)ε_{h}^{n+1}|

using (3.5) and the result of Lemma 3.3, we obtain the following estimate

δ|(∫_{t_{n}}^{t_{n+1}} u_{h}^{n}.∇(\frac{dσ}{dt} − ˜ (\frac{dσ}{dt}))(s) ds, ε_{hu_{h}^{n},λ}^{n+1})|

≤ δC_{8}C_{7}Mδkh|∇(u_{h}^{n} − u(t_{n}))||ε_{h}^{n+1}|

+ \{δλ^{−1}C_{7}Mh^{2} + C_{9} \frac{C7}{C_{6}}Mhδ_{0}δh^{−1}|u_{h}^{n}|\}k|λ(u_{h}^{n}.∇)ε_{h}^{n+1}|

≤ C_{7}C_{8}Mδkh|∇(u_{h}^{n} − u(t_{n}))|ε_{h}^{n+1}|

+ \{C_{7}δλ^{−1}M \frac{h^{2}}{\sqrt{δ_{0}}} + C_{9} \frac{C7}{C_{6}}Mδ\sqrt{δ_{0}}(M + max_{0≤n≤m−1} |u_{h}^{n} − u(t_{n})|)\}

× \sqrt{δ_{0}}|λ(u_{h}^{n}.∇)ε_{h}^{n+1}|

and finally

λ| ∑_{n=0}^{m−1} k(\frac{σ_{u_{h}^{n},δ}(t_{n+1}) − σ_{u_{h}^{n},δ}(t_{n+1})}{k} − \frac{σ_{u_{h}^{n},δ}^{∼}(t_{n+1}) − σ_{u_{h}^{n},δ}^{∼}(t_{n+1})}{k}, ε_{hu_{h}^{n},λ}^{n+1})|

≤ λC_{7}Mh^{2}( ∑_{n=0}^{m−1} k) max_{0≤n≤m−1} |ε_{hu_{h}^{n},λ}^{n+1}|

+ λC_{7}C_{8}Mδkh( ∑_{n=0}^{m−1} k|∇(u_{h}^{n} − u(t_{n}))|^{2})^{1/2}( ∑_{n=0}^{m−1} k|ε_{h}^{n+1}|^{2})^{1/2}  (3.15)

+ δM\{C_{7}λ^{−1} \frac{h^{2}}{\sqrt{δ_{0}}} + \frac{C7}{C_{6}}C_{9}\sqrt{δ_{0}}(M + max_{0≤n≤m−1} |u_{h}^{n} − u(t_{n})|))\}

× ( ∑_{n=0}^{m−1} k)^{1/2}( ∑_{n=0}^{m−1} kδ_{0}|λu_{h}^{n}.∇ε_{h}^{n+1}|^{2})^{1/2}.

□

Now, we return to the first part of term (3.10).

|k(\frac{dσ}{dt} (t_{n+1}) − \frac{σ_{u_{h}^{n},δ}(t_{n+1}) − σ_{u_{h}^{n},δ}(t_{n})}{k}, ε_{hu_{h}^{n},λ}^{n+1})|

≤ |k(\frac{dσ}{dt} (t_{n+1}) − \frac{σ(t_{n+1}) − σ(t_{n})}{k}, ε_{hu_{h}^{n},λ}^{n+1})| + δ|(u_{h}^{n}.∇(σ(t_{n+1}) − σ(t_{n})), ε_{hu_{h}^{n},λ}^{n+1})|

≤ |(∫_{t_{n}}^{t_{n+1}}(s − t)\frac{d^{2}σ}{dt^{2}} (s) ds, ε_{hu_{h}^{n},λ}^{n+1})| + δ|(∫_{t_{n}}^{t_{n+1}} u_{h}^{n}.∇\frac{dσ}{dt} (s) ds, ε_{hu_{h}^{n},λ}^{n+1})|

≤ k^{2} max_{t_{n}≤s≤t_{n+1}} |\frac{d^{2}σ}{dt^{2}} (s)||ε_{hu_{h}^{n},λ}^{n+1}| + δk|u_{h}^{n}|_{0,∞} max_{t_{n}≤s≤t_{n+1}} |\frac{dσ}{dt} (s)|_{1,2}|ε_{hu_{h}^{n},λ}^{n+1}| .

Using the regularity of σ, we have

|k(\frac{dσ}{dt} (t_{n+1}) − \frac{σ_{u_{h}^{n},δ}(t_{n+1}) − σ_{u_{h}^{n},δ}(t_{n})}{k}, ε_{hu_{h}^{n},λ}^{n+1})| ≤ M(k^{2} + kδ|u_{h}^{n}|_{0,∞})|ε_{hu_{h}^{n},λ}^{n+1}|.

On the other hand, by the imbedding result W^{1,4} ⊂ L^{∞} (see Lemma 3.3) and the inverse inequality result | · |_{0,∞} ≤ C_{10}h^{\frac{−1}{2}} | · |_{0,4} (see Lemma 3.4) we can prove,

|u_{h}^{n}|_{0,∞} ≤ C_{10}(M + h∥u∥_{3,2} + C_{1}Mh^{\frac{3}{2}} + h^{− \frac{1}{2}} |∇(u(t_{n}) − u_{h}^{n})|).

