a priori error estimates for semidiscrete finite element approximations

c 2011 Institute for Scientific

Computing and Information

INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 8, Number 2, Pages 324–352

A PRIORI ERROR ESTIMATES FOR SEMIDISCRETE FINITE ELEMENT APPROXIMATIONS TO EQUATIONS OF MOTION ARISING IN OLDROYD FLUIDS OF ORDER ONE DEEPJYOTI GOSWAMI AND AMIYA K. PANI Abstract. In this paper, a semidiscrete finite element Galerkin method for the equations of motion arising in the 2D Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in L∞ in time, is analyzed. A step-by-step proof of the estimate in the Dirichlet norm for the velocity term which is uniform in time is derived for the nonsmooth initial data. Further, new regularity results are obtained which reflect the behavior of solutions as t → 0 and t → ∞. Optimal L∞ (L2 ) error estimates for the velocity which is of order O(t−1/2 h2 ) and for the pressure term which is of order O(t−1/2 h) are proved for the spatial discretization using conforming elements, when the initial data is divergence free and in H01 . Moreover, compared to the results available in the literature even for the Navier-Stokes equations, the singular behavior of the pressure estimate as t → 0, is improved by an order 1/2, from t−1 to t−1/2 , when conforming elements are used. Finally, under the uniqueness condition, error estimates are shown to be uniform in time. Key Words. Viscoelastic fluids, Oldroyd fluid of order one, uniform a priori bound in Dirichlet norm, uniform in time and optimal error estimates, nonsmooth initial data.

1. Introduction In this paper, we consider semi-discrete Galerkin approximations to the following system of equations of motion arising in the Oldroyd fluids (see, J. G. Oldroyd ([23])) of order one: Z t ∂u (1.1) + u · ∇u − µ∆u − β(t − τ )∆u(x, τ ) dτ + ∇p = f (x, t), ∂t 0 with x ∈ Ω, t > 0 and incompressibility condition (1.2)

∇ · u = 0, x ∈ Ω, t > 0,

and initial and boundary conditions (1.3)

u(x, 0) = u0 in Ω, u = 0, on ∂Ω, t ≥ 0.

Here, Ω is a bounded domain in R2 with boundary ∂Ω, µ = 2κλ−1 > 0 and the kernel β(t) = γ exp(−δt), where γ = 2λ−1 (ν − κλ−1 ) > 0 and δ = λ−1 > 0. We note that ν is the kinematic coefficient of viscosity. λ is the relaxation time, and is characterized by the fact that after instantaneous cessation of motion, the stresses Received by the editors June 7, 2010 and, in revised form, November 25, 2010. 2000 Mathematics Subject Classification. 35L70; 65N30; 76D05. 324

ERROR ESTIMATE FOR THE OLDROYD FLUID OF ORDER ONE

325

in the fluid do not vanish instantaneously, but die out like exp(−λ−1 t). Moreover, the velocities of the flow, after instantaneous removal of the stresses, die out like exp(−κ−1 t), where κ is the retardation time. For further details of the physical background and its mathematical modeling, we refer to [14], [23] and [24]. There is considerable amount of literature devoted to Oldroyd model by Russian mathematicians such as A.P.Oskolkov, Kotsiolis, Karzeeva and Sobolevskii etc, see [24, 1, 8, 15, 18] and references, therein. Based on the analysis of Ladyzhenskaya [19] for the Navier-Stokes equations, Oskolkov [24] has proved existence of a unique ’almost’ classical solution in finite time interval [0, T ] for the 2D problem (1.1)-(1.3). In the proof, the constant appeared in a priori bounds depends exponentially on T and therefore, it is not possible to extend the results for large time. Subsequently, Agranovich and Sobolevskii [1] have extended the analysis of Oskolkov and have derived global existence of solutions for all t ≥ 0 when f ∈ L2 (L2 ) with smallness conditions on data for 3D problem. The solvability on the semi-axis t ≥ 0, for the problem (1.1)-(1.3), is discussed in [18, 8] when f , ft ∈ L∞ (R+ ; L2 (Ω)) in [18] and f , ft ∈ S 2 (R+ ; L2 (Ω)) in [8], where S 2 is a subspace of L2loc . We observe that results in [18, 8] hold true only for finite time (T < ∞), that is, for f , ft ∈ L∞ (0, T ; L2 (Ω)), with estimate depending on T , but there seems to have some difficulties in extending these results for all t ≥ 0, when f , ft ∈ L∞ (R+ ; L2 (Ω)). For example, in [18], it is difficult to derive the estimate (20) from (17) on page 2780 by applying integral version of the Gronwall’s Lemma and the estimate (12). Unfortunately, this is further carried over to subsequent articles, see Theorem 2 of [8]. In the context of dynamical system generated by Oldroyd model when f ∈ L∞ (L2 ), see, [15] and [17], it is not quite clear that the conclusion of Theorem 1.2 of [15] for s = 1 or Theorem 1 of [17] for ℓ = 1 holds true. In fact a more careful observation in both these articles R1 demands an estimate of 0 kφk2E1 , which is difficult to establish prior to this result. In the context of 2D Navier-Stokes equations, a standard tool for deriving uniform Dirichlet norm for the velocity term is to apply uniform Gronwall’s Lemma. Due to the presence of the integral term in (1.1), it is difficult to apply uniform Gronwall’s Lemma (see Remark 3.3(i)). To be more precise, on the right-hand side of (17) on R t PL ˜ l k2 dτ is not available from (12) page 2780 of [18], the estimate of 0 l=1 βl k∆u 2,Ωt (for notations, see [18]), which is crucial in applying uniform Gronwall’s Lemma. This is exactly a similar problem faced in article [15] and [17]. In [30], Sobolevskii has examined the behavior of the solution as t → ∞ under some stabilization conditions like positivity of the first eigenvalue of a selfadjoint spectral problem introduced therein and H¨older continuity of the function Φ = eδ0 t (f (x, t) − f∞ (x)), where f∞ = limt→∞ f and δ0 > 0, using energy arguments and positivity of the integral operator, see also Kotsiolis and Oskolkov [16]. Recently, He et al. [10] have proved similar results under milder conditions on f and weaker regularity assumptions on the initial data u0 . In fact, in their analysis, they have shown both the power and exponential convergence of the solutions to a steady state solution, when Φ ∈ L∞ (L2 ) only. For the numerical approximations to the problem (1.1)-(1.3), we refer to Akhmatov and Oskolkov [2], Cannon et al. [5], He et al. [11] and Pani et al. [28]. In [2], stable finite difference schemes are discussed without any discussion on convergence. Cannon et al. [5] have proposed a modified nonlinear Galerkin scheme for (1.1)-(1.3) with periodic boundary condition using a spectral Galerkin method and have discussed convergence analysis while keeping time variable continuous. In [11], local optimal error estimates for the velocity in L∞ (H1 )-norm and the pressure in L∞ (L2 )-norm are established. Moreover, these estimates are shown to be uniform

