# Stability Analysis of the Crank-Nicolson Finite Element Method for the Navier-Stokes Equations Driven by Slip Boundary Conditions

Authors: M. Mbehou; M. S. Daoussa Haggar; H. Olei Tahar
Journal: Journal of Applied Mathematics
 (2022)

---

## Abstract

This paper is devoted to the study of numerical approximation for a class of two-dimensional Navier-Stokes equations with slip boundary conditions of friction type. The objective is to establish the well-posedness and stability of the numerical scheme inL2-norm and H1-norm for all positive time using the Crank-Nicholson scheme in time and the finite element approximation in space. The resulting variational structure dealing with is in the form of inequality, and obtaining H1-estimate is more involved because of the presence of the nondifferentiable term appearing at the boundary where slip occurs. We prove that the numerical scheme is stable in L2 and H1-norms with the aid of different versions of discrete Grownwall lemmas, under a CFL-type condition. Finally, some numerical simulations are presented to illustrate our theoretical analysis.


---

## Body

## 1. Introduction

We consider the Navier-Stokes equations of viscous incompressible fluids:(1)ut+u·∇u−νΔu+∇p=finQ=Ω×ℝ+,(2)divu=0inQ,with the impermeability boundary condition
(3)un=u·n=0onS×ℝ+,and the slip boundary condition
(4)σnτ≤g,σnτ<g⇒uτ=0,σnτ=g⇒uτ≠0,−σnτ=guτuτ,onS×0,∞.On the remaining part of the boundary,Γ, we assume Dirichlet boundary condition, i.e.,
(5)u=0onΓ×ℝ+.Finally, the initial condition is given by(6)ux,0=u0xonΩ¯.Here,Ω⊂ℝ2 is a bounded domain, with boundary ∂Ω. It is assumed that ∂Ω is made of two components S, and Γ with ∂Ω¯=S∪Γ¯, and S∩Γ=∅. ν is a positive quantity representing the viscosity coefficient, u0:Ω⟶ℝ2 is the initial velocity, and g:S×0,∞⟶0,∞ is the barrier or threshold function. The velocity of the fluid is u and p stands for the pressure, while f is the external force. Furthermore, n is the outward unit normal to the boundary ∂Ω of Ω, uτ=u−unn is the tangential component of the velocity u, and σnτ=σn−n·σnn is the tangential traction. Of course, σ=−pI+2νεu is the Cauchy stress tensor, where I is the identity matrix, and εu=1/2∇u+∇uT.It can easily be shown that (4) is equivalent to
(7)−σnτ∈g∂uτonS×0,∞,where the symbol ∂. is the subdifferential of the real value function ., with u2=u·u. We recall that if X is the Hilbert space with x0∈X, then,
(8)y∈∂Ψx0⇔Ψx−Ψx0≥y·x−x0,forallx∈X.It should be mentioned that different boundary conditions describe different physical phenomena. The slip boundary condition of friction type can be justified by the fact that frictional effects of the fluid at the pores of the solid can be very important. Many studies have focused on the properties of the solution of the resulting boundary value problem, for example, existence, uniqueness, regularity, and continuous dependence on data, for Stokes, Navier-Stokes, and Brinkman–Forchheimer equations under such boundary condition. Details can be found in [1–4] among others. In [5], a generalization of the boundary condition (4) is formulated and analyzed for the steady Stokes flow, while the case of Navier-Stokes equations has been examined in [6]. There are numerous works devoted to the development of efficient schemes for the nonstationary Navier-Stokes problem dealing with Dirichlet or periodic boundary conditions; some works can be found in [7–11]. It should also be mentioned that there are other works dealing with Navier-Stokes equations with time fractional derivatives (see for instance [12] and references therein). For the time fractional operators, details can be found in [13, 14].The subject of the present work is to establish the well-posedness and stability of the numerical scheme onL2-norm and H1-norm for all positive times of the two-dimensional problems (1)-(4) using the Crank-Nicholson scheme in time and the finite element approximation in space. The resulting variational structure dealing with is in the form of inequality, and obtaining H1-estimate is more involved because of the presence of the nondifferentiable term appearing at the boundary where slip occurs.


