On a three level two-grid finite element method for the 2D-transient ...

On a three level two-grid finite element method for the 2D-transient

arXiv:1401.5540v1 [math.NA] 22 Jan 2014

Navier-Stokes equations Saumya Bajpai TIFR Centre for Applicable Mathematics, Post Bag No. 6503, GKVK Post Office, Sharada Nagar, Chikkabommasandra, Bangalore 560065, India, Amiya K. Pani Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India January 23, 2014

Abstract In this paper, an error analysis of a three steps two level Galekin finite element method for the two dimensional transient Navier-Stokes equations is discussed. First of all, the problem is discretized in spatial direction by employing finite element method on a coarse mesh TH with mesh size H. Then, in step two, the nonlinear system is linearized around the coarse grid solution, say, uH , which is similar to Newton’s type iteration and the resulting linear system is solved on a finer mesh Th with mesh size h. In step three, a correction is obtained through solving a linear problem on the finer mesh and an updated final solution is derived. Optimal error estimates in L∞ (L2 )-norm, when h = O(H 2−δ ) and in L∞ (H1 )-norm, when h = O(H 4−δ ) for the velocity and in L∞ (L2 )-norm, when h = O(H 4−δ ) for the pressure are established for arbitrarily small δ > 0. Further, under uniqueness assumption, these estimates are proved to be valid uniformly in time. Then based on backward Euler method, a completely discrete scheme is analyzed and a priori error estimates are derived. Finally, the paper is concluded with some numerical experiments. Keywords: Two-grid method, 2D-Navier-Stokes system, semidiscrete scheme, backward Euler method, optimal error estimates, order of convergence, uniform-in-time estimates, uniqueness assumption, numerical experiments.

1

Introduction

Consider the 2D-transient Navier-Stokes system: ∂u − ν∆u + u · ∇u + ∇p = f (x, t) x ∈ Ω, t > 0, ∂t

(1.1)

and incompressibility condition ∇ · u = 0 x ∈ Ω, t > 0,

1

(1.2)

with initial and boundary conditions u(x, 0) = u0 in Ω,

u = 0 on ∂Ω, t ≥ 0,

(1.3)

where, Ω is a bounded and convex polygonal domain in R2 with boundary ∂Ω and f is given external force. Here, u = u(x, t) is the velocity vector, p = p(x, t) denotes the pressure and ν > 0 is the kinematic coefficient of viscosity. In this article, a three level two-grid finite element Galerkin method for the problem (1.1)-(1.3) is analyzed. The algorithm used here is a suitable modification of the algorithms in [5, 15] and it is composed of the following three steps: • Step 1: solve a nonlinear problem over a coarse mesh with mesh size H which provides an approximate solution, say uH . • Step 2: linearize the nonlinear system around the coarse grid solution uH and solve the resulting linearized problem over a fine mesh with mesh size h and denote its solution as u∗h . • Step 3: correct the solution u∗h obtained in Step 2 over fine mesh which provides an updated final solution uh . As a result of the above mentioned three steps algorithm, the error ku − uh k is of the same order ˜ h k, where u ˜ h is the solution of the standard Galerkin system on a fine mesh h with an as ku − u appropriate scaling between h and H. The two grid method has been extensively studied for Navier-Stokes equations by Layton [18], Layton and Tobiska [15], Layton and Lenferink [16]-[17], Girault and Lions [8, 9], Dai et al. [5], Abboud et al. [3]-[4], Frutos et al. [7]. In [15], Layton et al. have examined a coarse mesh correction in the third step for a steady state Navier-Stokes equations. But, this correction fails to improve the results obtained in Step 2 and as a result, optimal error estimate in L2 -norm for the velocity is obtained when h = O(H 3/2 ). Based on stream function formulation, a two-grid finite element method has been studied by Fairag [6]. All the above results have been discussed for the steady state Navier-Stokes equations on a convex polyhedra or on a convex polygon. Subsequently, Girault et al [8] in their work on steady state Navier-Stokes equations have analyzed a two level two-grid algorithm and have obtained optimal H1 -norm error estimate for the velocity vector with a choice h = O(H 2 ), when the problem is defined on a Lipschitz polyhedron or on a convex polyhedron. The analysis is further extended to the transient Navier-Stokes equations in [9], and optimal error estimate in L∞ (H1 )-norm is established with a choice h = O(H 2 ), when Ω is a Lipschitz polyhedron or a convex polyhedron. In both of these articles, the key approach is to exploit the contribution of the coarse grid solution in L3 (Ω)-norm. In the context of nonlinear Galerkin method, two grid method is applied to the 2D-transient Navier-Stokes equations by Ait Ou Amni and Marion in [1]. They have shown that the nonlinear Galerkin solution has the same accuracy as that of the standard Galerkin solution, both for velocity in H 1 -norm and for pressure in L2 -norm with a choice h = O(H 2 ). Further, they have penalized their two-grid algorithm to get rid of the coupling between velocity and pressure with penalization parameter ǫ and have recovered the same accuracy for the penalized two-grid Galerkin solution as that of the standard Galerkin solution with h = O(H 2 ) and h = O(ǫ1/2 ). Garc´ıa-Archilla and Titi in [2] have applied Post-Processed method to the semilinear scaler elliptic equations in any dimensions and have derived optimal error bounds in H1 -norm for the

2

post-processed solution with a choice h = O(H r+1 |log(H)|)1/r )), where the post-processed solution is approximated by the polynomials of degree r with r ≥ 2. Recently, Frutos et al [7] have applied the two-grid scheme to the incompressible NavierStokes equations using mixed-finite elements, the mini-element, the quadratic and the cubic HoodTaylor elements for spatial discretization and a backward Euler method and a two step backward difference scheme for time discretization and have derived the rate of convergence of the fine mesh in the H1 -norm by taking h = O(H 2 ), which is an improvement over h = O(H 3/2 ) obtained in [4]. In [20], a fully discrete two-level method consisting of Crank-Nicolson extrapolation method with solution (uH,τ0 , pH,τ0 ) on a space-time coarse grid JH,τ0 and a backward Euler method with solution (uh,τ , ph,τ ) on a space-time fine grid Jh,τ is discussed. They have obtained convergence rate for the two level solution (uh,τ , ph,τ ), which is of same order as that of the one level standard 3/2 Crank-Nicolson extrapolation solution if τ0 + H 3/2 = O(τ ) for t ∈ [0, 1] and τ02 + H 2 = O(τ ) for t ∈ [1, T ]. An attempt has been made in this article to discuss optimal error estimates in L∞ (L2 ) and ∞ L (H1 )-norms for the velocity and L∞ (L2 )-norm for the pressure using a three level two-grid finite element method for the 2D-transient Navier-Stokes equations. The major contributions are given in terms of the following two tables. In Table 1, we present the order of convergences for the two-grid algorithm (3.3)-(3.5) stated in Section 3 for the pair of finite element spaces (Hµ , Lµ ), µ = H, h satisfying the approximation properties mentioned in (B1)-(B2). Table 2 provides the largest scaling between coarse and fine meshes for which the desired fine mesh accuracy is obtained for both velocity and pressure.

Solution

Velocity in L2 -norm

Velocity in H1 -norm

Pressure in L2 -norm

(u − uH , p − pH )

H2

H

H

(u −

u∗h , p

−

p∗h )

(u − uh , p − ph )

h2

+ H 3−δ

h2 + H 4−2δ

H 3−δ

h + H 3−δ

h + H 4−δ

h + H 4−δ

h+

Table 1: Error estimates obtained from the two-grid algorithm for arbitrarily small δ > 0. Solution

Velocity in L2 -norm

Velocity in H1 -norm

Pressure in L2 -norm

(u − uH , p − pH )

H2

H

H

(u −

u∗h , p

− p∗h )

(u − uh , p − ph )

h∼

H (3−δ)/2

h ∼ H 2−δ

H 3−δ

h ∼ H 3−δ

h ∼ H 4−δ

h ∼ H 4−δ

h∼

Table 2: The largest scaling for optimal error estimates. It is observed from Tables 1 and 2 that the introduction of Step 3 leads to a good improvement in scaling between H and h for both H1 -norm for the velocity and L2 -norm for the pressure, that is, the scaling improves from h ∼ H 3−δ to h ∼ H 4−δ . It also improves the scaling for L2 -norm of the velocity from Step 2 to Step 3 from h ∼ H (3−δ)/2 to h ∼ H (2−δ) for arbitrarily small δ > 0. The main contributions of this paper can be summarized as follows:

3

(i) Based on the steady state Oseen projection and Sobolev estimates in Lemma 3.1 involving δ, optimal error estimates for the two-grid Galerkin approximations to the velocity in L∞ (H1 )norm and to the pressure in L∞ (L2 )-norm with the largest scaling between H and h, h ∼ H 3−δ and h ∼ H 4−δ for Step 2 and Step 3, respectively are derived. The result obtained in Step 2 is an improvement over the result obtained by Frutos et al [7]. They have obtained using first order mini-elements the largest scaling between H and h, as h ∼ H 2 for both L∞ (H1 )-norm for the velocity and L∞ (L2 )-norm for the pressure. (ii) A use of linearized backward Oseen problem with related estimates yields optimal L∞ (L2 )norm estimates for the velocity in Step 2 with a choice h = O(H (3−δ)/2 ) and Step 3 with a choice h = O(H 2−δ ) for δ > 0 arbitrarily small. (iii) Under the assumption of uniqueness condition, a priori error estimates are obtained which hold uniformly in time. The remaining part of the paper consists of the following sections. In Section 2, some preliminaries to be used in the subsequent sections are presented. In Section 3, semidiscrete two-grid finite element approximations are introduced. Optimal error estimates for velocity and pressure are established in Section 4. Section 5 deals with the backward Euler method applied to the semidiscrete two grid system. Finally, in Section 6, the results of some numerical examples which confirm our theoretical results are presented.

2

Preliminaries

We denote R2 -valued function spaces using bold face letters, that is, H10 = (H01 (Ω))2 , L2 = (L2 (Ω))2 and Hm = (H m (Ω))2 . The standard notations for Lebesgue and Sobolev spaces with their norms are employed in the paper. The space H10 is equipped with a norm k∇vk = P 1/2 2 (∂ v , ∂ v ) . Given a Banach space X endowed with norm k · kX , let Lp (0, T ; X) j i j i i,j=1 Z T be the space of all strongly measurable functions φ : [0, T ] → X satisfying k φ(s) kpX ds < ∞ 0

and for p = ∞, ess sup k φ(t) kX < ∞. Also, define t∈[0,T ]

J = J1

=

{φ ∈ L2 : ∇ · φ = 0 in Ω, φ · n|∂Ω = 0 holds weakly} {φ ∈ H10 : ∇ · φ = 0},

where n is the unit 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 [21]. Let H m /IR be the quotient space with norm kφkH m /IR = inf c∈IR kφ + ckm . For m = 0, it is denoted by L2 /IR. Throughout this paper, we make the following assumptions: (A1). For g ∈ L2 , let (v, q) ∈ J1 × L2 /IR be the unique solution to the steady state Stokes problem −∆v + ∇q = g, ∇ · v = 0 in Ω,

v|∂Ω = 0

satisfying the regularity result [21]: kvk2 + kqkH 1 /IR ≤ Ckgk.

4

It is easy to show that kvk2 ≤ λ1−1 k∇vk2 ∀v ∈ H10 (Ω),

(2.4)

where λ1 is the minimum eigenvalue of the Laplacian with zero Dirichlet boundary condition. (A2). There exists a positive constant M0 such that the initial velocity u0 and external force f satisfy for t ∈ (0, T ] with 0 < T < ∞ u0 ∈ J1 ∩ H2 , f , ft ∈ L∞ (0, T ; L2 ) with ku0 k2 ≤ M0 , ess sup {kf (·, t)k, kft (·, t)k} ≤ M0 . 0 0 ) (ut , φ) + ν a(u, φ) + b(u, u, φ) − (p, ∇ · φ) = (f , φ) ∀φ ∈ H10 , (2.5) (∇ · u, χ) = 0 ∀χ ∈ L2 . Equivalently, find u(t) ∈ J1 such that for u(0) = u0 , t > 0, (ut , φ) + ν a(u, φ) + b(u, u, φ) = (f , φ) ∀φ ∈ J1 .

(2.6)

We recall below, the following regularity results. Lemma 2.1. [10, pp. 285, 302] Let the assumptions (A1)-(A2) hold true. Then, for any T with 0 < T < ∞, for any fixed α > 0 and for some constant C = C(M0 ), the solution of (2.5) satisfies sup {ku(t)k2 + kut (t)k + kp(t)kH1 /R } ≤ C,

0 0 (uht , φh ) + νa(uh , φh ) + b(uh , uH , φh ) + b(uH , uh , φh ) = (f , φh ) + b(uH , u∗h , φh ) + b(u∗h , uH − u∗h , φh ).

6

(3.5)

The following inequality will be used frequently in our error analysis: inf sup φh ∈Jh vh ∈Jh

ν a(φh , vh ) + b(uH , φh , vh ) + b(φh , uH , vh ) ≥ γ > 0. k∇φh kk∇vh k

(3.6)

For a proof, see [15]. For uniform estimates in time, we shall further assume the following uniqueness condition: N b(u, v, w) kf kL∞ (0,∞;L2 ) < 1 and N = sup , 1 ν2 k∇ukk∇vkk∇wk u,v,w∈H0 (Ω)

(3.7)

The main results of this section are stated in the following theorems. Theorem 3.1. Let Ω be a convex polygon and let assumptions (A1)-(A2) and (B1)-(B2) hold true. Further, let the discrete initial velocity u0h ∈ Jh with u0h = Ph u0 . Then, there exists a positive constant C, independent of h, such that for t ∈ (0, T ] with 0 < T < ∞, the following estimates hold true:

and


k(u − uh )(t)k ≤ K(t)(h2 + H 4−2δ , k∇(u − uh )(t)k ≤ K(t)(h + H 4−δ ),

(3.8)


k(p − ph )(t)k ≤ K(t)(h + H 4−δ ,

(3.9)

where δ > 0 is arbitrarily small and K(t) = CeCt . Under uniqueness condition (3.7), K(t) = C and the estimates in Theorem 3.1 are valid uniformly in time. The remaining part of this paper is devoted to the derivation of results, which will lead to the proof of Theorem 3.1.

4

Error Estimates

This section deals with optimal error estimates of the semidiscrete two-grid algorithm. Since Jh is not a subspace of J1 , the weak solution u satisfies (ut , φh ) + ν a(u, φh ) + b(u, u, φh ) = (f , φh ) + (p, ∇ · φh ) ∀φh ∈ Jh .

(4.10)

Define eH := u − uH , e∗ := u − u∗h and eh := u − uh . Then, a use of (3.4) and (4.10) yields (e∗t , φh ) + ν a(e∗ , φh ) + b(e∗ , uH , φh ) + b(uH , e∗ , φh ) = −b(eH , eH , φh ) + (p, ∇ · φh )

∀φh ∈ Jh .

(4.11)

Subtract (3.5) from (3.4) and then add the resulting equation to (4.11) to arrive at (eht , φh ) + ν a(eh , φh ) + b(eh , uH , φh ) + b(uH , eh , φh ) = −b(eH , e∗ , φh ) − b(e∗ , eH , φh ) + b(e∗ , e∗ , φh ) + (p, ∇ · φh )

∀φh ∈ Jh .

(4.12)

For analyzing optimal error estimates of eh in L∞ (L2 ) and L∞ (H1 )-norms, define an auxiliary ˜ h (t) ∈ Jh , 0 < t ≤ T, for a given u, as a solution of the following modified steady projection u state Oseen problem: ˜ h , φh ) + b(uH , u − u ˜ h , φh ) + b(u − u ˜ h , uH , φh ) = (p, ∇ · φh ) for all φh ∈ Jh . (4.13) ν a(u − u

7

Now split eh as ˜ h ) + (˜ eh =: (u − u uh − uh ) := ζ + Θ,

(4.14)

˜ h and Θ := u ˜ h − uh . where ζ := u − u A use of (4.12)-(4.14) leads to (Θt , φh ) + ν a(Θ, φh ) + b(Θ, uH , φh ) + b(uH , Θ, φh ) = −(ζ t , φh ) − b(eH , e∗ , φh ) − b(e∗ , eH , φh ) + b(e∗ , e∗ , φh ).

(4.15)

To seek estimates for Θ, we need estimates for ζ, eH and e∗ , which appear on the right hand side of (4.15). Lemma 4.1. [11, see page 362, proposition 3.2] Let uH (t) be the solution of (3.3) in [0, T ), 0 < T < ∞ satisfying u0H = PH u0 , and let the assumptions (A1)-(A2) hold true. Then, there exists a positive constant C = C(γ, ν, α, λ1 , M0 ) such that the following hold true for all t > 0 Z t ˜ H (s)k2 + k∇uHt (s)k2 )ds ≤ C, sup τ (t)kuHt (t)k21 ≤ C. kuH (t)k2 + kuHt (t)k + e−2αt e2αs (k∆u 0 0 is arbitrarily small and K(t) = CeCt . If, in addition, uniqueness condition (3.7) holds true, then K(t) = K, that is, estimates are bounded uniformly with respect to time. Proof. Since e∗ = ζ + ρ and estimates of ζ are known from Lemma 4.3, it is enough to derive estimates of ρ. A use of (4.11) with (4.13) and (4.16) leads to (ρt , φh ) + ν a(ρ, φh ) + b(uH , ρ, φh ) + b(ρ, uH , φh ) = −(ζ t , φh ) + b(eH , eH , φh ) for all φh ∈ Jh .

(4.39)

Multiplying (4.39) by σ(t), substitute φh = ρ and use (3.6) to arrive at 1 d 1 (σkρk2 ) + γσk∇ρk2 ≤ −σ (ζ t , ρ) + σ b(eH , eH , ρ) + σt kρk2 , 2 dt 2 where σ(t) = min{t, 1} e2αt . Integrate with respect to time from 0 to t and obtain Z t Z t σ(τ )(ζ τ (τ ), ρ)dτ σ(τ )k∇ρ(τ )k2 dτ ≤ − σ(t)kρ(t)k2 + γ 0 0 Z Z t 1 t στ (τ )kρ(τ )k2 dτ. + σ(τ ) b(eH (τ ), eH (τ ), ρ)dτ + 2 0 0

(4.40)

Using the Cauchy-Schwarz inequality and Lemma 4.3, the first term on the right hand side of (4.40) can be bounded as Z t σ(τ )(ζ τ (τ ), ρ)dτ ≤ σ(τ )kζ τ (τ )kkρkdτ 0 0 Z t σ(τ )h(h + keH (τ )k)(K(τ )k∇uHτ (τ )k + Kτ (τ ))kρkdτ. ≤

Z

t

(4.41)

0

A use of The Young’s inequality with estimates of k∇uHt k, Lemmas 2.1 and 4.2 in (4.41) leads to  Z t Z t e2ατ K2 (τ )dτ σ(τ )(ζ τ (τ ), ρ)dτ ≤ C(λ1 , ǫ)h2 (h2 + keH k2L∞ (L2 ) ) sup (τ (t)k∇uHt k2 ) 0 : φh ∈ Hh , φh 6= 0 . k∇φh k

14

(4.52)

Since the estimate of k∇e∗ k is known from Lemma 4.5, we now derive estimate of ke∗t k−1;h . As Hh ⊂ H10 , we note that   < e∗t , φh > ∗ : φh ∈ Hh , φh 6= 0 ket k−1;h = sup k∇φh k   < e∗t , φ > 1 ≤ sup : φ ∈ H0 , φ 6= 0 = ke∗t k−1 . (4.53) k∇φk Lemma 4.7. The error e∗ = u − u∗h satisfies for 0 < t < T ke∗t k−1 ≤ K(t) (h + H 3−δ ).

(4.54)

Proof. For any Ψ ∈ H10 , use orthogogal projection Ph : L2 → Jh and (4.11) with φ = Ph Ψ to obtain (e∗t , Ψ) = (e∗t , Ψ − Ph Ψ) + (e∗t , Ph Ψ) = (e∗t , Ψ − Ph Ψ) − ν a(e∗ , Ph Ψ) − b(e∗ , uH , Ph Ψ) − b(uH , e∗ , Ph Ψ) + b(eH , eH , Ph Ψ) + (p, ∇ · Ph Ψ).

(4.55)

Apply approximation property of Ph to find that (e∗t , Ψ − Ph Ψ) = (ut , Ψ − Ph Ψ) ≤ Chkut kk∇Ph Ψk.

(4.56)

A use of the Cauchy-Schwarz’s inequality with (4.50), (4.56), the discrete incompressibility condition and (4.53) in (4.55) leads to ke∗t k−1 ≤ C(hkut k + k∇e∗ k + k∇eH kk∇e∗ k + keH k1−δ k∇eH k1+δ + hk∇pk). Using Lemmas 2.1, 4.2 and 4.5, we arrive at the desired result. Thus, we have proved the following theorem.

(4.57) 

Theorem 4.2. Under the hypotheses of Lemma 4.2, there exists a positive constant C depending on ν, γ, M0 , such that, for all t > 0, it holds: k(p − p∗h )(t)kL2 /Nh ≤ K(t)(h + H 3−δ ), where δ > 0 is arbitrarily small and K(t) = CeCt . Under the uniqueness condition (3.7), K(t) = C and the estimate is uniform in time.

4.2

Error estimates for Step 3

This section is devoted to the derivation of semidiscrete error estimates in Step 3. Lemma 4.8. Under the hypotheses of Lemma 4.4, the following estimate holds true: Z t   τ (t)ke∗t k2 + β σ −1 (t) σ1 (τ )k∇e∗t (τ )k2 ≤ K(t) h2 + H 6−2δ , 0

where σ1 (t) = τ 2 (t)e2αt and β = γ − αλ−1 1 > 0.

15

Proof. Differentiate (4.11) with respect to time and substitute φh = Ph e∗t in the resulting equation and use discrete incompressibility condition to arrive at 1 d ∗ 2 ke k + γk∇e∗t k2 ≤ (e∗tt , ut − Ph ut ) + ν a(e∗t , ut − Ph ut ) − b(e∗t , uH , ut − Ph ut ) 2 dt t − b(uH , e∗t , ut − Ph ut ) − b(e∗ , uHt , Ph e∗t ) − b(uHt , e∗ , Ph e∗t ) + b(eHt , eH , Ph e∗t ) + b(eHt , eH , Ph e∗t ) + (pt − jh pt , ∇ · Ph e∗t ).

(4.58)

A use of (3.1)with Lemma 3.1 and Theorem 4.2 yields |b(eHt , eH , Ph e∗t ) +b(eH , eHt , Ph e∗t )| ≤ CkeH k1−δ k∇eH kδ k∇eHt kk∇Ph e∗t k ≤ C(ǫ) keH k2(1−δ) k∇eH k2δ k∇eHt k2 + ǫk∇e∗t k2 .

(4.59)

Apply (4.19)-(4.22) and (4.59) along with Cauchy-Schwarz’s inequality in (4.58) to arrive at  1 d 1 d ∗ 2 ˜ H kk∆u ˜ t k2 ket k + γk∇e∗t k2 ≤ kut − Ph ut k2 + C(ν, γ) h2 Kt2 + h2 k∇uH kk∆u 2 dt 2 dt 
(4.60) +K(t) (h2 + H 6−2δ )k∇uHt k2 + H 4−2δ k∇eHt k2 .

Use similar analysis to (4.60) as applied to (4.24) to arrive at (4.25) and Lemmas 2.1, 4.2, 4.5 to conclude the proof.  Lemma 4.9. Under the hypotheses of Lemma 4.4, the following estimate holds true: Z t −1 σ (t) e2ατ k∇eh (τ )k2 dτ ≤ K(t)(h2 + H 10−4δ ). 0

Proof. Consider (4.12) with φh = Ph eh = (Ph u − u) + eh . Then, use (3.6) to arrive at 1 d keh k2 + γ k∇eh k2 ≤ (eht , u − Ph u) + ν a(eh , u − Ph u) + b(uH , eh , u − Ph u) 2 dt + b(eh , uH , u − Ph u) − b(eH , e∗ , Ph eh ) − b(e∗ , eH , Ph eh ) + b(e∗ , e∗ , Ph eh ) + (p, ∇ · Ph eh ).

(4.61)

The first four terms in the right hand side of (4.61) can be bounded using (4.19)-(4.21) (with e∗ replaced by eh ). Now, from (3.1) and Lemma 3.1, we arrive at |b(eH , e∗ , Ph eh ) − b(e∗ , eH , Ph eh ) + b(e∗ , e∗ , Ph eh )|
≤ C keH k1−δ k∇eH kδ k∇e∗ k + k∇e∗ k2 k∇eh k.

(4.62)

A use of (4.19)-(4.22) and (4.62) leads to

1 d 1 d keh k2 + γk∇eh k2 ≤ ku − Ph uk2 + C(ν)hKk∇eh k 2 dt 2 dt + C(ν)(keH k1−δ k∇eH kδ k∇e∗ k + k∇e∗ k2 )k∇eh k.

(4.63)

The proof can be concluded by using the similar set of arguments now to (4.63) as applied to (4.24) leading to (4.25) and Lemmas 4.2, 4.4 and 4.5. This completes the proof of Lemma 4.9.  Below, we state a lemma that provides L∞ (H1 )-norm estimate for eh . The proof is obtained in the same lines as the proof of Lemma 4.5, starting with φh = Ph eht in (4.12), using Lemmas 4.8 and 4.9 and hence, is skipped.

16

Lemma 4.10. Under the hypotheses of Lemma 4.4, the following estimate holds true: Z t −1 σ (t) σ(τ )kehτ (τ )k2 dτ + k∇eh (t)k ≤ K(t) (h2 + H 8−2δ ), 0

where δ > 0 is arbitrarily small.



Lemma 4.11. Under the hypotheses of Theorem 4.2, the following is satisfied: Z t −1 σ (t) e2ατ keh (τ )k2 dτ ≤ K(t)(h4 + h2 H 4 ). 0

Proof. Proceeding in a similar way as in the proof of Lemma 4.6, we arrive at keh k2

=

d (eh , Ph φ) − (u − Ph u, φt ) − a(eh , φ − Ph φ) − b(uH , eh , φ − Ph φ) dt −b(eh , uH , φ − Ph φ) − b(eH , eh , φ) − b(eh , eH , φ) + b(eH , e∗ , Ph φ − φ) +b(eH , e∗ , φ) + b(e∗ , eH , Ph φ − φ) + b(e∗ , eH , φ) − b(e∗ , e∗ , Ph φ − φ) −b(e∗ , e∗ , φ) − (p − jh p, ∇ · (Ph φ − φ)) + (ψ − jh ψ, ∇ · eh ).

(4.64)

Use (3.1), Lemmas 3.1 and 4.2 to bound |b(eH , e∗ , Ph φ − φ)| + |b(e∗ , eH , Ph φ − φ)| + |b(eH , e∗ , φ)| + |b(eH , e∗ , φ)| ≤ C(hkeH k1−δ k∇eH kδ + keH k)k∇e∗ kkφk2 ≤ K(t)(hH 2−δ + H 2 )k∇e∗ kkφk2 .

(4.65)

An application of (3.1), Lemmas 3.1 and 4.5 yields |b(e∗ , e∗ , Ph φ − φ)| + |b(e∗ , e∗ , φ)| ≤ C(hk∇e∗ k2 + ke∗ kk∇e∗ k)kφk2 ≤ K(t)(h2 + H 3−δ )k∇e∗ kkφk2 .

(4.66)

Multiply (4.64) by e2αt and integrate with respect to time from 0 to t. Then, apply (4.34)-(4.36) with e∗ replaced by eh and (4.65)-(4.66) to obtain Z t Z t Z t
e2ατ keh (τ )k2 dτ ≤ C h4 e2ατ k∇eh (τ )k2 (h2 + keH (τ )k2 )dτ e2ατ K2 (τ )dτ + 0 0 0 Z t +K(t)(h2 + H 4−2δ )eCt e2ατ k∇e∗ (τ )k2 dτ 0 Z t Z t
4 2ατ 2 ≤ C h e2ατ k∇eh (τ )k2 dτ e K (τ )dτ + (h2 + keH k2L∞ (L2 ) ) 0 0 Z t 2 4−2δ 2ατ ∗ 2 +K(t)(h + H ) e k∇e (τ )k dτ. (4.67) 0

A use of Lemmas 2.1, 4.2, 4.4 and 4.9 completes the rest part of the proof. Proceeding in a similar way as in Lemma 4.7, we arrive at the following estimate.



Lemma 4.12. The error eh = u − uh satisfies for 0 < t < T keht k−1 ≤ K(t)(h + H 4−δ ). For the pressure error estimates corresponding to the correction in Step 3 of two-grid algorithm, consider the equivalent form of (3.5): seek (uh (t), ph (t)) ∈ Hh × Lh such that uh (0) = u0h and for t > 0,   (uht , φh ) + νa(uh , φh ) + b(uh , uH , φh ) + b(uH , uh , φh ) = (f , φh )  (4.68) +b(uH , u∗h , φh ) + b(u∗h , uH − u∗h , φh ) + (ph , ∇ · φh ) ∀φh ∈ Hh ,   (∇ · uh , χh ) = 0 ∀χh ∈ Lh .

17

For pressure equation, subtract (4.68) from (4.10) to obtain (p − ph , ∇ · φh ) = (eht , φh ) + νa(eh , φh ) + b(eh , uH , φh ) + b(uH , eh , φh ) − b(eH , e∗ , φh ) − b(e∗ , eH , φh ) − b(e∗ , e∗ , φh ) ∀φh ∈ Hh .

(4.69)

Armed with these estimates, next we derive proof of main Theorem 3.1. Proof of Theorem 3.1. Multiply (4.15) by σ(t), substitute φh = Θ and integrate the resulting equation from 0 to t to obtain Z t Z t Z t σ(t)kΘ(t)k2 + γ σ(τ )k∇Θ(τ )k2 dτ ≤ − σ(τ ) (ζ τ (τ ), Θ)dτ + στ (τ )kΘ(τ )k2 dτ 0 0 0 Z t σ(τ ) (−b(eH (τ ), e∗ (τ ), Θ) − b(e∗ (τ ), eH (τ ), Θ) + b(e∗ (τ ), e∗ (τ ), Θ)) dτ. (4.70) + 0

The first term on the right hand side of (4.70) can be tackled as in (4.41). Write Θ = eh − ζ and use Lemmas 2.1, 4.3 to obtain Z t Z t 2 στ (τ )kΘ(τ )k dτ ≤ e2ατ (keh (τ )k2 + kζ(τ )k2 )dτ ≤ K(t)(h4 + h2 H 4−2δ )σ. (4.71) 0

0

Use Young’s inequality and Lemmas 3.1, 4.2, 4.4 to bound Z t σ(τ )(b(eH (τ ), e∗ (τ ), Θ) + b(e∗ (τ ), eH (τ ), Θ))dτ 0 Z t Z t 2−2δ 2δ ≤ C(ǫ)keH kL σ(τ )k∇e∗ (τ )k2 dτ + ǫ σ(τ )k∇Θ(τ )k2 dτ ∞ (L2 ) k∇eH kL∞ (L2 ) 0 0 Z t σ(τ )k∇Θ(τ )k2 dτ. (4.72) ≤ K(t)(h2 H 4−2δ + H 10−4δ )σ + ǫ 0

An application of Lemmas 3.1, 4.4 and 4.5 leads to Z t Z t ∗ ∗ σ(τ )b(e (τ ), e (τ ), Θ)dτ ≤ C σ(τ )k∇e∗ (τ )k2 k∇Θkdτ 0 0 Z t Z t ∗ 2 ∗ 2 ≤ Ck∇e kL∞ (L2 ) σ(τ )k∇e (τ )k dτ + ǫ σ(τ )k∇Θk2 dτ 0 0 Z t 4 12−4δ ≤ K(t)(h + H )σ + ǫ σ(τ )k∇Θk2 dτ.

(4.73)

0

Apply Lemma 4.2 to obtain

kΘ(t)k2 + σ −1 (t)

Z

t

σ(τ )k∇Θ(τ )k2 dτ ≤ K(t)(h4 + h2 H 4−2δ + H 10−2δ ).

(4.74)

0

A use of Lemmas 4.3, 4.10 with (4.74) completes the proof of Theorem 3.1. The uniform estimates in Theorem 3.1 can be achieved by using Lemma 4.2 under uniqueness condition. For the pressure estimate (3.9), a use of boundedness of k∇uH k and Lemmas 3.1, 4.2, 4.5, 4.10 leads to |b(eh , uH , φh ) + b(uH , eh , φh ) − b(eH , e∗ , φh ) − b(e∗ , eH , φh ) − b(e∗ , e∗ , φh )| ≤ C(k∇eh kk∇uH k + k∇eH kk∇e∗ k + k∇e∗ k2 )k∇φh k.

(4.75)

A use of Cauchy-Schwarz’s inequality and (4.75) in (4.69) leads to
(p − ph , ∇ · φh ) ≤ C keht k−1 + k∇eh k + k∇eh kk∇uH k + k∇eH kk∇e∗ k + k∇e∗ k2 k∇φh k.

A use of Lemmas 4.2, 4.10, 4.12 completes the proof of the pressure estimate (3.9) and this concludes the rest of the proof of Theorem 3.1. 

18

5

Backward Euler Method

For a complete discretization, we apply a backward Euler method for the time discretization. Let {tn }N n=0 be a uniform partition of the time interval [0, T ] and tn = nk, with time step k > 0. For
a sequence {φn }n≥0 ∈ Jh defined on [0, T ], set φn = φ(tn ), ∂¯t φn = φn − φn−1 /k. The backward Euler method applied to (3.3)-(3.5) is stated in terms of the following algorithm: Step 1. Solve the nonlinear system (1.1) on TH : find UnH ∈ JH , such that for all φH ∈ JH for U0H = PH u0 and t > 0 (∂¯t UnH , φH ) + ν a(UnH , φH ) + b(UnH , UnH , φH ) = (f n , φH ).

(5.1)

Step 2. Update on Th with one Newton iteration: find Un ∈ Jh , such that for all φh ∈ Jh for U0 = Ph u0 and t > 0 (∂¯t Un , φh ) + ν a(Un , φh ) + b(Un , UnH , φh ) + b(UnH , Un , φh ) = (f n , φh ) + b(UnH , UnH , φh ).

(5.2)

Step 3. Correct on Th : find Unh ∈ Jh such that, for all φh ∈ Jh for U0h = Ph u0 and t > 0 (∂¯t Unh , φh ) + ν a(Unh , φh ) + b(Unh , UnH , φh ) + b(UnH , Unh , φh ) = (f n , φh ) + b(UnH , Un , φh ) + b(Un , UnH − Un , φh ).

(5.3)

The results in Lemmas 5.1-5.6 will play an important role in the derivation of error estimates in this section. Lemma 5.1. Let u∗h be the solution of (3.4) on some interval [0, T ), 0 < T < ∞ satisfying u∗0h = γλ1 Ph u0 . Then, there exists a positive constant C = C(γ, ν, α, λ1 , M0 ), such that for 0 ≤ α < 2 for all t > 0, the following holds true: Z t ˜ ∗h (s)k2 )ds ≤ C. ku∗h (t)k2 + k∇u∗h (t)k2 + e−2αt e2αs (k∇u∗h (s)k2 + k∆u 0

αt

ˆ ∗h = eαt u∗h . Substitute φh = u ˆ ∗h and use Proof. Multiply (3.4) by e for some α > 0 and set u (2.4) (kˆ u∗h k2 ≤ λ−1 u∗h k2 ) and (3.6) to obtain 1 k∇ˆ   α 1 d ∗ 2 ˆ ∗h ) + e−αt b(ˆ ˆH , u ˆ ∗h ). k∇ˆ u∗h k2 ≤ (ˆf , u uH , u (5.4) kˆ u k + γ− 2 dt h λ1 An application of Cauchy-Schwarz’s inequality and Young’s inequality leads to ˆ ∗h )| ≤ C(λ1 , ǫ)kˆf k2 + ǫk∇ˆ |(ˆf , u u∗h k2 .

(5.5)

A use of Lemma 3.1, k∇uH k ≤ C and Young’s inequality yields ˆH , u ˆ ∗h )| ≤ Ce−αt k∇ˆ |e−αt b(ˆ uH , u uH k2 k∇ˆ u∗h k ≤ C(ǫ)k∇ˆ uH k2 + ǫk∇ˆ u∗h k2 .

(5.6)

Apply (5.5)-(5.6) in (5.4) with ǫ = γ2 and integrate the resulting equation with respect to time to obtain Z t  Z t  2α k∇ˆ u∗h (s)k2 ds ≤ ku0 k2 + C k∇ˆ uH (s)k2 + kˆf(s)k2 ds. kˆ u∗h (t)k2 + γ − λ1 0 0

19

Multiply above equation by e−2αt , use assumption (A2) and the fact that e−2αt

Z

t

e2αs ds =

0

1 (1 − e−2αt ) to arrive at 2α ku∗h (t)k2

+e

−2αt

Z

0

t

e2αs k∇u∗h (s)k2 ds ≤ C.

(5.7)

Next, multiply (3.4) by eαt and rewrite it as ˜ hu ˆ ∗h , φh ) = α(ˆ ˆ ∗h , φh ) u∗h , φh ) − e−αt b(ˆ uH , u (ˆ u∗ht , φh ) − νa(∆ ˆ H , φ ) − b(ˆ ˆ H , φ )) + (ˆf , φ ). − e−αt (b(ˆ u∗ , u uH , u h

h

h

h

(5.8)

˜ u∗ ) = 1 d k∇ˆ ˜ hu ˆ ∗h in (5.8), note the fact that −(ˆ u∗ht , ∆ˆ Substitute φh = −∆ u∗h k2 and integrate the h 2 dt resulting equation with respect to time to obtain Z t Z t Z t ∗ ∗ 2 ∗ 2 ∗ 2 ∗ ˜ ˜ hu ˜ ˆ ∗h )ds ˆ h )ds + 2 e−αs b(ˆ ˆ ∗h , ∆ uH , u k∇ˆ uh (t)k + 2ν k∆ˆ uh (s)k ds = k∇u0h k − 2α (ˆ uh , ∆h u 0 0 0 Z t Z t   ∗ ∗ ∗ −αs ˜ hu ˜ ˜ ˆ ∗h )ds. ˆ h ) ds − 2 (ˆf , ∆ ˆ h ) − b(ˆ ˆ H , ∆h u ˆ H , ∆h u (5.9) uH , u b(ˆ uh , u +2 e 0

0

An application of Lemmas 3.1, 4.1 with Young’s inequality yields Z t   ˜ hu ˜ hu ˜ hu ˆ ∗h )| ds ˆ ∗h )| + |b(ˆ ˆH , ∆ ˆ ∗h )| + |b(ˆ ˆH, ∆ ˆ ∗h , ∆ uH , u u∗h , u uH , u 2 e−αs |b(ˆ 0 Z t Z t 2 2 ∗ −2αs ˜ hu ˜ ˆ ∗h (s)k2 ds. ˆ H (s)k + k∇ˆ uh (s)k )ds + ǫ k∆ (k∆H u ≤ C(ǫ) e

(5.10)

0

0

A use of (5.10) along with Cauchy-Schwarz’s inequality leads to k∇ˆ u∗h (t)k2

+ν

Z

t 0

˜ hu ˆ ∗h (s)k2 ds ≤ k∇u∗0h k2 + C k∆

Z

0

t

˜ Hu ˆ H (s)k2 )ds. (kˆf (s)k2 + k∇ˆ u∗h (s)k2 + k∆

An application of (5.7), assumption (A2) and Lemma 4.1 completes the proof.



Lemma 5.2. Under the assumption of Lemma 5.1, the following holds true: Z t −2αt e e2αs (ku∗ht (s)k2 + ku∗htt (s)k2−1 )ds ≤ C. 0

Proof. Substitute φh = e2αt u∗ht in (3.4) and write it as ˜ h u∗ , u∗ ) − e2αt (b(uH , u∗ , u∗ ) + b(u∗ , uH , u∗ )) e2αt ku∗ht k2 = ν e2αt (∆ h ht h ht h ht + e2αt b(uH , uH , u∗ht ) + e2αt (f , u∗ht ).

(5.11)

Apply Lemmas 3.1, 4.1, 5.1 with Cauchy-Schwarz’s inequality and Young’s inequality to obtain ˜ H uH k2 + k∆u ˜ ∗h k2 + kf k2 ). e2αt ku∗ht k2 ≤ Ce2αt (k∆

(5.12)

Integrate (5.12) with respect to time and use Lemmas 4.1 and 5.1, assumption (A2) to arrive at e

−2αt

Z

0

t

e2αs ku∗ht (s)k2 ds ≤ C.

20

(5.13)

Next, differentiate (3.4) with respect to time and obtain (u∗htt , φh ) + ν a(u∗ht , φh ) + b(uH , u∗ht , φh ) + b(u∗ht , uH , φh ) = (ft , φh )

− b(uHt , u∗h , φh ) + b(u∗h , uHt , φh ) + b(uH , uHt , φh ) + b(uHt , uH , φh ) .

(5.14)

Substitute φh = e2αt u∗ht in (5.14) and use (3.6) to obtain

e2αt d ∗ 2 ku k + γe2αt k∇u∗ht k2 ≤ e2αt (ft , u∗ht ) + I, say. 2 dt ht

(5.15)

A use of Lemma 3.1 yields |I| ≤ e2αt |b(uHt , u∗h , u∗ht )| + |b(u∗h , uHt , u∗ht )| + |b(uHt , uH , u∗ht )| + |b(uH , uHt , u∗ht )| ˜ h u∗ k + k ∆ ˜ H uH k)k∇u∗ k. ≤ Ce2αt kuHt k(k∆ (5.16) h ht Apply (2.4), (5.16), Cauchy-Schwarz’s inequality and Young’s inequality in (5.15) to arrive at     2α 2αt d 2αt ∗ 2 ˜ h u∗ k 2 + k ∆ ˜ H uH k 2 ) . e k∇u∗ht k2 ≤ Ce2αt kft k2 + kuHt k2 (k∆ e kuht k + γ − h dt λ1 An integration with respect to time, a use of assumption (A2), (5.12) and Lemmas 4.1, 5.1 leads to Z t ∗ 2 −2αt kuht (t)k + e e2αs k∇u∗ht (s)k2 ds ≤ C. (5.17) 0

Next, choose φh =

˜ −1 u∗ −e2αt ∆ htt h

in (5.14) and use Lemma 3.1 to arrive at

˜ −1 u∗ )| + |b(uH , uHt , ∆ ˜ −1 u∗ )| ≤ Ck∇uH kk∇uHt kku∗ k−1 , |b(uHt , uH , ∆ htt htt htt h h −1 ∗ ∗ ˜ −1 ∗ ∗ ∗ ∗ ˜ |b(uHt , uh , ∆ uhtt ) + |b(uh , uHt , ∆ uhtt )| ≤ Ck∇uHt kk∇uh kkuhtt k−1 , h ∗ ˜ −1 ∗ |b(uH , uht , ∆h uhtt )

+

h ∗ ∗ ˜ |b(uht , uH , ∆−1 h uhtt )|

≤

(5.18)

Ck∇u∗ht kk∇uH kku∗htt k−1 .

Integrate with respect to time from 0 to t, use (5.12), (5.18) and Lemma 5.1 to obtain Z t Z t ˜ H u0H k2 + k∆ ˜ h u∗ k 2 ) + C e2αt ku∗ht (t)k2 + e2αs ku∗htt (s)k2−1 ds ≤ (k∆ e2αs ku∗ht (s)k2 ds 0h 0 0  Z t Z t 2αs 2 ∗ 2 2αs 2 + e (k∇uHt (s)k + k∇uht (s)k )ds + e kft (s)k ds . 0

0

A use of (5.13), (5.17), (A2) and Lemma 4.1 concludes the proof of Lemma 5.2.



Lemma 5.3. (a priori bounds for UnH ) With α > 0, choose k0 small so that for 0 < k ≤ k0 ,   γλ1 k ≥ eαk . (5.19) 1+ 2 Further, let U0H = PH u0H . Then, discrete solution UnH , n ≥ 1 of (5.1) satisfies the following estimates: kUnH k2 + e−2αtn k

n X


e2αti k∇UiH k2 ≤ C(γ, ν, α, λ1 ) e−2αtn kU0H k2 + kf k2∞ ,

i=1 n X

k∇UnH k2 + e−2αtn k

i=1


˜ H Ui k2 ≤ C(γ, ν, α, λ1 ) e−2αtn k∇U0 k2 + kf k2 , e2αti k∆ H ∞ H

where kf k∞ = kf kL∞ (L2 ) .



21

Lemma 5.4. (estimates for enH ) Let the assumptions of Lemma 5.3 be satisfied. Also, let uH (t) be a solution of (3.3) and enH = UnH − unH , for n ≥ 1. Then, for some positive constant KT , that depends on T , there holds kenH k2 + ke−2αtn

n X

e2αti k∇eiH k2 ≤ KT k 2 .

i=1

Lemma 5.5. (a priori bounds for Un ) Under the hypotheses of Lemma 5.3, the discrete solution Un , n ≥ 1 of (5.2) satisfies kUn k2 + e−2αtn k

n X

e2αti k∇Ui k2 ≤ C(γ, ν, α, λ1 , T )(e−2αtn kU0 k2 + kf k2∞ ).

i=1 n X

k∇Un k2 + e−2αtn k

˜ h Ui k2 ≤ C(γ, ν, α, λ1 , T )(e−2αtn k∇U0 k2 + kf k2∞ ). e2αti k∆

i=1

ˆi − Proof. For n = i, multiply (5.2) by eαti , use eαti ∂¯t Ui = eαk ∂¯t U



eαk −1 k



ˆ i and divide the U

resulting equation by eαk to obtain   1 − e−αk i ˆ i , φ ) + ν eαti e−αk (a(Ui , φ ) + b(ui , Ui , φ ) + b(Ui , ui , φ )) ˆ ¯ (U (∂t U , φh ) − h H h H h h k ˆi ,U ˆ i , φ ). (5.20) ˆ i, e ˆ i , φ ) − e−αti+1 b(U ˆi , φ ) + e−αti+1 b(U ei , U = e−αk (ˆf i , φ ) − e−αti+1 b(ˆ h

H

H

h

h

H

H

h

Observe that (∂¯t φi , φi ) =


1 1 φi − φi−1 ≥ ∂¯t kφi k2 . 2k 2

ˆ i in (5.20), use (3.2) and (5.21) to arrive at Substitute φh = U     1¯ ˆi 2 1 − e−αk −1 −αk ˆ i k2 k∇U λ1 ∂t kU k + γe − 2 k ˆi ,U ˆi ,U ˆ i ). ˆ i, e ˆ i ) + e−αti+1 b(U ˆ i ) − e−αti+1 b(U ˆiH , U ≤ e−αk (ˆf i , U H H

(5.21)

(5.22)

Applying (2.4) and Cauchy-Schwarz’s inequality, the first term on the right hand side of (5.22) can be bounded as ˆ i )| ≤ Ce−αk kˆf i kkU ˆ i k ≤ C(λ1 , ǫ)e−αk kˆf i k2 + ǫe−αk k∇U ˆ i k2 . |e−αk (ˆf i , U

(5.23)

A use of Lemma 3.1 with Young’s inequality yields ˆ i k4 + k∇ˆ ˆ i k2 ) ˆi ,U ˆi ,U ˆ i )| + |b(U ˆ i, e ˆ i )|) ≤ C(ǫ)e−2αti e−αk (k∇U ˆiH , U eiH k2 kU e−αti+1 (|b(U H H H ˆ i k2 . + ǫe−αk k∇U (5.24) Apply (5.23)-(5.24) with ǫ = γ/2 in (5.22) to obtain     −αk ˆ i k2 + γ e−αk − 2 1 − e ˆ i k2 ∂¯t kU k∇U λ−1 1 k   ˆ i k2 . ˆ i k4 + e−2αti k∇ˆ eiH k2 kU ≤ Ce−αk kˆf i k2 + e−2αti k∇U H

22

(5.25)

  −αk   λ−1 We choose k0 > 0, such that 1+ γλ2 1 k ≥ eαk . This guarantees that γe−αk −2 1−ek 1 ≥ 0. Multiply (5.25) by k and then sum over i = 1 to n to obtain     X  n n X 1 − e−αk −1 i 2 0 2 −αk 2 n 2 −αk ˆ ˆ kf k∞ e2αti kU k + γe −2 λ1 k k∇U k ≤ kU k + C ke k i=1 i=1  n n X X i 2 ˆi 2 −2αti i 4 −2αti ˆ k∇ˆ eH k k U k . k∇UH k + k e +k e i=1

i=1

An application of Gronwall’s lemma with Lemmas 5.3 and 5.4 leads to the desired result. ˆ i to arrive at ˜ hU For n = i, multiply (5.2) by e2αti and choose φh = −∆   −αk 1¯ ˆ i) ˆ i, ∆ ˜ hU ˆ i) − 1 − e ˆ i k2 ≤ −e−αk (ˆf i , ∆ ˜ hU ˆ i k2 + νe−αk k∆ ˜ hU (U ∂t k∇U 2 k ˆ i ). ˆi ,U ˆi ,∆ ˆ i )) − e−αti+1 b(U ˆ i ) + b(U ˆ i, U ˆi ,∆ ˆi ,U ˆ i, ∆ ˜ hU ˜ hU ˜ hU (5.26) + e−αti+1 (b(U H H H H A use of Lemma 3.1 leads to ˆ i )| ˆ iH , ∆ ˜ hU ˆ i )| + |b(U ˆ iH , U ˆ i )| + |b(U ˆ i, U ˆ iH , ∆ ˜ hU ˆ iH , U ˆ i, ∆ ˜ hU |b(U ˆ i k. ˆ i k)k∆ ˜ hU ˆ i kk∇U ˆ i k + k∇U ˆ i kk∆ ˜ HU ˜ HU ≤ C(k∆ H

H

H

(5.27)

Multiply (5.26) by k and then sum over i = 1 to n and use (2.4), (5.27) to arrive at ˆ n k2 + νe−αk k k∇U

n X

ˆ i k2 ≤ k∇U0 k2 + C(λ1 )k ˜ hU k∆

+ e−αk k

ˆ i k2 ) e−αk (kˆf i k2 + k∇U

i=1

i=1

n X

n X

ˆ i k2 ). ˆ i k2 k∇U ˆ i k2 + k∇U ˆ i k2 k∆ ˜ HU ˜ HU e−2αti (k∆ H H H

(5.28)

i=1

An application of Gronwall’s lemma with (A2), Lemmas 5.3 and 5.5 concludes the proof.



Lemma 5.6. (a priori bounds for Unh ) Under the hypotheses of Lemma 5.3, the discrete solution Unh , n ≥ 1 of (5.3) satisfies kUnh k2 + e−2αtn k

n X

e2αti k∇Uih k2 ≤ C(γ, ν, α, λ1 , T )(e−2αtn kU0 k2 + kf k2∞ ),

i=1 n X

k∇Unh k2 + e−2αtn k

˜ h Uih k2 ≤ C(γ, ν, α, λ1 , T )(e−2αtn k∇U0 k2 + kf k2∞ ). e2αti k∆

i=1

ˆ i , use (3.2) and (5.21) to arrive at Proof. For n = i, multiply (5.3) by eαti , substitute φh = U h     1¯ ˆi 2 1 − e−αk ˆ ih k2 ≤ e−αk (ˆf i , U ˆ ih ) k∇U λ−1 ∂t kUh k + γe−αk − 1 2 k   ˆi ,U ˆ i, U ˆ i ) + b(U ˆ i, U ˆi −U ˆ i, U ˆi) . ˆi ,e ˆ i ) + e−αti+1 b(U ˆi , U − e−αti+1 b(U h

H

h

H

h

H

h

(5.29)

The first two terms on the right hand side of (5.29) can be tackled similar to (5.23)-(5.24). To bound the third term, use Lemmas 3.1, 5.3, 5.5 and Young’s inequality and arrive at ˆ i k2 + k∇U ˆ i k2 ) ˆi ,U ˆ i, U ˆ i ) + b(U ˆ i, U ˆi −U ˆ i , Ui )| ≤Ce−2αti e−αk (k∇U e−αti+1 |b(U H H h H h ˆ i k2 . + ǫe−αk k∇U (5.30) h

23

A use of (5.23)-(5.24) and (5.30) in (5.29) yields     1 − e−αk −1 i 2 −αk ¯ ˆ i k2 ≤ Ce−αk × ˆ k∇U λ1 ∂t kUh k + γe −2 h k   ˆ ih k2 . ˆ iH k2 + k∇U ˆ i k2 ) + e−2αti k∇ˆ eiH k2 kU kˆf i k2 + e−2αti (k∇U

Multiply (5.31) by k, sum over i = 1 to n and use Lemmas 5.3-5.5 to complete the proof.