326

DEEPJYOTI GOSWAMI AND AMIYA K. PANI

provided the given data satisfy the uniqueness condition. In [28], optimal error bounds in L∞ (L2 ) as well as in the L∞ (H1 )-norms for the velocity and for the pressure in the L∞ (L2 )-norm are derived which are valid uniformly in time t > 0 under the condition that f ≡ 0. In fact, Pani et al. [28] have obtained new regularity results, which are valid for all time t > 0, without nonlocal compatibility conditions. Based on Stokes-Volterra projection and duality arguments, they have proved optimal error estimates, when the initial data u0 ∈ H2 ∩ J1 . Subsequently in [29], a backward Euler method is used to discretize in temporal direction and semi-group theoretic approach is then employed to establish a priori error estimates. Our present investigation is a continuation of [28]. In this paper, we obtain regularity results which are uniform in time under realistically assumed regularity on the exact solution, when f 6≡ 0 with f , ft ∈ L∞ (L2 ). As is pointed out in [12] and [28], some of the regularity results depend on the non-local compatibility conditions on the data at t = 0, which are either very hard to verify or difficult to meet in practice. We have, in this article, obtained new regularity results under realistically assumed conditions on initial data so that we can avoid non-local compatibility conditions. At this point, we would like to stress that for 2D Oldroyd fluids of order one, a step-by-step proof of the Dirichlet norm estimate which is uniform in time is missing in the literature. Following the analysis of 2D Navier-Stokes equations, it is hard to apply the uniform Gronwall’s Lemma [32] or the proof techniques of Ladyzhenskaya [19] for deriving uniform estimate in the Dirichlet norm for the velocity term. Hence, we hope that, our present analysis will also fill this missing link. Under the uniqueness condition (see, Section 5), we have shown uniform (in time) optimal error estimates for both velocity and pressure terms. This is an improvement over the results obtained in [11], where the uniform error estimate for velocity is not optimal in L∞ (L2 ). We have also improved the error estimation of the pressure term, in the sense that, the estimate now reads O(t−1/2 h) instead of O(t−1 h), which is again an improvement over the results observed in [11] for the nonsmooth data, i.e., u0 ∈ J1 and [28], when u0 ∈ J1 ∩ H2 . In [12], Heywood and Rannacher have noted that this singular behavior of the pressure estimate for the Navier-Stokes problem is due to a difficulty which appears technical, but which may be inherent to the problem. Therefore, our present analysis will improve the result of Heywood and Rannacher [12] for the Navier-Stokes equations using conforming elements. The main contributions of the present article are as follows : (i) step-by-step proof of uniform H1 -bound for the velocity. (ii) proof of regularity results for the solution which reflect the behavior as t → 0 and as t → ∞ when f , ft ∈ L∞ (L2 ) and u0 ∈ J1 (see, for definition, Section 2). (iii) proof of optimal error L∞ (L2 ) estimates for semidiscrete Galerkin approximations to the velocity and pressure for the nonsmooth initial data u0 , that is, u0 ∈ J1 . (iv) proof of uniform optimal error estimates for both velocity and pressure terms under the assumption of the uniqueness condition. (v) improvement in the singular behavior (as t → 0) of the pressure estimate, i.e., k(p − ph )(t)k ≤ Kht−1/2 instead of Kht−1 . For related papers on finite element approximations to parabolic partial integrodifferential equations, we may refer to [7, 20, 22, 25, 33] for smooth solutions and

ERROR ESTIMATE FOR THE OLDROYD FLUID OF ORDER ONE

327

[21, 26, 27, 33] for the nonsmooth initial data. The smoothing properties proved via energy argument in [27] will be useful for deriving the regularity results for the present problem without nonlocal compatibility assumptions on the data at t = 0. The remaining part of this paper is organized as follows. In Section 2, we discuss some notations, weak formulation, basic assumptions and statement of positivity and Gronwall’s Lemma. Section 3 focuses on uniform estimates in L∞ (L2 ) and L∞ (H1 )-norms and new regularity results without nonlocal compatibility conditions. In Section 4, a semidiscrete Galerkin method is discussed. Section 5 is devoted to optimal L∞ (L2 ) error estimates of the velocity term, for the nonsmooth initial data. In Section 6, optimal error bound for the pressure term is derived. It is also shown that under the uniqueness assumption, uniform estimates in time t > 0 are also established. Finally, we summarize our results in the Section 7. 2. Preliminaries For our subsequent use, we denote by bold face letters the R2 -valued function space such as H10 = [H01 (Ω)]2 , L2 = [L2 (Ω)]2 and Hm = [H m (Ω)]2 , where H m (Ω) is the standard Hilbert Sobolev space of order m. Note that H10 is equipped with a norm  1/2 !1/2 2 2 X X   k∇vk = (∂j vi , ∂j vi ) = (∇vi , ∇vi ) . i,j=1

i=1

Further, we introduce some more function spaces for our future use: J1

= {φ ∈ H10 : ∇ · φ = 0}

J = {φ ∈ L2 : ∇ · φ = 0 in Ω, φ · n|∂Ω = 0 holds weakly}, where n is the outward normal to the boundary ∂Ω and φ · n|∂Ω = 0 should be understood in the sense of trace in H−1/2 (∂Ω), see [31]. Let H m /R be the quotient space consisting of equivalence classes of elements of H m differing by constants, which is equipped with norm kpkH m /R = kp + ckm . For any Banach space X, let Lp (0, T ; X) denote the space of measurable X -valued functions φ on (0, T ) such that Z T

0

kφ(t)kpX dt < ∞ if 1 ≤ p < ∞,

and for p = ∞

ess sup kφ(t)kX < ∞ if p = ∞. 0 0 and φ ∈ L2 (0, t∗ ), the following positive definite property holds  Z t∗ Z t exp [−α(t − s)]φ(s) ds φ(t) dt ≥ 0. 0