## 2. Preliminaries and Variational Formulation

In this section, we introduce notations and some results that will be used throughout our work. We also formulate various weak formulations and discuss (recall) some existence results. For the mathematical setting of the problem, we need to introduce the following spaces:(9)V=v∈H1Ω2,vΓ=0,v·nS=0,Vσ=v∈V,divv=0,H=v∈Ł2Ω2,divv=0,vΓ=0,v·nS=0,M=L02Ω.V′ is the dual space of V, and the duality pairing between V′ and V is denoted by ·. Throughout the paper, we assume that Ω is bounded, convex planar domain with polygonal boundary. As usual, ϕt stands for the function x∈Ω↦ϕx,t. Next, we introduce the Stokes operator A by following the approach adopted in [15, 16]. We denote by P:L2Ω2⟶H the Helmholtz projection operator, which is bounded projection associated to the Helmholtz decomposition of L2Ω2. We define the Stokes operator as follows A:V⟶V′ such that A=−PΔ, with domain given as follows, DA=v∈V,suchthatAv∈H. Now, assuming that Γ is C2 and S is C3, then DA⊂H2Ω2 since w2≤CAw, and one has
(10)λ1∫Ωv2dx≤∫Ω∇v2dx,forallv∈V,(11)λ1∫Ω∇v2dx≤∫ΩAv2dx,forallv∈DA,where λ1 is the first eigenvalue of the Stokes operator A. It should be noticed that thanks to (10), ∇v is a norm on V equivalent to the usual H1-norm.The Stokes operatorA:DA⟶H is self-adjoint, positive with a compact inverse A−1 which is self-adjoint as a mapping from H to H.We recall some classical bilinear and trilinear forms (see [17, 18])
(12)b:V×M⟶ℝwithbu,p=divu,p,a:V×V⟶ℝwithau,v=νεu,εv=2νu,v,d:V×V×V⟶ℝwithdu,v,w=u·∇v,w.We denote byB a bilinear operator from V×V into V′ such that
(13)Bu,v,w=du,v,w,forallu,v,w∈V.The bilinear formb·,· satisfies the inf-sup conditions; i.e., there exists a positive constant β such that
(14)βp≤supu∈Vbu,pu1,forallp∈L02Ω.As a readily obtainable consequence of Korn’s inequality (11), there exists a positive constant α such that
(15)av,v≥αv12,forallv∈V.The trilinear formd·,·,· is continuous on H1Ω3 and enjoys the following properties:
(16)du,v,w≤cdu1/2Au1/2v1w,forallu∈DA,v∈V,w∈H,(17)du,v,w≤cdu1/2u11/2v1w1/2w11/2,forallu,v,w∈V,(18)du,v,v=0,forallu,v∈Vσ,(19)du,v,w=−du,w,v,forallu,v,w∈Vσ.We will make reference to the following inequalities:(20)2u−v,u=u2−v2+u−v2,forallu,v∈L2Ω,(21)ab≤εpap+1qεq/pbq,foralla,b,ϵ>0and1p+1q=1.We assume thatf∈L∞0,∞;L2Ω2, and we set f∞≔fL∞0,∞;L2Ω2. We also assume that u0∈L2Ω2. With the above notations, we introduce the following variational formulation for (1)-(6): Find ut,pt∈V×M such that
(22)u0=u0inΩ,and for a.e. t, with t≥0(23)forallv,q∈V×M,u′t,v−ut+aut,v−ut+dut,ut,v−ut−bv−ut,pt+Jv−Jut≥ft,v−ut,but,q=0,where Jv=g,vτS.Note that since the bilinear formb·,· satisfies the inf-sup condition (14), the variational inequality problem (23) is equivalent to the following:Findut∈Vσ such that
(24)u0=u0inΩ,and for a.e.t, with t≥0(25)forallv∈Vσ,u′t,v−ut+aut,v−ut+dut,ut,v−ut+Jv−Jut≥ft,v−ut.The problem of existence and uniqueness of (24)-(25) can be stated as follows and has been proved in Kashiwabara [3].Theorem 1.
Assume(26)f∈H1ℝ+,L2Ω2,g∈H1ℝ+,L2S2withg0∈H1S,u0∈H2Ω2∩Vσ,andslipboundarycondition1.4issatisfiedatt=0,i.e.,στu0≤g0andστu0u0τ+gu0τ=0a.e.onS.
Then, there exists a unique solutionu of problem (24)-(25) such that
(27)u∈L∞ℝ+,Vσ,andu′∈L∞ℝ+,L2Ω2∩L2ℝ+,Vσ.
Lettn/tn=nk,n∈ℕ be a uniform partition of ℝ+ with a given time step k. We consider a time discretization of (24)-(25) using the Crank-Nicolson scheme. Find un∈V such that
(28)u0=u0inΩ,and for all n≥1(29)forallv∈Vσ,un−un−1k,v−un+12aun+un−1,v−un+14dun+un−1,un+un−1,v−un+Jv−Jun≥fn,v−un,where fn=1/k∫tn−1tnftdt. We want to show that the solution un of (28)-(29) is uniformly bounded for all n≥0, in both the L2- and H01-norms. In what follows, we discretize in space and derived such a result assuming some kind of stability condition.


## 3. Numerical Scheme

For the spatial discretization, we introduce the general framework as in, e.g., [18, 19]. We consider a family of finite element spaces Vσh⊂L2Ω2, each of which is endowed with two scalar products, ·,·h and ·,·h, with the corresponding norms, ·h and ·1,h which mimic the L2- and H01-norms. These norms are related as follows:
(30)uhh≤K1uh1,h,foralluh∈Vσh,(31)uh1,h≤Shuhh,foralluh∈Vσh,where K1 is independent of h and Sh is such that
(32)Sh⟶∞ash⟶0.We assume that the operatorA satisfies the same properties on Vσh as on V. We also assume that a trilinear continuous form d·,·,· enjoys the same properties on Vσh×Vσh×Vσh with the constant cd independent of h.We introduce the so-called restriction operatorsrh:Vσ⟶Vσh and assume that, if u0∈Vσ∩C1Ω¯2, then,
(33)rhu0h≤K2u0C1Ω¯2,with the constant K2 being independent of h (see, e.g., [18]).As for the temporal discretization, we consider the following scheme, a discrete version of (24)-(25): