A second order accuracy in time for a full discretized time-dependent Navier-Stokes equations by a two-grid scheme

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A second order accuracy in time for a full discretized time-dependent Navier-Stokes equations by a two-grid

scheme

Hyam Abboud, Vivette Girault, Toni Sayah

To cite this version:

Hyam Abboud, Vivette Girault, Toni Sayah. A second order accuracy in time for a full discretized

time-dependent Navier-Stokes equations by a two-grid scheme. 2007. �hal-00176003�

A SECOND ORDER ACCURACY IN TIME FOR A FULL DISCRETIZED TIME-DEPENDENT NAVIER-STOKES EQUATIONS BY A TWO-GRID SCHEME

HYAM ABBOUD^†, VIVETTE GIRAULT^‡AND TONI SAYAH^⋆.

Abstract. We study a second-order two-grid scheme fully discrete in time and space for solving the Navier-Stokes equations. The two-grid strategy consists in discretizing, in the first step, the fully non- linear problem, in space on a coarse grid with mesh-sizeH and time step ∆tand, in the second step, in discretizing the linearized problem around the velocityu

H computed in the first step, in space on a fine grid with mesh-sizehand the same time step. The two-grid method has been applied for an analysis of a first order fully-discrete in time and space algorithm and we extend the method to the second order algorithm.

Keywords Two-grid scheme, Non-linear problem, Incompressible flow, Time and Space discretizations, Taylor-Hood finite element, Duality argument, “ superconvergence“.

1. Introduction.

Let Ω be a bounded domain of IR²with a polygonal boundary∂Ω and let ]0, T[ be a given time-interval.

Consider the following Navier-Stokes problem for an incompressible fluid

∂u

∂t(x, t)−ν∆u(x, t) +u(x, t)· ∇u(x, t) +∇p(x, t) =f(x, t) in Ω×]0, T[, (1.1) with the incompressibility condition

divu(x, t) = 0 in Ω×]0, T[, (1.2)

the homogeneous Dirichlet boundary condition

u(x, t) = 0 on∂Ω×]0, T[, (1.3)

and the initial condition

u(x,0) = 0 in Ω, (1.4)

where u and p represent respectively the velocity and the pressure of the fluid. All the quantities are taken at the point (x, t) where x= (xi)1≤i≤2 ∈ R² denotes the position and t ∈ [0, T] the time. We suppose that the fluid density is a constant (ρ = 1); f denotes the external forces applied to the fluid andν is the viscosity. The notationsu· ∇u,∆uand divumean :

u· ∇u= X2 i=1

ui

∂u

∂xi

,∆u= X2

i=1

∂²u

∂²xi

and divu= X2 i=1

∂ui

∂xi

.

The termu· ∇uis the convection term andν∆uis the diffusion one.

The purpose of this article is to solve by a second-order, in time and space, two-grid scheme, on a coarse grid and a fine grid, the non-stationary incompressible Navier-Stokes problem and to show that the two-grid algorithm’s global error is similar to the error of the direct resolution of the non-linear prob- lem on a fine grid. The two-grid strategy is a general method for solving a non-linear Partial Differential Equation (PDE), depending or not in time, with solutionu. This technique consists on what follows : In a first step, we discretize the fully non-linear PDE on a coarse grid of mesh-sizeH and we compute an approximate solutionuH.Then, in a second step, we linearize the PDE arounduH and we discretize

26th September 2007.

†Actual address: Facult´e des Sciences et de G´enie Informatique, Universit´e Saint-Esprit de Kaslik, B.P. 446 Jounieh, Liban.

‡Laboratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie (Paris 6), Boˆite Courrier 187, 4, place Jussieu, 75252 Paris cedex 05, France.

⋆Facult´e des Sciences, Universit´e Saint-Joseph, B.P 11-514 Riad El Solh, Beyrouth 1107 2050, Liban.

E-mail :[email protected].

1

the linearized problem on a fine grid of mesh-sizeh; letu^lin_h be the corresponding solution. Then, under suitable assumptions, we can prove that ifh, H and the time step ∆t are well-chosen, the global error of the two-grid algorithmku−u^lin_h k has the same order as the errorku−uh k that would have been obtained if the non-linear problem had been directly discretized on the fine grid.

Two-grid discretizations have been widely applied to linear and non-linear elliptic boundary value problems: J. Xu in [22], [23], [24] has pioneered their development. These methods have been extended to the steady Navier-Stokes equations, cf. for instance the work of W. Layton in [13], W. Layton & W.

Lenferink in [14] and V. Girault & J.-L. Lions in [7]. Also, this method has been applied to the time- dependent Navier-Stokes problem, cf. V. Girault & J.-L. Lions [8] in which they analyze a semi-discrete algorithm, H. Abboud & T. Sayah in [2] and H. Abboud, V. Girault & T. Sayah in [3] for an analysis of a first order fully-discrete in time and space algorithm and in [1] for a numerical analysis of a second-order totally discrete in time and space scheme.

Setting L²₀(Ω) = {q ∈ L²(Ω);

Z

Ω

q dx = 0} and assuming that f belongs to L²(0, T;H⁻¹(Ω)²), it is well-known that (1.1)–(1.2) has the following variational formulation in ]0, T[: Findu(t)∈H₀¹(Ω)², such that in the sense of distributions on ]0, T[,

∀v∈H₀¹(Ω)², d

dt(u(t), v) +ν(∇u(t),∇v) + (u(t)· ∇u(t), v)−(p(t),divv) =hf(t), vi, (1.5)

∀q∈L²₀(Ω),(q,divu(t)) = 0, (1.6) and

u(0) = 0, (1.7)

whereu(t) =u(x, t).

Furthermore, this problem has one and only one solutionuinL^∞(0, T;L²(Ω)²)∩L²(0, T;H¹(Ω)²) and pin the dual space ofW₀^1,1(0, T;L²₀(Ω)) (see e.g. J.-L. Lions in [15] and O.A. Ladyzenskaya in [12]).

In addition, we have the following regularity result:

Theorem 1.1. If Ωis convex andf ∈L²(0, T;L²(Ω)²), then

u∈L^∞(0, T;H¹(Ω)²)∩L²(0, T;H²(Ω)²)andp∈L²(0, T;H¹(Ω)). (1.8) For discretizing (1.5)–(1.7),letη >0 be a discretization parameter in space and for each η,letT^η be a corresponding regular (or non-degenerate) family of triangulations of Ω, consisting of triangles such that any two triangles are either disjoint or share a vertex or an entire side. For an arbitrary triangleκ,we denote by ηκ the diameter ofκand byρκ the diameter of the circle inscribed inκ. Thenη denotes the maximum of ηκ and we assume thatTη is regular in the sense of Ciarlet [6] : there exists a constant σ independent ofη such that

sup

κ∈Tη

η_κ ρκ

=σκ≤σ. (1.9)

LetXηandMη be a “stable” pair of finite-element spaces for discretizing the velocityuand the pressure p,stable in the sense that it satisfies a uniform discrete inf-sup condition: there exists a constantβ^⋆≥0, independent ofη,such that

∀qη∈Mη, sup

vη∈Xη

1

|vη|H¹(Ω)

Z

Ω

qηdivvηdx≥β^⋆kqηkL²(Ω). (1.10) Let IPκ denote the space of polynomials with total degree less than or equal to κ.For a second-order two-grid scheme, we choose the Taylor-Hood finite-element, where in each triangleκ,each component of the velocity is a polynomial of IP2 and the pressurepis a polynomial of IP1.Therefore, the finite-element spaces are:

Xη =

vη ∈C⁰(Ω)²; ∀κ∈ T^η, v_η|κ ∈IP²₂, v_η|∂Ω = 0 , (1.11)

Mη =

qη ∈C⁰(Ω); ∀κ∈ T^η, q_η|κ∈P1, Z

Ω

qηdx= 0

. (1.12)

SECOND-ORDER TWO-GRID SCHEME FOR THE FULLY DISCRETE TRANSIENT NAVIER-STOKES EQUATIONS

There exists an approximation operatorPη∈ L(H₀¹(Ω)²;Xη) such that (see V. Girault and P.-A. Raviart in [9]):

∀v∈H₀¹(Ω)², ∀qη ∈Mη, Z

Ω

qηdiv(Pη(v)−v)dx= 0, (1.13) and fork= 0,1 or 2,

∀v∈[H^1+k(Ω)∩H₀¹(Ω)]², kPη(v)−vkL²(Ω)≤ Cη^1+k|v|H^1+k(Ω), (1.14) and forallr≥2, k= 0,1 or 2,

∀v∈[W^1+k,r(Ω)∩H₀¹(Ω)]², |P_η(v)−v|W^1,r(Ω) ≤ C_rη^k|v|W^1+k,r(Ω). (1.15) In addition, as Mη contains all polynomials of degree one, there exists an operator rη ∈ L(L²₀(Ω);Mη), such that for any real numbers∈[0,2],

∀q∈H^s(Ω)∩L²₀(Ω),krη(q)−qkL²(Ω)≤Cη^s|q|H^s(Ω). (1.16) To discretize in time, we divide the interval [0, T] into N subintervals of equal length k = T

N, with grid-pointstⁿ=nk,0≤n≤N.

With these spaces, we propose the following two-grid scheme for discretizing (1.5)–(1.7). We use two regular triangulations of Ω : a coarse triangulationT^H and a fine oneT^h,that for practical purposes, is a refinement ofT^H.On each of these, we define the same stable pair of finite-element spaces, (XH, MH) and (Xh, Mh) such thatXH⊂XhandMH⊂Mh.At each time step, we solve (1.17)–(1.18) and (1.19)–(1.20) below. The two-grid algorithm reads :

• Step One (non-linear problem on coarse grid): Knowinguⁿ⁻¹_h and uⁿ_h, find (uⁿ⁺¹_H , pⁿ⁺¹_H ) with values inXH×MH,solution of

∀vH∈XH, 1

2∆t(3uⁿ⁺¹_H −4uⁿ_H+uⁿ⁻¹_H , vH) + ν(∇uⁿ⁺¹_H ,∇vH) + (uⁿ⁺¹_H · ∇uⁿ⁺¹_H , vH) +1

2(divuⁿ⁺¹_H , uⁿ⁺¹_H ·vH)− (pⁿ⁺¹_H ,divvH) = (fⁿ⁺¹, vH),

(1.17)

∀qH ∈MH, (qH,divuⁿ⁺¹_H ) = 0. (1.18)

• Step Two (linearized problem on fine grid): Knowing (uⁿ⁺¹_H , pⁿ⁺¹_H ),find (uⁿ⁺¹_h , pⁿ⁺¹_h ) with values in Xh×Mh solution of

∀vh∈Xh, 1

2∆t(3uⁿ⁺¹_h −4uⁿ_h+uⁿ⁻¹_h , vh) + ν(∇uⁿ⁺¹_h ,∇vh) + (uⁿ⁺¹_H · ∇uⁿ⁺¹_h , vh)

−(pⁿ⁺¹_h ,divvh) = (fⁿ⁺¹, vh), (1.19)

∀qh∈Mh, (qh,divuⁿ⁺¹_h ) = 0. (1.20) By assumption,u⁰_H =u⁰_h= 0 andu¹_H andu¹_hare computed by solving one iteration of an Euler scheme.

In both (1.17) and (1.19), fⁿ⁺¹is a suitable approximation off at timetⁿ⁺¹.The purpose of this two-grid algorithm is to reduce the time of computation for both velocity and pressure.

In the sequel, we shall take (∆t)² of the order of H³ : there exist constants α1 and α2 that do not depend onH and ∆tsuch that

α1H³≤(∆t)²≤α2H³.

The remainder of this article is organized as follows : In Section 2, we present some conventions and notations that will be used throughout the article. In Section 3, we present a first error estimate for the fully-discrete Step One, then in section 4 we establish a duality argument based on the backward semi-discrete Stokes system and we derive some uniform bounds that allow us to prove the Stokes prob- lem’s error estimate inL²(Ω×]0, T[)², then we apply it to the Navier-Stokes problem. We also prove a

“superconvergence” result for the non-linear part. Finally, the pressure is estimated in section 5 and the error estimation for the solution of Step Two is studied in section 6.

2. Preliminaries.

To begin with, we present some conventions and notations that will be used throughout the article.

As usual, for handling time-dependent problems, it is convenient to consider functions defined on a time interval ]a, b[ with values in a functional space, say X (cf. Lions and Magenes [16]). More precisely, let k.kX denote the norm ofX; then for any r, 1≤r≤ ∞,we define

L^r(a, b;X) =n

f mesurable in ]a, b[;

Z b

a kf(t)k^rX dt <∞o equipped with the norm

kf kL^r(a,b;X)= Z ^b

a kf(t)k^rXdt1/r

,

with the usual modifications ifr=∞. It is a Banach space ifX is a Banach space.

Let (k1, k2) denote a pair of non-negative integers, set|k|=k1+k2 and define the partial derivative∂^k by∂^kv = ∂^|k|v

∂x^k₁¹∂x^k₂².Here X is usually a Sobolev space, such as (cf. Adams [4] or Neˇcas [17]): for any non-negative integerm and numberr≥1,

W^m,r(Ω) ={v∈L^r(Ω);∂^kv∈L^r(Ω),∀|k| ≤m}. This space is equipped with the seminorm

|v|W^m,r(Ω)=h X

|k|=m

Z

Ω|∂^kv|^rdxi1/r

,

and is a Banach space for the norm

kvkW^m,r(Ω)=h X

0≤|k|≤m

|v|^rW^k,r(Ω)

i1/r

,

with the usual extension whenr=∞. Whenr= 2,this space is the Hilbert spaceH^m(Ω).In particular, the scalar product of L²(Ω) is denoted by (. , . ). Similarly, L²(a, b;H^m(Ω)) is a Hilbert space and in particularL²(a, b;L²(Ω)) coincides with L²(Ω×]a, b[).

For functions that vanish on the boundary, we recall Poincar´e’s inequality: there exists a constantP such that

∀v∈H₀¹(Ω),kvkL²(Ω)≤ P|v|H¹(Ω). (2.1) More generally, recall the inequalities of Sobolev imbeddings in two dimensions: for eachr∈[2,∞[,there exits a constantSr such that

∀v∈H₀¹(Ω), kvkL^r(Ω)≤Sr|v|H¹(Ω), (2.2) where

|v|H¹(Ω)=k ∇vkL²(Ω). (2.3)

When r = 2, (2.2) reduces to Poincar´e’s inequality and S2 is Poincar´e’s constant. The case r =∞ is excluded and is replaced by: for anyr >2,there exists a constantMr such that

∀v∈W₀^1,r(Ω), kvkL^∞(Ω)≤Mr|v|W^1,r(Ω). (2.4) We also have in dimension 2,

kgkL⁴(Ω)≤2^1/4kgk^1/2_L²_(Ω)k ∇gk^1/2_L²_(Ω). (2.5) Owing to (2.1), we use the seminorm|.|H¹(Ω) as a norm on H₀¹(Ω) and we use it to define the norm of the dual spaceH⁻¹(Ω) ofH₀¹(Ω):

kf kH⁻¹(Ω)= sup

v∈H₀¹(Ω)

hf, vi

|v|H¹(Ω)

,

SECOND-ORDER TWO-GRID SCHEME FOR THE FULLY DISCRETE TRANSIENT NAVIER-STOKES EQUATIONS

whereh·,·idenotes the duality pairing betweenH⁻¹(Ω) andH₀¹(Ω).Also, we recall the spaces we intro- duced at the beginning:

V ={v∈H₀¹(Ω)²; divv= 0 in Ω} and L²₀(Ω) ={q∈L²(Ω);

Z

Ω

q dx= 0}, and the orthogonal complement ofV inH₀¹(Ω)²:

V^⊥={v∈H₀¹(Ω)²;∀w∈V,(∇v,∇w) = 0}. The results of this article are based on the identity:

2(aⁿ⁺¹,3aⁿ⁺¹−4aⁿ+aⁿ⁻¹) =|aⁿ⁺¹|²+|2aⁿ⁺¹−aⁿ|²+|δ²aⁿ|²− |aⁿ|²− |2aⁿ−aⁿ⁻¹|², (2.6) where

δ²aⁿ=aⁿ⁺¹−2aⁿ+aⁿ⁻¹. (2.7)

3. Error estimates for the solution of Step One The results in this paragraph are written for the non-linear scheme (1.17)–(1.18).

To simplify, we denote byη the mesh parameter. First of all, we prove the existence and the uniqueness of the solution of (1.17)–(1.18).

Lemma 3.1. (Stability) Letuⁿ⁺¹_η be a solution of (1.17)–(1.18)with the initial datasu⁰_η andu¹_η ∈Vη; We have

sup

2≤n≤Nkuⁿ_η kL²(Ω)+ sup

2≤n≤N k2uⁿ_η−uⁿ⁻¹_η kL²(Ω)+√ 2νX^N

n=2

∆tk ∇uⁿ_η k²L²(Ω)

1/2

+^N−1X

n=1

kδ²uⁿ_η k²L²(Ω)

1/2

≤C2S₂² ν

XN n=2

∆tkfⁿ k²L²(Ω)+ku¹_ηk²L²(Ω)+k2u¹_η−u⁰_η k²L²(Ω)

1/2

.

Proof. We take the scalar product of (1.17) by 4∆tuⁿ⁺¹_η , use (2.6) and sum the result over 1 ≤ n ≤

m−1.

The stability of (1.17)–(1.18) results from the following a priori estimation:

Lemma 3.2. (Uniqueness) The scheme (1.17)–(1.18) has a solution for all ν > 0, all initial datas u⁰_η, u¹_η∈Vη and for all dataf ∈ C⁰([0, T];L²(Ω)²).The solution is unique for∆tsufficiently small.

Proof. For all 1≤n≤N−1,the problem (1.17)–(1.18) is a square system of algebric non-linear equations in finite dimension. Due to the anti-symetrisation of the non-linear term, we prove, by the theorem of the seddle point of Brouwer and the inf-sup condition, that for all 1≤n≤N −1, the problem has at least a solution (uⁿ_η, pⁿ_η).For the unicity, we consider two solutions (u⁽¹⁾η , p⁽¹⁾η ) and (u⁽²⁾η , p⁽²⁾η ). Their difference (wⁿ_η, pⁿ_η) satisfies:

∀vη∈Vη, 1

2∆t(3wⁿ⁺¹_η −4wⁿ_η +w_ηⁿ⁻¹, vη) + ν(∇wⁿ⁺¹_η ,∇vη) + (w_ηⁿ⁺¹· ∇u⁽¹⁾ⁿ⁺¹_η , vη) + (u⁽²⁾ⁿ⁺¹η ·w_ηⁿ⁺¹, vη) +1

2(divwⁿ⁺¹_η , u⁽¹⁾ⁿ⁺¹_η ·vη) +1

2(divu⁽²⁾ⁿ⁺¹_η , wⁿ⁺¹_η ·vη) = 0.

By using the identity (2.6) and choosingvη =wⁿ⁺¹_η ,we obtain 1

4∆t

kwⁿ⁺¹_η k²L²(Ω)+k2w_ηⁿ⁺¹−wⁿ_η k²L²(Ω)+kδ²wⁿ_η k²L²(Ω)− kwⁿ_η k²L²(Ω)− k2w_ηⁿ−wⁿ⁻¹_η k²L²(Ω)

+ν|wⁿ⁺¹_η |²H¹(Ω)− kwⁿ⁺¹_η k²L⁴(Ω)|u⁽¹⁾ⁿ⁺¹η |H¹(Ω)−1

2|wⁿ⁺¹_η |H¹(Ω)kwⁿ⁺¹_η kL⁴(Ω)ku⁽¹⁾ⁿ⁺¹_η kL⁴(Ω)≤0.

Due to the fact that in finite dimension, all the norms are equivalent, summing the precedent inequality fromn= 1 tom−1,and using Lemma 3.1 andw_η⁰=w_η¹= 0,we obtain

kw^m_η k²_L²_(Ω)+ν Xm n=2

∆t|w_ηⁿ|²H¹(Ω)+

m−1X

n=1

kδ²w_ηⁿk²L²(Ω)+k2w^m_η −w^m−1_η k²L²(Ω)≤C∆t Xm n=1

kwⁿ_η k²L²(Ω), (3.1) with a constant C that depends on η but does not depend on ∆t.For the last term of the sum of the right-hand side, we write:

w^m_η =δ²w_η^m−1+ 2w^m−1_η −w^m−2_η . Then

kw_η^mk²L²(Ω)≤2

kδ²w^m−1_η k²L²(Ω)+k2w^m−1_η −w^m−2_η k²L²(Ω)

, and for ∆t sufficiently small, the term

2C∆t

kδ²w^m−1_η k²L²(Ω)+k2w^m−1_η −w^m−2_η k²L²(Ω)

can be absorbed by the term in the left-hand side of the inequality. Applying Gronwall’s lemma, we obtainw_ηⁿ=0then, the inf-sup condition impliespⁿ_η = 0,2≤n≤N.

In the next proposition, we will establish the error estimate for the solution computed by one iteration of Euler’s scheme (u¹_η−u(∆t), p¹_η−p(∆t)):

Proposition 3.3. Suppose that u^′′ ∈ C⁰(0, T;L²(Ω)²), u(∆t) ∈H³(Ω)² andp(∆t)∈H²(Ω), the error of the solution computed by one iteration of Euler’s scheme satisfies the following estimations, for ∆t≤ k0>0 sufficiently small,

1

2 ku¹_η−u(∆t)k²L²(Ω)+ν∆t

2 |u¹_η−u(∆t)|²H¹(Ω)

≤ (∆t)⁴

4 ku^′′k²L^∞(0,T;L²(Ω)²)+C(∆t)η⁴

|u(∆t)|²H³(Ω)+|p(∆t)|²H²(Ω)

+Cη⁶|u(∆t)|²H³(Ω),

(3.2)

and

(∆t)^1/2kp(∆t)−p¹_η kL²(Ω)≤C

(∆t)^3/2+η²+ η³

√∆t

. (3.3)

Proof. Due to the regularity assumption ofu, there existsθ∈]0,1[ such that 0 =u0=u(∆t)−(∆t)u^′(∆t) +1

2(∆t)²u^′′(θ∆t), andu¹_η satisfies the following error equation

∀vη ∈Vη, 1

∆t(u¹_η−u(∆t), vη) +ν(∇(u¹_η−u(∆t)),∇vη) =∆t

2 (u^′′(θ∆t), vη)

−(p(∆t)−rηp(∆t),divvη) + (u(∆t)· ∇u(∆t)−u¹_η· ∇u¹_η, vη)−1

2(divu¹_η, u¹_η·vη).

(3.4)

Settingvη=v_η¹=u¹_η−Pηu(∆t) andϕ¹_η =Pηu(∆t)−u(∆t),we obtain 1

∆t kv_η¹k²L²(Ω)+ν|v_η¹|²H¹(Ω)=∆t

2 (u^′′(θ∆t), v¹_η) + (rηp(∆t)−p(∆t),divv_η¹)−(v¹_η· ∇u¹_η, v¹_η)

−1

2(divv¹_η, u¹_η·v¹_η)−(ϕ¹_η· ∇u¹_η, v¹_η)−1

2(divϕ¹_η, u¹_η·v¹_η)

−(u(∆t)· ∇ϕ¹_η, v¹_η)− 1

∆t(ϕ¹_η, v_η¹)−ν(∇ϕ¹_η,∇v¹_η).

(3.5)

SECOND-ORDER TWO-GRID SCHEME FOR THE FULLY DISCRETE TRANSIENT NAVIER-STOKES EQUATIONS

Then (3.2) follows readily by applying the error approximation ofPη. For the pressure, we have

(rηp(∆t)−p(∆t),divv_η) + (p¹_η−r_ηp(∆t),divv_η) = 1

∆t(u¹_η−u(∆t), vη) +ν(∇(u¹_η−u(∆t)),∇v_η)

−∆t

2 (u^′′(θ∆t), vη)−(u(∆t)· ∇u(∆t)−u¹_η· ∇u¹_η, vη) +1

2(divu¹_η, u¹_η·vη), (3.6) and owing to the inf-sup condition (1.10),there existsvη∈V_η^⊥ such that

(p¹_η−rηp(∆t),divvη) =kp¹_η−rηp(∆t)k²L²(Ω) and|vη|H¹(Ω)≤ 1

β^⋆ kp¹_η−rηp(∆t)kL²(Ω),

withβ^⋆>0 independent ofη.Then, by applying (3.2),we obtain (3.3).

The next result, stated in Lemma 3.4, is a standard error estimate. We give the proof for the sake of completeness.

Lemma 3.4. LetXη andMη be defined by(1.11)and(1.12)and approximatefⁿ⁺¹ byfⁿ⁺¹=f(tⁿ⁺¹).

At each time step,(1.17)–(1.18)has a solutionuⁿ⁺¹_η and this solution is unique if∆tis sufficiently small.

Suppose thatu∈L²(0, T;H³(Ω)²), u^′∈L²(0, T;H²(Ω)²), u⁽³⁾∈L²(Ω×]0, T[)²andp∈L²(0, T;H²(Ω)), there exist a constant C that does not depend onη and ∆t and a constant k0>0 that does not depend onη such that, for all∆t≤k0,

sup

1≤n≤N kuⁿ_η −u(tⁿ)kL²(Ω)+^NX⁻¹

n=1

kδ²(uⁿ_η−u(tⁿ))k²L²(Ω)

1/2

+√ ν^NX⁻¹

n=1

∆t|uⁿ⁺¹_η −u(tⁿ⁺¹)|²H¹(Ω)

1/2

≤C(η²+ (∆t)²).