0

In order to deal with the integral term, we present the following Lemma. See [28]. Lemma 2.2. Let g ∈ L1 (0, t∗ ) and φ ∈ L2 (0, t∗ ) for some t∗ > 0. Then the following estimate holds ! Z ∗ !1/2  2 !1/2 Z t∗ Z s Z t∗ t g(s − τ )φ(τ ) dτ ds ≤ |g(s)| ds |φ(s)|2 ds . 0

0

0

0

Lemma 2.3 (Gronwall’s lemma). Let g, h, y be three locally integrable non-negative functions on the time interval [0, ∞) such that for all t ≥ 0 Z t Z t y(t) + G(t) ≤ C + h(s) ds + g(s)y(s) ds, 0

0

where G(t) is a non-negative function on [0, ∞) and C ≥ 0 is a constant. Then, Z t   Z t  y(t) + G(t) ≤ C + h(s) ds exp g(s) ds . 0

0

ERROR ESTIMATE FOR THE OLDROYD FLUID OF ORDER ONE

329

3. A Priori Estimates In this Section, we discuss a priori bounds for the solution {u, p} of (1.1)-(1.3). Here, we present a step-by-step proof of uniform estimate (in time) in H 1 , when d = 2. Below, we derive a priori bounds following the proof techniques of [28]. Lemma 3.1. Let 0 < α < min (δ, λ1 µ), and let the assumption (A2) hold. Then, there is a positive constant K0 = K0 (M0 , µ, δ, λ1 ) such that the solution u of (2.2) satisfies, for t > 0 Z t Z t 1 2 2 2 ku(t)k + µ k∇u(s)k ds ≤ ku0 k + kf (s)k2 ds µλ1 0 0  t  (3.1) , ≤ M02 1 + λ1 µ Z t α ku(t)k2 + (µ − )e−2αt e2ατ k∇u(τ )k2 dτ ≤ e−2αt ku0 k2 λ1 0 (1 − e−2αt ) (3.2) kf k2L∞ (L2 ) = K0 . + 2α(λ1 µ − α) Moreover, (3.3)

lim k∇u(t)k ≤

t→∞

kf k2L∞ (L2 ) λ1 µ2

2 = K01 .

Proof. We easily modify the proof of Lemma 4.1 in [28](pg 758) to derive estimates (3.1)-(3.2). For the estimate (3.3), we again modify the technique of [28] to obtain Z t Z t 2 −2αt 2ατ 2 −2αt 2 −2αt ku(t)k + µe e k∇u(τ )k dτ ≤ e ku0 k + 2e α e2ατ ku(τ )k2 dτ 0

0

+

(3.4)

kf k2L∞ (L2 ) 2αλ1 µ

(1 − e−2αt ).

Now, taking limit supremum as t → ∞, the second term on the left-hand side (3.4) becomes R t 2ατ Z t e k∇u(τ )k2 dτ −2αt 2ατ 2 (3.5) lim µe e k∇u(τ )k dτ = µ lim 0 t→∞ t→∞ e2αt 0 µ = lim k∇u(t)k2 , 2α t→∞ and therefore, we find that kf k2L∞ (L2 ) µ lim k∇u(t)k2 ≤ . 2α t→∞ 2αλ1 µ This completes the rest of the proof.



Remark 3.1. We obtain the final result (3.3) by using L’Hospital rule in (3.5), Rt under the assumption that 0 e2αt k∇uk2 ds is not bounded as t → ∞. This allows us to claim the uniform (in time) Dirichlet norm of u in the next lemma. On the Rt contrary, if 0 e2αt k∇uk2 ds is bounded for all time, then we can obtain uniform (in time) Dirichlet norm of u directly using Gronwall’s Lemma in (3.8)(see, Lemma 3.2).

330

DEEPJYOTI GOSWAMI AND AMIYA K. PANI

Lemma 3.2. Let 0 < α < min (δ, λ1 µ) and let assumption (A2) hold. Then there is a positive constant M3 = M3 (α, µ, λ1 , M0 ) such that for all t > 0 Z t 2 −2αt ˜ k∇u(t)k + e e2ατ k∆u(τ )k2 dτ ≤ M32 . 0

˜ we rewrite (2.2) as ˆ = e u and using the Stokes operator ∆, Proof. Setting u Z t ˜ u, φ) − ˜ u(τ ), φ) dτ (ˆ ut , φ) − α(ˆ u, φ) − µ(∆ˆ β(t − τ )eα(t−τ ) (∆ˆ αt

0

(3.6)

=

−e−αt (ˆ u · ∇ˆ u, φ) + (ˆf , φ),

˜ u in (3.6), we note that With φ = −∆ˆ ˜ u) = 1 d k∇ˆ −(ˆ ut , ∆ˆ uk 2 . 2 dt Thus, d k∇ˆ uk 2 dt

˜ uk 2 + 2 + 2µk∆ˆ

Z

t

˜ u(τ ), ∆ˆ ˜ u(t)) dτ β(t − τ )eα(t−τ ) (∆ˆ

0

˜ u) + 2e−αt (ˆ ˜ u) − 2(ˆf , ∆ˆ ˜ u). = −2α(ˆ u, ∆ˆ u · ∇ˆ u, ∆ˆ On integration with respect to time and using Lemma 2.1 with definition of β, it follows for 0 < α < min (δ, λ1 µ) that Z t Z t 2 2 2 ˜ ˜ u) dτ k∇ˆ u(t)k + 2µ k∆ˆ u(τ )k dτ ≤ k∇u0 k − 2α (ˆ u, ∆ˆ 0 0 Z t Z t −ατ ˜ ˜ u) dτ (ˆf , ∆ˆ + 2 e (ˆ u · ∇ˆ u, ∆ˆ u) dτ − 2 0

0

= k∇u0 k2 + I1 + I2 + I3 .