5.1

(5.31) 

A Priori Error Estimates

Consider (3.3)-(3.5) at t = tn and subtract the resulting equations from (5.1)-(5.3), respectively, to arrive at the following error equations: Step 1. for all φH ∈ JH n n n n n n (∂¯t en H , φH ) + ν a(eH , φH ) + b(uH , eH , φH ) + b(eH , uH , φH ) = (σH , φH ) + ΛH (φH ),

(5.32)

n ¯ n where ΛH (φH ) = b(unH , enH , φH ) − b(UnH , enH , φH ) and σH = un Ht − ∂t uH . Step 2. for all φh ∈ Jh n n n n ∗ (∂¯t en , φh ) + ν a(en , φh ) + b(un H , e , φh ) + b(e , uH , φh ) = (σ , φh ) + Λ (φh ),

(5.33)

∗ ¯ ∗n where σ n = u∗n ht − ∂t uh , Λ (φh ) = Λ1 (φh ) + Λ2 (φh ) + Λ3 (φh ) with n Λ1 (φh ) = −b(en H , U , φh ),

Λ2 (φh ) = −b(Un , en H , φh ), Λ3 (φh ) =

n b(Un H , UH , φ h )

−

n b(un H , uH , φh ).

Step 3. for all φh ∈ Jh

  

(5.34)

 

n n n n n n (∂¯t en h , φh ) + ν a(eh , φh ) + b(uH , eh , φh ) + b(eh , uH , φh ) = (σh , φh ) + Λh (φh ),

(5.35)

where σhn = unht − ∂¯t unh , Λh (φh ) = Λ1h (φh ) + Λ2h (φh ) + Λ3h (φh ) + Λ4h (φh ) with n Λ1h (φh ) = −b(en H , Uh , φh ), n Λ2h (φh ) = −b(Un h , eH , φh ), ∗n n n Λ3h (φh ) = b(Un H , U , φh ) − b(uH , uh , φh ), ∗n n n ∗n Λ4h (φh ) = b(Un , Un H − U , φh ) − b(uh , uH − uh , φh ).

     

(5.36)

    

The main result of this section is stated as: Theorem 5.1. (fully discrete error estimates) Under the assumptions of Theorem 3.1 and Lemma 5.3, the following hold true: ku(tn ) − Unh k ≤ C(h2 + H 4−2δ + k),

k∇(u(tn ) − Unh )k ≤ C(h + H 4−δ + k),

kp(tn ) − Phn k ≤ C(h + H 4−δ + k 1/2 ), where δ > 0 is arbitrarily small. Below, we prove a lemma which will be used subsequently. Lemma 5.7. Assume that (A1)-(A2) and (B1)-(B2) hold true. Let for some fixed h, u∗h satisfies (3.4). Then, there is a positive constant KT that depends on T such that kei k2 + ke−2αtn

n X

e2αti k∇ei k2 ≤ KT k 2 .

i=1

24

Proof. For n = i, substitute φh = ei in (5.33) and use (5.21) to obtain ∂¯t kei k2 + 2γk∇ei k2 ≤ 2(σ i , ei ) + 2Λ∗ (ei ).

(5.37)

Multiply (5.37) by e2αik and sum over i = 1 to n, where T = nk. Use the fact n X

ke2αik ∂¯t kei k2 =

n X

e2αik (kei k2 − kei−1 k2 )

i=1

i=1

= e2αnk ken k2 −

n−1 X

e2αik (e2αk − 1)kei k2

(5.38)

i=1

to arrive at e2αnk ken k2 + 2kγ

n X

e2αik k∇ei k2 ≤

n−1 X

e2αik (e2αk − 1)kei k2

i=1

i=1

n n X X + 2k e2αik (σ i , ei ) + 2k e2αik Λ∗ (ei ).

(5.39)

i=1

i=1

A use of Taylor’s series expansion in the interval (ti−1 , ti ) with use of Cauchy-Schwarz’s and Young’s inequalities yields Z 2 ti i i |2(σ , e )| ≤ (t − ti−1 )ku∗htt k−1 dtk∇ei k k ti−1 (Z )1/2 ti 1/2 ∗ 2 ≤Kk kuhtt k−1 dt k∇ei k. (5.40) ti−1

From Lemma 5.2, observe that Z ti n Z n X X ku∗htt k2−1 dt = e2αik

e2α(ti −t) e2αt ku∗htt k2−1 dt

i=1 ti−1 Z tn 2αt 2αk

ti−1

i=1

ti

e

≤e

0

ku∗htt k2−1 dt ≤ Ke2α(n+1)k .

(5.41)

Apply (5.34) to obtain |2k

n X i=1

e2αik Λ∗ (ei )| ≤ 2k

n X e2αik (|Λ1 (ei )| + |Λ2 (ei )| + |Λ3 (ei )|).

(5.42)

i=1

An application of Lemma 3.1 with Young’s inequality and Lemmas 5.4 and 5.5 leads to 2k

n n X X e2αik (|Λ1 (ei )| + |Λ2 (ei )|) ≤ Ck e2αik k∇eiH kk∇Ui kk∇ei k i=1

i=1

≤ C(ǫ)k

n X

e2αik k∇eiH k2 k∇Ui k2 + ǫk

n X

e2αik k∇ei k2

i=1

i=1

n X ≤ C(T, ǫ)k 2 e2αnk + ǫk e2αik k∇ei k2 . i=1

A use of boundedness of k∇uH k ≤ C and Lemmas 3.1 and 5.3 to obtain |Λ3 (ei )| = |b(eiH , uiH , ei ) + b(UiH , eiH , ei )| ≤ C(k∇UiH k + k∇uiH k)k∇eiH kk∇ei k ≤ Ck∇eiH kk∇ei k.

25

(5.43)

Now, a use of Young’s inequality and Lemma 5.4 yields |2k

n n n X X X e2αik Λ3 (ei )| ≤ C(ǫ)k e2αik k∇eiH k2 + ǫk e2αik k∇ei k2 i=1

i=1

i=1

≤ C(T, ǫ)k 2 e2αnk + ǫk

n X

e2αik k∇ei k2 .

(5.44)

i=1

With the help of (5.40)-(5.44), (5.39) can be written as e

2αnk

n 2

ke k + k

n X

e

2αik

i 2

2

k∇e k ≤ Ck (e

2α(n+1)k

+e

2αnk

) + Ck

n−1 X

e2αik kei k2 .

i=1

i=1

A use of discrete Gronwall’s Lemma leads to ken k2 + ke−2αnk

n X

e2αik k∇ei k2 ≤ KT k 2

i=1

and this completes the rest of the proof. ˜ h )−1 ∂¯t en in (5.33) and use Lemma 3.1 to arrive at Now, substitute φh = (−∆



k∂¯t en k2−1 ≤ C(ν)(k∇en k + kσ n k + k∇unH kk∇en k + (k∇UnH k + k∇unH k + k∇Un k)k∇enH k)k∂¯t en k−1 .

(5.45)

An application of (5.40) and Lemmas 4.1, 5.2, 5.3, 5.4, 5.5, 5.7 leads to k∂¯t en k2−1 ≤ Ck.

(5.46)

Next, to derive pressure error estimates, we consider the equivalent form of semidiscrete approximations (3.4) as: find (u∗h (t), p∗h (t)) ∈ Hh × Lh such that u∗h (0) = u0h and for t > 0,   (u∗ht , φh ) + νa(u∗h , φh ) + b(u∗h , uH , φh )  (5.47) + b(uH , u∗h , φh ) = b(uH , uH , φh ) + (p∗h , ∇ · φh ) ∀φh ∈ Hh ,   ∗ (∇ · uh , χh ) = 0 ∀χh ∈ Lh . The equivalent form of fully discrete approximations (5.2) is as follows: ∀(φh , χh ) ∈ Hh × Lh , seek a sequence of functions (Un , P n )n≥1 ∈ Hh × Lh as solutions of the following equations:   (∂¯t Un , φh ) + ν a(Un , φh ) + b(Un , UnH , φh )  (5.48) +b(UnH , Un , φh ) = b(UnH , UnH , φh ) + (P n , ∇ · φh ),   n (∇ · U , χh ) = 0.

Subtract (5.47) from (5.48) and write ρn = P n − p∗n h to obtain

(ρn , ∇ · φh ) = (∂¯t en , φh ) + νa(en , φh ) − Λ∗ (φh ) − (σ n , φh ). A use of the Cauchy-Schwarz’s inequality along with (5.40), (5.46) and Lemmas 4.1, 5.2, 5.3, 5.4, 5.5, 5.7 yields kρn k ≤ C(κ, ν, λ1 , M )k 1/2 . A combination of (5.49) and Theorem 4.2 leads to the following pressure estimate. kp(tn ) − P n k ≤ C(h + H 3−δ + k 1/2 ).