(3.7) Proof. Settingv_ηⁿ =uⁿ_η −Pηu(tⁿ) andϕⁿ_η =Pηu(tⁿ)−u(tⁿ),0≤n≤N, we substruct (1.17) and (1.1) taken att=tⁿ⁺¹ and by using the following second-order backward finite difference scheme

∂u

∂t(tⁿ⁺¹) = 3u(tⁿ⁺¹)−4u(tⁿ) +u(tⁿ⁻¹)

2∆t +O((∆t)²), (3.8)

we have

u^′(t+ ∆t)−3u(t+ ∆t)−4u(t) +u(t−∆t) 2∆t

≤(∆t)^3/2 2√

3 ku⁽³⁾kL²(t−∆t;t+∆t), (3.9) and by summing the result over 1≤n≤m−1,we obtain :

kv_η^mk²L²(Ω)+k2v^m_η −v_η^m−1k²L²(Ω)+

m−1X

n=1

kδ²vⁿ_η k²L²(Ω)+4ν

m−1X

n=1

∆t|v_ηⁿ⁺¹|²H¹(Ω)

≤

kv_η¹k²L²(Ω)+k2v_η¹−v⁰_ηk²L²(Ω)

+ 2

m−1X

n=1

(3ϕⁿ⁺¹_η −4ϕⁿ_η +ϕⁿ⁻¹_η , vⁿ⁺¹_η ) + 4ν

m−1X

n=1

∆t(∇ϕⁿ⁺¹_η ,∇vⁿ⁺¹_η ) +4

m−1X

n=1

∆t(p(tⁿ⁺¹)−rηp(tⁿ⁺¹),divvⁿ⁺¹_η ) + 4

m−1X

n=1

∆t(∆t)^3/2 2√

3 ku⁽³⁾kL²(tⁿ⁻¹;tⁿ⁺¹)kv_ηⁿ⁺¹kL²(Ω)

+4

m−1X

n=1

∆t(u(tⁿ⁺¹)· ∇u(tⁿ⁺¹)−uⁿ⁺¹_η · ∇uⁿ⁺¹_η , vⁿ⁺¹_η )−1 2

m−1X

n=1

∆t(divuⁿ⁺¹_η , uⁿ⁺¹_η .vⁿ⁺¹_η ) .

(3.10) Let us study the terms of the right hand side of (3.10) denoted by ((trhs)i)1≤i≤7.The first term (trhs)1

is bounded as in Proposition 3.3.

To study the second term, we have

3ϕⁿ⁺¹_η −4ϕⁿ_η +ϕⁿ⁻¹_η

2∆t =Pηu^′(tⁿ⁺¹)−u^′(tⁿ⁺¹) +R2,

with

|R2| ≤ (∆t)^3/2 2√

3 kPηu⁽³⁾−u⁽³⁾kL²(tⁿ⁻¹;tⁿ⁺¹).

Hence, by assuming thatPη is stable in the normL² (cf. Girault and Lions [8]), we have

|(trhs)2|= 4

m−1X

n=1

∆t3ϕⁿ⁺¹_η −4ϕⁿ_η+ϕⁿ⁻¹_η

2∆t , v_ηⁿ⁺¹

≤Cη⁴

2ε2 ku^′k²L^∞(0,T;H²(Ω)²)+C(∆t)⁴

2ε2 ku⁽³⁾k²L²(Ω×]0,T[)² +ε2

2

m−1X

n=1

∆t|vⁿ⁺¹_η |²H¹(Ω).

The third term is bounded as follows :

|(trhs)3|= 4ν

m−1X

n=1

∆t(∇ϕⁿ⁺¹_η ,∇v_ηⁿ⁺¹)

≤2Cνη⁴

ε3 kuk²L²(0,T;H³(Ω)²)+2νε3 m−1X

n=1

∆t|v_ηⁿ⁺¹|²H¹(Ω).

For the pressure contribution, we have :

|(trhs)4|= 4

m−1X

n=1

∆t(p(tⁿ⁺¹)−r_ηp(tⁿ⁺¹),divv_ηⁿ⁺¹)

≤ 2Cη⁴

ε4 kpk²L²(0,T;H²(Ω))+2ε4 m−1X

n=1

∆t|v_ηⁿ⁺¹|²H¹(Ω).

The fifth term is treated as follows :

|(trhs)5|= 4

m−1X

n=1

∆t(∆t)^3/2 2√

3 ku⁽³⁾kL²(tⁿ⁻¹;tⁿ⁺¹)kv_ηⁿ⁺¹kL²(Ω)

≤ (∆t)⁴S₂²

3ε5 ku⁽³⁾k²L²(Ω×]0,T[)² +ε5 m−1X

n=1

∆t|v_ηⁿ⁺¹|²H¹(Ω). Let us consider now the non-linear terms, (trhs)6+ (trhs)7, which are treated like follows :

(−u(tⁿ⁺¹)· ∇u(tⁿ⁺¹) +uⁿ⁺¹_η · ∇uⁿ⁺¹_η , vⁿ⁺¹_η ) +1

2(divuⁿ⁺¹_η , uⁿ⁺¹_η ·vⁿ⁺¹_η )

=−(vⁿ⁺¹_η · ∇v_ηⁿ⁺¹, Pηu(tⁿ⁺¹))−1

2(divv_ηⁿ⁺¹, Pηu(tⁿ⁺¹)·vⁿ⁺¹_η )−(ϕⁿ⁺¹_η · ∇vⁿ⁺¹_η , Pηu(tⁿ⁺¹))

−1

2(divϕⁿ⁺¹_η , Pηu(tⁿ⁺¹)·v_ηⁿ⁺¹)−(u(tⁿ⁺¹)· ∇v_ηⁿ⁺¹, ϕⁿ⁺¹_η ).

(3.11)

The study of the three terms in the right-hand side of (3.11),denoted by ((trhs)67).j, j= 1,2,3,will end the proof. Setting

C1= sup

n |u(tⁿ)|H¹(Ω), applying on one hand

Z

Ω

div(v_ηⁿ⁺¹−u(tⁿ⁺¹))u(tⁿ⁺¹)·ϕⁿ⁺¹_η dx = − Z

Ω

(v_ηⁿ⁺¹−u(tⁿ⁺¹))· ∇u(tⁿ⁺¹)·ϕⁿ⁺¹_η dx

− Z

Ω

(v_ηⁿ⁺¹−u(tⁿ⁺¹))· ∇ϕⁿ⁺¹_η ·u(tⁿ⁺¹)dx,

(3.12)

and on the other hand

ab≤ a^p p +b^p′

p^′ , avec 1 p+ 1

p^′ = 1, (3.13)

we have

|((trhs)67).1|= 4

m−1X

n=1

∆t

(v_ηⁿ⁺¹· ∇vⁿ⁺¹_η , Pηu(tⁿ⁺¹)) +1

2(divvⁿ⁺¹_η , Pηu(tⁿ⁺¹)·v_ηⁿ⁺¹)

≤3S4C₁√ 2 2ε⁴₆

m−1X

n=1

∆tkv_ηⁿ⁺¹k²L²(Ω)+9S4C₁ε^4/3₆ 2√

2

m−1X

n=1

∆t|vⁿ⁺¹_η |²H¹(Ω),

SECOND-ORDER TWO-GRID SCHEME FOR THE FULLY DISCRETE TRANSIENT NAVIER-STOKES EQUATIONS

|((trhs)67).2|= 4

m−1X

n=1

∆t

(ϕⁿ⁺¹_η · ∇vⁿ⁺¹_η , P_ηu(tⁿ⁺¹)) +1

2(divϕⁿ⁺¹_η , P_ηu(tⁿ⁺¹)·v_ηⁿ⁺¹)

≤3S₄²CC1

nη⁴

ε7 kuk²L²(0,T;H³(Ω)²)+ε7 m−1X

n=1

∆t|v_ηⁿ⁺¹|²H¹(Ω)

o ,

and

|((trhs)67).3|= 4

m−1X

n=1

∆t

u(tⁿ⁺¹)· ∇vⁿ⁺¹_η , ϕⁿ⁺¹_η

≤S²₄C1

2

nC²η⁴

ε8 kuk²L²(0,T;H³(Ω)²)+ε8 m−1X

n=1

∆t|v_ηⁿ⁺¹|²H¹(Ω)

o .

After a suitable choice ofεi,(3.10) becomes kv_η^mk²L²(Ω)+k2v^m_η −v_η^m−1k²L²(Ω)+

m−1X

n=1

kδ²vⁿ_η k²L²(Ω)+ν

m−1X

n=1

∆t|vⁿ⁺¹_η |²H¹(Ω)

≤ α(∆t)⁴+βη⁴+ξ

m−1X

n=1

∆tkv_ηⁿ⁺¹k²L²(Ω),

(3.14)

whereα, β andξare constants that do not depend onη and ∆t.

Then after applying Gronwall’s lemma and for ∆t sufficiently small, the result follows from this in- equality:

sup

1≤n≤Nkv_ηⁿkL²(Ω)+ sup

1≤n≤N k2vⁿ_η −vⁿ⁻¹_η kL²(Ω)+^NX⁻¹

n=1

kδ²vⁿ_η k²L²(Ω)

1/2

+√ ν^NX⁻¹

n=1

∆t|v_ηⁿ⁺¹|²H¹(Ω)

1/2

≤α^′(∆t)²+β^′η².

(3.15)

Finally, (3.7) follows by applying a triangular inequality and thePη’s properties.

Remark 3.5. We suppose that there exist two constantsα^′ andγ^′ >0that do not depend on η and∆t such that

α^′η³≤(∆t)²≤γ^′η³, (3.16)

which means that(∆t)² is of the same order ofη³.