(3.7)

1 2 To estimate |I1 | and |I3 |, we apply Cauchy-Schwarz inequality with ab ≤ 2ǫ a + 2ǫ b2 , a, b ≥ 0, ǫ > 0 to obtain Z Z Z t α2 t 1 t ˆ ˜ u(τ )k2 dτ. |I1 | + |I3 | ≤ kˆ u(τ )k2 dτ + kf(τ )k2 dτ + 2ǫ k∆ˆ ǫ 0 ǫ 0 0

To estimate I2 , a use of H¨older’s inequality shows that ˜ u)| ≤ kˆ ˜ uk. |(ˆ u · ∇ˆ u, ∆ˆ ukL4 (Ω) k∇ˆ ukL4 (Ω) k∆ˆ

Now, we appeal to the following Sobolev inequality(d = 2)(see [31]) kφkL4 (Ω) ≤ 21/4 kφk1/2 k∇φk1/2 , φ ∈ H01 (Ω), and, therefore, |I2 | ≤

Z

t

˜ uk3/2 dτ e−ατ kˆ uk1/2 k∇ˆ uk k∆ˆ Z t Z t ˜ uk2 dτ. C(ǫ) e−4ατ kˆ uk2 k∇ˆ uk4 dτ + ǫ k∆ˆ C

0

≤

0

0

Substituting the estimates of I1 , I2 and I3 in (3.7), we find that Z t Z t ˜ u(τ )k2 dτ ≤ k∇u0 k2 + C(α, µ) k∇ˆ u(t)k2 + µ k∆ˆ (kˆ u(τ )k2 + kˆf (τ )k2 ) dτ 0 0 Z t (3.8) + C(µ) kuk2 k∇uk2 k∇ˆ uk2 dτ . 0

ERROR ESTIMATE FOR THE OLDROYD FLUID OF ORDER ONE

331

Using Gronwall’s lemma, we arrive at Z t Z t n o 2 2 2 ˜ k∇ˆ u(t)k + µ k∆ˆ u(τ )k dτ ≤ k∇u0 k + C(α, µ) (kˆ u(τ )k2 + kˆf (τ )k2 ) dτ 0 0 Z t   kuk2 k∇uk2 dτ . × exp C(µ) 0

Finally, we use Lemma 3.1 with condition (A2) to obtain the following integral inequality Z t   2 −2αt ˜ k∇u(t)k + µe e2αs k∆u(τ )k2 dτ ≤ e−2αt M02 + C(α, µ){K0 + M02 } 0  t  × exp C(µ, M0 , K0 )(1 + ) . λ1 µ This inequality along with lim supt→∞ k∇uk ≤ K01 would lead us to the following conclusion k∇u(t)k < M3 t > 0, for some positive constant M3 . Now (3.8) along with this estimate of k∇u(t)k would provide us the desired result.  Remark 3.2. Although Lemma 3.2 provides a uniform Dirichlet norm estimate in time, it is difficult to obtain a precise bound for the estimate k∇uk for all t > 0. We present below another proof of the uniform estimate in L∞ (H10 ) norm which enables us to obtain a precise bound depending on the initial data, forcing term and the smoothness of the domain. Lemma 3.3. (Uniform estimate in L2 ). Let the assumption (A2) hold. Then, for 0 < α < min (2δ, λ1 µ), the solution u of (2.2) satisfies the following estimates for t > 0 (3.9)

ku(t)k2 +

kf k2L∞ (L2 ) 1 k∇˜ uβ k2 ≤ e−αt ku0 k2 + (1 − e−αt ) = M1 , γ λ1 µα

and for fixed T0 > 0  Z t+T0  2δ M 2 T0 (3.10) µk∇u(s)k2 + k∇˜ uβ k2 ds ≤ M1 + 0 = M2 , γ λ1 µ t Rt ˜ β (t) = 0 β(t − s)u(s) ds. where u Proof. With

˜ β (t) = u

Z

t

β(t − s)u(s) ds,

0

we rewrite the equation (2.2) as

(3.11) (ut , φ) + µ(∇u, ∇φ) + (u · ∇u, φ) + (∇˜ uβ , ∇φ) = (f , φ), ∀φ ∈ J1 , t > 0. ˜ β,t + δ u ˜ β = γu, to obtain Take φ = u in (3.11) and use u (3.12)

d 1 2δ 1 (kuk2 + k∇˜ uβ k2 ) + µk∇uk2 + k∇˜ uβ k 2 ≤ kf k2 . dt γ γ µλ1

We use the Poincar´e inequality λ1 kuk2 ≤ k∇uk2 and multiply (3.12) by eαt to find that  d  αt 1 1 αt e (kuk2 + k∇˜ uβ k 2 ) ≤ e kf k2 . dt γ µλ1

332

DEEPJYOTI GOSWAMI AND AMIYA K. PANI

Now, integrate with respect to time and multiply by e−αt to conclude (3.9). Next, we integrate (3.12) from t to t + T0 , for fixed T0 > 0 and use (3.9) to obtain (3.10). This completes the rest of the proof.  Lemma 3.4. (Uniform estimates in H1 ). Let 0 < α < min (δ, λ1 µ). Under assumption (A2), there exists a positive constant M3 = M3 (α, µ, λ1 , M1 ) such that for T0 > 0 with t ∈ [T0 , ∞), the following estimates hold: 1 ˜ k∇u(t)k2 + k∆˜ uβ k2 ≤ M32 , t ≥ T0 , ∀ T0 > 0. γ and Z t ˜ k∇u(t)k2 + e−2αt e2ατ k∆u(τ )k2 dτ ≤ M32 , t > 0. 0

˜ in (3.11) and use u ˜ β,t + δ u ˜ β = γu, to obtain Proof. Put φ = −∆u (3.13) 1 d 1 ˜ ˜ 2 + δ k∆˜ ˜ uβ k2 ≤ kf kk∆uk ˜ ˜ (k∇uk2 + k∆˜ uβ k2 ) + µk∆uk + |(u · ∇u, −∆u)|. 2 dt γ γ

Recollecting the estimates of I2 , I3 of (3.7), we find that µ ˜ 2 3 ˜ (3.14) kf kk∆uk ≤ k∆uk + kf k2 , 6 µ and µ ˜ 2 9 ˜ (3.15) |(u · ∇u, −∆u)| ≤ k∆uk + ( )3 kuk2 k∇uk4 . 6 2µ Using (3.14)-(3.15) in (3.13), we obtain   d 1 ˜ + (3.16) k∇uk2 + k∆˜ uβ k 2 dt γ ≤