26

(5.49)

Proof of Theorem 5.1. Write u(tn )−Unh = (u(tn )−uh (tn ))−enh . The estimate of u(tn )−uh (tn ) is obtained in Theorem 3.1. Next, we proceed to derive the estimates for enh . For n = i, substitute φh = eih in (5.35) and use (5.21). Multiply the resulting equation by e2αik and sum over i = 1 to n to obtain e2αnk kenh k2 + 2kγ

n X

e2αik k∇eih k2 ≤

n−1 X

e2αik (e2αk − 1)keih k2

i=1

i=1

n n X X + 2k e2αik (σhi , eih ) + 2k e2αik Λh (eih ).

(5.50)

i=1

i=1

The second term in the right hand side of (5.50) can be bounded similar to (5.40)-(5.41). Also, from (5.36) observe that 2k

n n X X
e2αik |Λh (eih )| ≤ 2k e2αik |Λ1h (eih )| + |Λ2h (eih )| + |Λ3h (eih )| + |Λ4h (eih )| .

(5.51)

i=1

i=1

An application of Lemmas 3.1, 5.4 and 5.6 yields 2k

n n X X e2αik (|Λ1h (eih )| + |Λ2h (eih )|) ≤ C(T, ǫ)k 2 e2αnk + kǫ e2αik k∇eih k2 .

(5.52)

i=1

i=1

Observe that i |Λ3h (eih )| ≤ |b(UiH , ei , eih )| + |b(eiH , u∗i h , eh )|.

(5.53)

With the help of Lemmas 3.1, 5.1, 5.3 and Young’s inequality, it follows that 2k

n n n X X X
e2αik |Λ3h (eih )| ≤ C(ǫ) e2αik k∇ei k2 + k∇eiH k2 + ǫk e2αik k∇eih k2 .

(5.54)

i=1

i=1

i=1

For the estimation of the fourth term on the right hand side of (5.51), rewrite it as i |Λ4h (eih )| = |b(Ui , eiH , eih ) − b(Ui , ei , eih ) + b(ei , uiH − u∗i h , eh )|.

(5.55)

Apply Lemma 3.1, 4.1, 5.1, 5.5 and Young’s inequality to obtain 2k

n n n X X X
e2αik |Λ4h (eih )| ≤ C(T, ǫ)k e2αik k∇ei k2 + k∇eiH k2 + ǫk e2αik k∇eih k2 .

(5.56)

i=1

i=1

i=1

A use of (5.52)-(5.56) in (5.50) leads to e2αnk kenh k2 + k

n X

e2αik k∇eih k2 ≤ Ck 2 (e2α(n+1)k + e2αnk ) + Ck

n X

e2αik keih k2 .

(5.57)

i=1

i=1

Use discrete Gronwall’s Lemma to arrive at kenh k2 + ke−2αnk

n X

e2αik k∇eih k2 ≤ KT k 2 .

(5.58)

i=1

A use of (5.58) along with Theorem 3.1 completes the proof of error estimates for velocity in Theorem 5.1. Using the similar techniques as to arrive at (5.49) and Theorem 3.1, the desired pressure estimate in Theorem 5.1 can be obtained and this will conclude the proof of Theorem 5.1. 

27

6

Numerical Experiments

In this section, numerical results are presented to support theoretical results in Theorem 5.1. For space discretization, P2 -P0 mixed finite element space is used. We choose the domain Ω = (0, 1) × (0, 1), time t = [0, 1], coefficients ν = 1 and h = O(H 2 ). Here, N denotes the number of unknowns in the system. Example 6.1. The right hand side function f is chosen in such a way that the exact solution (u, p) = ((u1 , u2 ), p) is u1 = 2et x2 (x − 1)2 y(y − 1)(2y − 1), u2 = −2et y 2 (y − 1)2 x(x − 1)(2x − 1), p = yet . Table 1 gives the numerical errors and convergence rates obtained on successively refined meshes for backward Euler scheme with k = O(h2 ) applied to two grid system (3.3)-(3.5). The theoretical analysis provides a convergence rate of O(h2 ) in L2 -norm, of O(h) in H1 -norm for velocity and of O(h) in L2 -norm for pressure with a choice of k = O(h). These results support the optimal theoretical convergence rates obtained in Theorem 5.1

h

ku(tn ) − Un k

1/4

0.009085

577

1/8

0.002651

1.777183

0.075081

0.898156

0.281244

0.963220

2433

1/16

0.000713

1.893768

0.038833

0.951145

0.142265

0.983237

9986

1/32

0.000184

1.950443

0.019731

0.976861

0.071518

0.992191

40449

1/64

0.000046

1.976824

0.009940

0.989066

0.035856

0.996088

N

Rate

ku(tn ) − Un kH1

Rate

0.139927

kp(tn ) − P n k

Rate

0.548331

Table 3: Errors and convergence rates for backward Euler method with k = O(h2 ). Example 6.2. In this example, we choose the right hand side function f in such a way that the exact solution (u, p) = ((u1 , u2 ), p) is: 2

u1 = te−t sin2 (3πx) sin(6πy),

2

u2 = −te−t sin2 (3πy) sin(6πx),

p = te−t sin(2πx) sin(2πy). In Table 2, we have shown the convergence rates for backward Euler method, respectively for L2 and H1 -norms in velocity and L2 -norm in pressure with k = O(h2 ). These results agree with the optimal theoretical convergence rates obtained in Theorem 5.1.

28

h

ku(tn ) − Un k

1/4

0.132916

577

1/8

0.028166

2.238442

1.537594

1.281009

0.186591

2.406260

2433

1/16

0.003717

2.921475

0.463199

1.730969

0.063322

1.559099

9986

1/32

0.000473

2.971736

0.124017

1.901084

0.016057

1.979463

40449

1/64

0.000063

2.894587

0.032022

1.953411

0.006437

1.318638

N

Rate

ku(tn ) − Un kH1

Rate

3.736491

kp(tn ) − P n k

Rate

0.989116

Table 4: Errors and convergence rates for backward Euler method with k = O(h2 ).

References [1] Ait Ou Ammi, A. and Marion, M., Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations, Numer. Math. 68 (1994), 189-213. [2] Garcia-Arch´ılla, B. and Titi, E. S., Postprocessing the Galerkin method: The finite element case, SIAM J. Numer. Anal. 37 (2000), 470-499. [3] Abboud, H. and Sayah, T., A full discretization of the time-dependent Navier-stokes equations by a two-grid scheme, M2AN Math. Model. Numer. Anal. 42 (2008), 141-174. [4] Abboud, H., Girault, V. and Sayah, T., A second order accuracy in time for a full discretized time-dependent Navier-Stokes equations by a two-grid scheme, Numer. Math. 114 (2009) 189-231. [5] Dai, Xiaoxia and Cheng, Xiaoliang, A two-grid method based on Newton iteration for the Navier-Stokes equations, J. Comput. Appl. Math. 220 (2008), 566-573. [6] Fairag, F. A., A Two-level Finite Element Discretization for the stream function form of the Navier-Stokes equations, Comput Math Appl. 36 (1998), 117-127. [7] Frutos, J. de, Garc?a-Archilla, B. and Novo, J., Optimal error bounds for two-grid schemes applied to the Navier-Stokes equations, Appl. Math. Comput. 218 (2012), 7034-7051. [8] Girault, V. and Lions, J. L., Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra, Portugal. Math. 58 (2001), 25-57. [9] Girault, V. and Lions, J. L., Two-grid finite-element schemes for the transient Navier-Stokes problem, M2AN Math. Model. Numer. Anal. 35 (2001), 945-980. [10] 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. [11] 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. [12] Hill, A. T. and S¨ uli E., Approximation of the global attractor for the incompressible NavierStokes equations, IMA J. Numer. Anal. 20 (2000), 633-667. [13] Xu, J., Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996), 1759–1777.

29

[14] Xu, J., A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1994), 231–237. [15] Layton, W. and Tobiska, L., A two level method with backtracking for the Navier-Stokes equations, SIAM J. Numer. Anal. 35 (1998), 2035-2054. [16] Layton, W. and Lenferink, W., Two-level Picard and modified Picard methods for the NavierStokes equations, Appl. Math. Comput. 69 (1995), 263–274. [17] Layton, W. and Lenferink, H. W. J., A multilevel mesh independence principle for the NavierStokes equations, SIAM J. Numer. Anal. 33 (1996), 17–30. [18] Layton, W., A two-level discretization method for the Navier-Stokes equations, Comput. Math. Appl. 26 (1993), 33–38. [19] Niemist¨o, A., FE-approximation of unconstrained optimal control like problems, Report No. 70, University of Jyvaskyla, (1995). [20] He, Y., Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal. 41 (2003), 1263-1285. [21] Temam, R., Navier-Stokes equations, theory and numerical analysis, North-Holland, Amsterdam, 1984.

30