4. Some error estimates for the Stokes problem

The error estimate of order two inL²(Ω×]0, T[)², that will be established in the next section, is based on a duality argument for the transient Stokes problem:

∂v

∂t(x, t)−ν∆v(x, t) +∇q(x, t) =g(x, t) in Ω×]0, T[, (4.1) divv(x, t) = 0 in Ω×]0, T[, v(x, t) = 0 on∂Ω×]0, T[, v(x,0) = 0 in Ω. (4.2) The fully-discrete scheme for (4.1)–(4.2) is: Find (vⁿ⁺¹_η , qⁿ⁺¹_η ) with values inXη×Mη, for each 1≤n≤ N−1, solution of:

∀zη ∈Xη, 1

2∆t(3vⁿ⁺¹_η −4vⁿ_η +vⁿ⁻¹_η , zη) +ν(∇v_ηⁿ⁺¹,∇zη)−(q_ηⁿ⁺¹,divzη) = (gⁿ⁺¹, zη), (4.3)

∀λη∈Mη, (λη,divvⁿ⁺¹_η ) = 0. (4.4) These equations are completed by initial conditions similar to the Navier-Stokes problem’s ones.

This linear problem (4.3)–(4.4) has a unique solution, owing to the inf-sup condition (1.10), without

any restriction on ∆t. This solution satisfies the following error estimates in normL^∞(0, T;L²(Ω)²) and L²(0, T;H¹(Ω)²): We prove, first of all, that the initial valuev¹_η,as in the Navier-Stokes problem, satisfies:

Ifv(∆t)∈H³(Ω)², q(∆t)∈H²(Ω) andv^′′∈ C⁰([0, T];L²(Ω)²),then kv_η¹−v(∆t)k²L²(Ω)+ν∆t|v¹_η−v(∆t)|²H¹(Ω)

≤ (∆t)⁴

2 kv^′′k²L^∞(0,T;L²(Ω)²)+C(∆t)η⁴

|v(∆t)|²H³(Ω)+|q(∆t)|²H²(Ω)

+Cη⁶|v(∆t)|²H³(Ω).

(4.5)

Secondly, in the general case, we have the following result (the proof is similar to the one of Lemma 3.4, but simpler because of the absence of the convection term).

Lemma 4.1. Let(v, q)and(v_ηⁿ, q_ηⁿ)be the respective solution of(4.1)–(4.2) and(4.3)–(4.4). In addition to the precedent hypotheses, we suppose that g is regular enough in space and in time,

v∈L²(0, T;H³(Ω)²), v^′ ∈L²(0, T;H²(Ω)²), v⁽³⁾∈L²(Ω×]0, T[)² andq∈L²(0, T;H²(Ω)). There exists a constant C that does not depend onη and∆t such that

sup

1≤n≤N kvⁿ_η −v(tⁿ)kL²(Ω)+ sup

1≤n≤N k2(v_ηⁿ−v(tⁿ))−(v_ηⁿ⁻¹−v(tⁿ⁻¹))kL²(Ω)

+^NX⁻¹

n=1

kδ²(vⁿ_η −v(tⁿ))k²L²(Ω)

1/2

+√ νX^N

n=1

∆t|v_ηⁿ−v(tⁿ)|²H¹(Ω)

1/2

≤C(η²+ (∆t)²). (4.6) In addition, the solution (v_ηⁿ⁺¹, q_ηⁿ⁺¹) of (4.3)–(4.4) satisfies an error estimate inL^∞(0, T;H¹(Ω)²). To simplify, we introduce the following notation

δ¹aⁿ= 3aⁿ⁺¹−4aⁿ+aⁿ⁻¹

2∆t . (4.7)

The proof is based on the following Stokes projection: ∀(u, p)∈V ×L²₀(Ω), Sη(u)∈V_η satisfies

∀vη ∈Vη, ν(∇(Sη(u)−u),∇vη) = −(p,divvη). (4.8) The operatorSη satisfies the following inequalities:

Lemma 4.2. Let (u, p)∈V ×L²₀(Ω). Sη(u) defined by(4.8) satisfies

|Sη(u)−u|H¹(Ω)≤2|Pη(u)−u|H¹(Ω)+ 1

ν krη(p)−pkL²(Ω). (4.9) If in additionΩ is convex, there exists a constantC that does not depend on η such that

kS_η(u)−ukL²(Ω)≤Cη(|S_η(u)−u|H¹(Ω)+kr_η(p)−pkL²(Ω)). (4.10)

Lemma 4.3. In addition to the hypotheses of Lemma 4.1, suppose thatv^′∈ C⁰(0, T;H²(Ω)²),

v^′′∈ L²(0, T;H¹(Ω)²), v⁽³⁾ ∈L²(Ω×]0, T[)², q^′ ∈ C⁰(0, T;H¹(Ω)) and q^′′ ∈L²(Ω×]0, T[). Then, if Ω is convex, there exists a constantC that does not depend onη and∆t such that

sup

1≤n≤N|v_ηⁿ−v(tⁿ)|H¹(Ω)+ sup

1≤n≤N−1|2(vⁿ⁺¹_η −v(tⁿ⁺¹))−(v_ηⁿ−v(tⁿ))|H¹(Ω)

+^NX⁻¹

n=1

∆tkδ¹(v_ηⁿ−v(tⁿ))k²L²(Ω)

1/2

+^NX⁻¹

n=1

|δ²(v_ηⁿ−v(tⁿ))|²H¹(Ω)

1/2

≤C(η²+ (∆t)^3/2+ η³

√∆t).

(4.11)