4µ ˜ 2 2δ ˜ k∆uk + k∆˜ uβ k 2 3 γ 9/2 3 3 kf k2 + ( ) kuk2 k∇uk4 . µ µ

Note that

˜ ≤ α0 kukk∆uk, ˜ α0 k∇uk2 = α0 (u, −∆u) where α0 > 0 is a constant to be chosen precisely at a later stage and then, we find that µ ˜ 2 3 2 (3.17) α0 k∇uk2 ≤ k∆uk + α kuk2 . 3 4µ 0 ˜ 2 ≥ λ1 k∇uk2 to arrive at Now, add (3.17) to (3.16) and use the inequality k∆uk     1 ˜ 9/2 3 2δ ˜ d k∇uk2 + k∆˜ uβ k 2 + α0 + µλ1 − ( ) kuk2 k∇uk2 k∇uk2 + k∆˜ uβ k 2 dt γ µ γ 3 3 2 ≤ kf k2 + α kuk2 . µ 4µ 0 As kuk2 ≤ M1 from Lemma 3.1, and kf kL∞(L2 ) ≤ M0 , we obtain     d 1 ˜ 9 2δ ˜ 2 2 k∇uk + k∆˜ uβ k + α0 + µλ1 − ( )3 kuk2 k∇uk2 k∇uk2 + k∆˜ uβ k 2 dt γ 2µ γ (3.18) ≤ K1 (M0 , µ, α0 , M1 ) = K1 . Setting (3.19)

n o 9 h(t) = min α0 + µλ1 − ( )3 kuk2 k∇uk2 , 2δ 2µ

ERROR ESTIMATE FOR THE OLDROYD FLUID OF ORDER ONE

333

˜ uβ k 2 , we obtain from (3.18) with E(t) = k∇uk2 + γ1 k∆˜ d E(t) + h(t)E(t) ≤ K1 . dt Hence, for 0 ≤ s < t, we find that  Rt d  R t h(τ ) dτ es E(t) ≤ K1 e s h(τ ) dτ . dt On integrating with respect to time over 0 to t, we arrive at Z t R Rt t (3.20) E(t) ≤ e− 0 h(τ ) dτ k∇u0 k2 + K1 e− s h(τ ) dτ ds. 0

Now, from (3.19), we note, for some T0 > 0 that Z Z t+T0 o n 9 3 t+T0 ku(s)k2 k∇u(s)k2 ds, 2δT0 . h(s) ds = min (α0 + µλ1 )T0 − ( ) 2µ t t

A use of Lemma 3.1 yields Z t+T0 Z 9 CM1 t+T0 CK0 ( )3 ku(s)k2 k∇u(s)k2 ds ≤ k∇u(s)k2 ds ≤ M2 = K2 . 3 2µ µ µ4 t t Therefore, we obtain Z t+T0 t

n o h(s) ds ≥ min (α0 + µλ1 )T0 − K2 , 2δT0 .

Now, choose α0 T0 = K2 to arrive at Z t+T0 (3.21) h(s) ds ≥ T0 min{µλ1 , 2δ} ≥ αT0 . t

Given s and t with 0 ≤ s < t, we choose two positive integers k and l such that kT0 ≤ s ≤ (k + 1)T0 , lT0 ≤ t ≤ (l + 1)T0 . Then, from (3.21) and −h(t) ≥ −2δ, t > 0 using (3.19), we find that Z t Z (l+1)T0 Z s Z (l+1)T0 h(τ ) dτ = h(τ ) dτ − h(τ ) dτ − h(τ ) dτ s

kT0

≥ ≥

kT0

t

(l + 1 − k)αT0 − 2δT0 − 2δT0 (t − s)α − 4δT0 .

Hence, −

Z

t

h(τ ) dτ ≤ 4δT0 − (t − s)α.

s

Now, without loss of generality, we have assumed that s ≤ (k +1)T0 ≤ lT0 ≤ t (with one of these ≤ is strict inequality to preserve the fact that 0 ≤ s < t). All other possible cases will simplify the present situation and hence, we skip the related analysis. From (3.20), we obtain Z t E(t) ≤ e−tα e4δT0 k∇u0 k2 + K1 e−(t−s)α ds e4δT0 0

≤ ≤

 K1 e−tα k∇u0 k2 + (1 − e−tα ) e4δT0 α   K 1 e4δT0 k∇u0 k2 + = M32 . α 

334

DEEPJYOTI GOSWAMI AND AMIYA K. PANI

This establishes the first estimate of Lemma 3.4. Use it in (3.8) to obtain the second estimate, and this completes the rest of the proof.  Remark 3.3. (i) It is difficult to apply the uniform Gronwall’s Lemma to (3.16) to obtain the desired result as we do not have an estimate of Z t+T0 1 ˜ (k∇u(s)k2 + k∆˜ (3.22) uβ (s)k2 ) ds ≤ M4 , T0 > 0, γ t

where M4 is some positive constant independent of t. Note that from Lemma 3.1, we only obtain the estimate Z t+T0 µ k∇u(s)k2 ds ≤ M2 . t

(ii) Instead of the assumption (A2), if we make the following assumption Z t e2αt kf (τ )k2 dτ ≤ M1 , and k∇u0 k2 ≤ M1 , 0

then a simple modification of the above Lemmas yields

k∇u(t)k ≤ C(α, µ, δ, λ1 , M1 )e−αt ∀t > 0, and Z

t

˜ e2ατ k∆u(τ )k2 dτ ≤ C(α, µ, δ, λ1 , M1 ) ∀t > 0.

0

Note that we have the exponential decay property for the gradient of u(t) in L∞ (L2 )norm. (iii) Following the arguments of Sobolevskii [30], it is possible to obtain similar asymptotic behavior for some 0 < δ0 < min(δ, λ1 µ) provided sup (eδ0 t kf (t)k) ≤ M, 0 0. (iii) For u0 ∈ J1 ∩ H2 , based on the analysis of [28], we can easily obtain the following error estimate for the velocity term k(u − uh )(t)k ≤ Ch2 eCt , t ∈ [0, T ]. Below, we derive uniform (in time) error estimate for the velocity term under the assumption of the uniqueness condition, that is, (5.31)

N b(u, v, w) kf kL∞ (0,∞;L2 (Ω)) < 1 and N = sup , ν2 1 k∇ukk∇vkk∇wk u,v,w∈H0 (Ω)

where ν = µ + γδ . Theorem 5.2. Under the assumptions of Theorem 5.1 and the uniqueness condition (5.31), there exists a positive constant C, independent of time and h, such that for all t > 0 k(u − uh )(t)k ≤ Ch2 t−1/2 .

ERROR ESTIMATE FOR THE OLDROYD FLUID OF ORDER ONE

347

Proof. We note that, in Lemma 5.3, it is shown that estimates of ξ are uniform in time, for all time away from zero. As e = ξ + η, uniform estimates of η would give us the desired result. In Theorem 5.1, we apply Gronwall’s Lemma, and hence, the constant depends on exponential in time. In order to have estimates which are uniformly valid for all t > 0, we need a different estimate of the nonlinear term Λh (ˆ η ) with the help of the uniqueness condition. We rewrite (5.32)

Λh (η) = −[b(ξ, uh , η) + b(η, uh , η) + b(u, ξ, η)].