So we have,

λ| ∑_{n=0}^{m−1} k(\frac{dσ}{dt} (t_{n+1}) − \frac{σ_{u_{h}^{n},δ}(t_{n+1}) − σ_{u_{h}^{n},δ}(t_{n})}{k}, ε_{hu_{h}^{n},λ}^{n+1})|

≤ λM[k + δMC_{10}(M + hM + C_{1}Mh^{\frac{3}{2}} )]( ∑_{n=0}^{m−1} k) × max_{0≤n≤m−1} |ε_{hu_{h}^{n},λ}^{n+1}|

+ λ(1 + \sqrt{δ_{0}})δh^{− \frac{1}{2}} ( ∑_{n=0}^{m−1} k|∇(u(t_{n}) − u_{h}^{n})|^{2})^{1/2}

× [( ∑_{n=0}^{m−1} k|ε_{h}^{n+1}|^{2})^{1/2} + ( ∑_{n=0}^{m−1} kδ_{0}|λu_{h}^{n}.∇ε_{h}^{n+1}|^{2})^{1/2}] .

Then Lemma 3.5 follows from the above inequality and (3.15).

Conclusion. We conclude this analysis with some comments. The proof given here can be extended to the more realistic rheological PTT model, to a quadrilateral FE approximation, following [1] and to the higher finite element methods(P_{k}, k ≥ 1). With a judicious coefficient choice of stabilization δ and δ_{0}, we find the error bound given respectively by Baranger and al. [16] and Ervin and al. [7].

The use of a decoupled fractional step scheme would be computationally cheaper, following [6]. Numerical analysis of such method is currently in progress.

References

[1] A. Bahar, J. Baranger, D. Sandri , Galerkin discontinuous approximation of the transport equation and viscoelastic fluid flow on Quadrilateral, Numer. Meth. Partial Diff. Equations, 14(1)(1998), 97-114.

[2] F. G. Basambrio , Flows in viscoelastic fluids treated by the method of characteristics, JNNFM 39 (1991), 17-34.

[3] M. Bensaada, D. Esselaoui, Stabilization method for continuous approximation of transient convection problem, In press: Numer. Methods for Partial Differential Equations, (2004).

[4] P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, (1978).

[5] D. Esselaoui, K. Najib, A. Ramadane, A. Zine, Analyse numérique et approche découplée d’un problème d’écoulements de fluides non newtoniens de type Phan-Thien-Tanner:contraintes discontinues, C.R. Acad. Sci. Paris série I (2000), 931-936.

[6] D. Esselaoui, A. Ramadane, A. Zine, Decoupled approch for the problem of viscoelastic fluid flow of PTT model I: continuous stresses, Compt. Methods Appl. Mech. Engrg. (2000), 543-560.

[7] V. J. Ervin and W. W. Miles, Approximation of time-dependent viscoelastic fluid flow: SUPG Approximation, SIAM J. Numer. Anal. 41 N0. 2 (2003) 457-486.

[8] M. Fortin, D. Esselaoui, A finite element procedure for viscoelastic flows. Int. J. Num. Meth. Fluids, 7 (1987), 1035-1052.

[9] C. Guillopé, J.C. Saut, Résultats d’existence pour des fluides viscoélastiques à lois de comportements de type différentiel, Nonlinear Analysis Theory, methods and Applications, 15 (1990), 849-869.

[10] V. Girault, P.A. Raviart, Finite element method for Navier-Stokes equation theory and algorithms, Springer, Berlin, (1986).

[11] R. A. Keiller, Numerical instability of time dependant flows. JNNFM 43 (1992), 229-246.

[12] V. Legat, M.J. Crochet, The consistent streamline upwind/Petrov-Galerkin method for viscoelastic flow recisited. In Problèmes non-linéaires appliqués. CEA. INRIA. EDF, (1992).

[13] M.J. Marchal, M.J. Crochet, A new finite element for calculating viscoelastic flows, J. N?. N?. F?. M?, 26 (1987), 77-114.

[14] P.A. Raviart, J.M. Thomas, Introduction à l’analyse numérique des E.D.P., MASSON, Paris, (1983).

[15] D. Sandri, Finite approximation of viscoelastic fluid flow: existence of approximate solutions and error bounds, continuous approximation of the stress. SIAM J. Numer. Anal., 31 (1994), 362-377.

[16] S. Wardi, J. Baranger, Numerical analysis of a FEM for a transient viscoelastic flow. Comput. Methods Appl. Mech. Engr. 125 (1995), 171-185.

Mohammed Bensaada Laboratoire des Sciences de l’Ingénieur, Analyse Numérique et Optimisation (SIANO), Faculté des Sciences, Université Ibn Tofail, B.P.133, 14000-Kénitra, Maroc

Driss Esselaoui Laboratoire des Sciences de l’Ingénieur, Analyse Numérique et Optimisation (SIANO), Faculté des Sciences, Université Ibn Tofail, B.P.133, 14000-Kénitra, Maroc

E-mail address: desselaoui@yahoo.fr   Fax:  212 37 37 27 70