Using the uniqueness condition, we find (5.33)

|b(η, uh , η)| ≤ N k∇ηk2 k∇uh k.

Using (4.7) and (4.8), we find (5.34) ˜ h uh k1/2 )k∇ηkkξk. |b(u, ξ, η)| + |b(ξ, uh , η)| ≤ C(k∇uk1/2 k∆uk1/2 + k∇uh k1/2 k∆ Substitute (5.33)-(5.34) in (5.32) and use Lemmas 3.1, 3.2, Theorem 3.1 and Lemmas 4.2, 5.3 to find that (5.35)

|Λh (η)| ≤ N k∇ηk2 k∇uh k + Ch2 [τ ∗]−3/4 k∇ηk.

We now modify the proof of Theorem 5.1 as follows: We choose Choose φh = e2αt η in (5.12) and use the estimate (5.35) to obtain Z t d 2 2 ˆ (t)) ds (5.36) kˆ η k + 2(µ − N k∇uh k)k∇ˆ ηk + 2 β(t − s)eα(t−s) a(ˆ η (s), η dt 0 ≤ 2αkˆ η k2 + Ch2 [τ ∗]−3/4 e2αt k∇ηk.

Integrate (5.36) with respect to time from 0 to t and multiply by e−2αt to arrive at Z t kη(t)k2 + e−2αt e2αs 2(µ − N k∇uh (s)k)k∇η(s)k2 ds 0 Z s Z t −2αt 2αs + 2e e β(s − τ )a(η(τ ), η(s)) dτ ds 0 0 Z t Z t h i −2αt 2 2αs 2 2 [τ ∗ (s)]−3/4 e2αs k∇η(s)k ds . ≤ e kη(0)k + 2α e kη(s)k ds + Ch 0

0

Take t → ∞ and use the following results from [11] Z t Z s γ lim 2e−2αt e2αs β(s − τ )a(φ(τ ), φ(s))dτ ds = lim k∇φ(t)k2 , t→∞ t→∞ αδ 0 0 lim k∇uh (t)k2 ≤ ν −1 kf kL∞ (0,∞;L2 (Ω))

t→∞

and lim (τ ∗ (t))−3/4 = 1

t→∞

to find that 1h γi Ch2 (µ − N ν −1 kf kL∞ (0,∞;L2 (Ω)) ) + lim k∇η(t)k2 ≤ lim k∇η(t)k. α δ t→∞ α t→∞ Therefore, we obtain 1 (1 − N ν −2 kf kL∞ (0,∞;L2 (Ω)) ) lim k∇η(t)k ≤ Ch2 . t→∞ ν From (5.31), we conclude that lim k∇η(t)k ≤ Ch2 .

t→∞

348

DEEPJYOTI GOSWAMI AND AMIYA K. PANI

Clearly, lim kη(t)k ≤ Ch2 .

t→∞

This, with uniform estimate of ξ from Lemma 5.3, leads to lim ke(t)k ≤ Ch2 t−1/2 .

t→∞

Note that the constant C is valid uniformly for all t > 0, and this completes the rest of the proof. 

6. A priori Error Estimates for the Pressure In this Section, we derive optimal error estimates for the Galerkin approximation ph of the pressure p. The main theorem of this Section is stated as follows. Theorem 6.1. In addition to the hypotheses of Theorem 5.1, assume that (B2′ ) holds. Then, there exists a positive constant C such that for 0 < t < T and for u0 ∈ J1 , k(p − ph )(t)kL2 /Nh ≤ CeCt ht−1/2 . Now, we prove Theorem 6.1 with the help of a series of Lemmas. As in [28], we obtain the following result. Lemma 6.1. The semi-discrete Galerkin approximation ph of the pressure satisfies for all t > 0 (6.1)   Z t ˜ k(p−ph )(t)kL2 /Nh ≤ Ck|et k|−1;h + (K + CkekL3 )k∇ek + β(t − s)k∇e(s)k ds , 0

where

k|gk|−1;h = sup

o n < g, φ > h , φh ∈ Hh , φh 6= 0 . k∇φh k

From Theorem 5.1, the estimate k∇ek is known and using Sobolev embedding Theorem for d = 2, the estimate kekL3 ≤ Ck∇ek ≤ K. In order to complete the proof of Theorem 6.1, we need to estimate k|et k|−1;h in (6.1). Since Hh ⊂ H10 , we note that n< e , φ > o t h k|et k|−1;h = sup , φh ∈ Hh , φh 6= 0 k∇φh k n< e , φ > o t ≤ sup , φ ∈ H10 , φ 6= 0 , k∇φk supremum being taken over a bigger set. Therefore, we obtain k|et k|−1;h ≤ ket k−1 . Lemma 6.2. The error e = u − uh in approximating the velocity satisfies for t > 0 ket (t)k−1 ≤ CeCt h(τ ∗ (t))−1/2 , where τ ∗ (t) = min{t, 1}. Proof. From (4.2) and (5.1), we write the equation in e as Z t (6.2) (et , φh ) + µa(e, φh ) + β(t − s)a(e(s), φh ) ds 0

=

Λh (φh ) + (p, ∇ · φh ) ∀φh ∈ Jh ,

ERROR ESTIMATE FOR THE OLDROYD FLUID OF ORDER ONE

349

where, from (5.14), we find Λh (φh ) = −b(e, uh , φh ) − b(u, e, φh ). 2

Since Ph : L → Jh , then, for any ψ ∈ H10 , we obtain using (6.2), (6.3) (et , ψ) = =

(et , ψ − Ph ψ) + (et , Ph ψ) (et , ψ − Ph ψ) − µa(e, Ph ψ) −

Z

t

β(t − s)a(e(s), Ph ψ) ds

0

+Λh (Ph ψ) + (p, ∇ · Ph ψ).

Using discrete incompressibility condition, we write (p, ∇ · Ph ψ) = (p − jh p, ∇ · Ph ψ) 1

and hence, H -stability of Ph yields (6.4)

|(p, ∇ · Ph ψ)| ≤ Chk∇pkk∇ψk.

Also, using the Cauchy-Schwarz inequality and H 1 -stability of Ph , we obtain (6.5)

|Λh (Ph ψ)| ≤ k∇ek(k∇uh k + k∇uk)k∇ψk.

Using approximation property of Ph , we find that (6.6)

(et , ψ − Ph ψ) = (ut , ψ − Ph ψ) ≤ Chkut kk∇ψk.

Substitute (6.4)-(6.6) in (6.3) and use the boundedness of a(·, ·) to obtain Z t n (et , ψ) ≤ Chkut k + Ck∇ek + β(t − s)k∇e(s)k ds 0 o +k∇ek(k∇uhk + k∇uk) + Chk∇pk k∇ψk, and therefore, (6.7)

ket k−1

= ≤

n< e , φ > o t , φ ∈ H10 , φ 6= 0 k∇φk Z t Chkut k + Ck∇ek + β(t − s)k∇e(s)k ds

sup

0

+k∇ek(k∇uhk + k∇uk) + Chk∇pk.

From Lemma 4.2, we find that k∇uh (t)k ≤ K, t > 0. Now, using Theorem 3.1 and 5.1, we obtain from (6.7) Z t Ct ∗ −1/2 (6.8) ket k−1 ≤ Ce h(τ (t)) + β(t − s)k∇e(s)k ds 0 Z t n o ≤ CeCt h(τ ∗ (t))−1/2 1 + (τ ∗ (s))−1/2 ds 0

Ct

∗

−1/2

≤ Ce h(τ (t))

,

and this completes the rest of the proof.

Proof of Theorem 6.1. Use (6.8) in (6.1) to obtain (6.9)

k(p − ph )(t)kL2 /Nh

≤ +

˜ + CkekL3 )k∇ek CeCt h(τ ∗ (t))−1/2 + (K Z t β(t − s)k∇e(s)k ds. 0



350

DEEPJYOTI GOSWAMI AND AMIYA K. PANI

The last term involving integral can be estimated as in (6.8). From Theorem 5.1, we find that k∇ek ≤ CeCt h(τ ∗ (t))−1/2 kekL3 ≤ Ck∇ek ≤ CeCt h(τ ∗ (t))−1/2 . Now, plugging these estimates in (6.9), we obtain k(p − ph )(t)kL2 /Nh ≤ CeCt h(τ ∗ (t))−1/2 , and this completes the rest of the proof.



Remark 6.1. Under the uniqueness condition (5.31), we establish error estimate for pressure, which is valid for all time t > 0. From the proof of Theorem 5.2 and using the inverse inequality along with the triangle inequality, we easily arrive at k∇ek ≤ Ch(τ ∗ (t))−1/2 , where generic constant C is independent of time. As the estimates in Theorem 3.1 are uniform in time, we obtain from Lemma 6.2 ket k−1 ≤ Ch(τ ∗ (t))−1/2 . Finally, Z

t

β(t − s)k∇e(s)k ds ≤Che−δt

0

Z

t

eδs (τ ∗ (s))−1/2 ds

0

=Che

−δt

nZ

1

eδs s−1/2 ds +

0

Z

t 1

eδs ds

o

1 ≤Ch{2 + }. (6.10) δ Note that the second integral on right-hand side of (6.10) vanishes for t ≤ 1. Now, an appeal to (6.1) yields k(p − ph )(t)kL2 /Nh ≤ Ch(τ ∗ (t))−1/2 , and we obtain optimal error estimate for the pressure term, which is uniform in time. 7. Conclusion In this paper, we have discussed optimal error estimates of the velocity and the pressure terms in L∞ (L2 ) and L∞ (L2 ), respectively, for the semidiscrete finite element approximations to the equations of motion arising in the 2D Oldroyd fluids of order one. We have also established uniform in time error estimates under the uniqueness assumption. All these results are proved when forcing term f in L∞ (L2 ) and initial velocity u0 in J1 . Optimal error estimates are also proved in [28], when f ≡ 0 and u0 in J1 ∩ H2 . The main difficulty we encounter here, in proving a priori estimate in H10 , which is uniform in time, is due to nonzero f , with f , ft ∈ L∞ (L2 ). Although a vast amount of literature is devoted to the problem (1.1)-(1.3) (mainly Oskolkov and his pupil, Sobolevskii and recently Lin et.al), to the best of our knowledge, a direct proof of uniform estimate in Dirichlet norm is missing in the literature. Therefore, in the first part of this article, we have discussed a stepby-step proof of this estimate, which is uniform in time, in Section 3. All the regularity results are obtained without the nonlocal compatibility conditions. In [11], authors have considered assumptions (A1)-(A2) and derived error estimate which is optimal in L∞ (H1 ), but suboptimal in L∞ (L2 ) of the velocity. We, in this paper, have improved their results in [11] and obtain optimal error estimate for the

ERROR ESTIMATE FOR THE OLDROYD FLUID OF ORDER ONE

351

velocity in L∞ (L2 )-norm. Under the uniqueness condition, uniform error estimates in time, for both velocity and pressure are also obtained. We have also managed to improve the pressure estimate in the following sense. For u0 ∈ J1 , we obtain kp−ph k ≤ Kht−1/2 , which exhibits similar singular behavior (as t → 0) as that of velocity estimate. In earlier articles (e.g. [28], [11]), this singularity is of order 1, i.e. kp − ph k ≤ Kht−1 . Finally, we would like to comment that, although Lemmas 3.1 and 3.2 yield estimates in Dirichlet norm which is uniform in time, they do not provide us with a concrete bound for the norm. Lemma 3.4 gives us a concrete bound, which in turn will be useful in the study of global attractors for the Oldroyd model of order one. Acknowledgement: The first author would like to express his gratitude to Council of Scientific and Industrial Research (CSIR) for the financial support. The second author acknowledges the support provided by DST (Department of Science and Technology),Government of India vide Project no 08DST012. Both the authors acknowledge the comments and suggestions provided by Neela Nataraj ( IIT Bombay), Jinyun Yuan ( UFPR, Brazil), Y.Lin ( Univ. Alberta, Canada) and E.Suli (Oxford Univ., UK) during the preparation of this manuscript. This publication is also based on work supported in part (AKP) by Award No.KUK-C1-013-04, made by the King Abdullah University of Science and Technology (KAUST). Both authors thank the referee for valuable comments and suggestions which improve our manuscript. References [1] Agranovich, Yu. Ya. and Sobolevskii, P. E. , Investigation of viscoelastic fluid mathematical model, RAC. Ukrainian SSR. Ser. A, 10 (1989), 71-74. [2] Akhmatov, M. M. and Oskolkov, A. P. , On convergent difference schemes for the equations of motion of an Oldroyd fluid, J. Soviet Math., 47 (1989), 2926-2933. [3] Brezzi, F. and Fortin, M. , Mixed and Hybrid finite element methods, Springer-Verlag, New York, 1991. [4] Bercovier, M. and Pironneau, O. , Error estimates for finite element solution of the Stokes problem in the primitive variables, Numer. Math., 33 (1979), 211-224. [5] Cannon, J. R., Ewing, R. E., He, Y. and Lin, Y. , An modified nonlinear Galerkin method for the viscoelastic fluid motion equations, Int. J. Engg. Sci., 37 (1999), 1643-1662. [6] Ciarlet, P. G. , The finite Element Method for Elliptic Problems, North-Holland Publ. Comp., Amsterdam, 1978. [7] Cannon, J. R. and Lin, Y. , A priori L2 error estimates for finite element methods for nonlinear diffusion equations with memory, SIAM J. Numer. Anal. 27 (1990), 595-607. [8] Emel’yanova, D. V. and Oskolkov, A. P. , Certain nonlocal problems for two-dimensional equations of motion of Oldroyd fluids, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 189 (1991), 101-121; English translation in J. Soviet Math. 62 (1992), 30043016. [9] Girault, V. and Raviart, P. A. , Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics No. 749, Springer, New York, 1980. [10] He, Y., Lin, Y, Shen, S. S. P. and Tait, R. , On the convergence of Viscoelastic fluid flows to a steady state, Adv. Diff. Eqns. 7 (2002), 717-742. [11] He, Y., Lin, Y., Shen, S. S. P., Sun, W. and Tait, R. , Finite element approximation for the viscoelastic fluid motion problem, J. Comp. Appl. Math. 155 (2003), 201-222. [12] Heywood, J. G. and Rannacher, R. , 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. [13] Heywood, J. G. and Rannacher, R. , Finite element approximation of the nonstationary Navier-Stokes problem: IV. Error analysis for second order time discretization, SIAM J. Numer. Anal. 27 (1990), 353-384. [14] Joseph, D. D. , Fluid Dynamics of Viscoelastic Liquids, Springer Verlag, New York, 1990.

352

DEEPJYOTI GOSWAMI AND AMIYA K. PANI

[15] Karzeeva, N. A., Kotsiolis, A. A. and Oskolkov, A. P. , On dynamical system generated by initial-boundary value problems for the equations of motion of linear viscoelastic fluids, (Russian) Trudy Mat. Inst. Steklov. 188 (1990), 59-87; English translation in Proc. Steklov Inst. Math. (1991), No. 3, 73-108. [16] Kotsiolis, A. A. and Oskolkov, A. P. , On the solvability of fundamental initial boundary value problem for the motion equations of Oldroyd’s fluid and behavior of solutions, when t 7→ ∞, Notes of Scientific Seminar of LOMI, 150, 6 (1986), 48-52. [17] Kotsiolis, A. A. and Oskolkov, A. P. , Dynamical system generated by the equations of motion of Oldroyd fluids, Zap. Nauch. Semin. LOMI 155 (1986), 136-141; English translation in J. Soviet Math. 41 (1988), pp. 967-970. [18] Kotsiolis, A. A., Oskolkov, A. P. and Shadiev, R. D. , A priori estimate on the semiaxis t ≥ 0 for the solutions of the equations of motion of linear viscoelastic fluids with an infinite Dirichlet integral, and their applications, (Russian. English summary) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. ( LOMI) 182 (1990), 86-101; English translation in J. Soviet Math. 62 (1992), No. 3, 2777-2788. [19] Ladyzenskaya, O. A. , The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. [20] Lin, Y., Thom´ ee, V. and Wahlbin, L. B. , Ritz-Volterra projections to finite element spaces and applications to integro-differential and related equations, SIAM J. Numer. Anal. 28 (1991), 1047-1070. [21] Larsson, S., Thom´ ee, V. and Wahlbin, L. B. , Numerical solution of parabolic integrodifferential equations by discontinuous Galerkin method, Math. Comp. 67, No. 221 (1998), 45-71. [22] McLean, W. and Thom´ ee, V., Numerical solution of an evolution equation with a positive type memory term, J. Austral. Math. Soc. Ser. B 35 (1993), 23-70. [23] Oldroyd, J. G. , Non-Newtonian flow of liquids and solids. Rheology: Theory and Applications, Vol. I (F. R. Eirich, Ed.), AP, New York (1956), 653-682. [24] Oskolkov, A. P. , Initial boundary value problems for the equations of motion of Kelvin-Voigt fluids and Oldroyd fluids, (Russian) Trudy Mat. Inst. Steklov. 179 (1988), 126-164; English translated in Proc. Steklov. Inst. Math. (1989), No. 2, 137-182. [25] Pani, A. K., Thom´ ee, V. and Wahlbin, L. B. , Numerical methods for hyperbolic and parabolic integro-differential equations, J. Integral Equations Appl. 4 (1992), 533-584. [26] Pani, A. K. and Peterson, T. E. , The effect of numerical quadrature on semidiscrete finite element methods for parabolic integro-differential equations, SIAM J. Numer. Anal. 33 (1996), 1084-1105. [27] Pani, A. K. and Sinha R. K. , Error estimates for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data, CALCOLO 37 (2000), 181-205. [28] Pani, A. K. and Yuan, J. Y. , Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model, IMA J. Numer. Anal. 25 (2005), 750-782. [29] Pani, A. K., Yuan, J. Y. and Damazio, P. , On a linearized backward Euler method for the equations of motion arising in the Oldroyd fluids of order one, SIAM J. Numer. Anal. 44 (2006), 804-825. [30] Sobolevskii, P. E. , Stabilization of viscoelastic fluid motion (Oldroyd’s mathematical model), Diff. Integ. Eq., 7(1994), 1597-1612. [31] Temam, R. , Navier-Stokes Equations, Theory and Numerical Analysis, North -Holland , Amsterdam, 1984. [32] Temam, R. , Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences 68, Springer-Verlag, 1988. [33] Thom´ ee, V. and Zhang, N. -Y. , Error estimates for semidiscrete finite element methods for parabolic integro-differential equations, Math. Comp. 53 (1989), 121-139. [34] Wheeler, M. F. , A priori L2 error estimates for Galerkin Approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723-759. Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India. E-mail : [email protected] and [email protected]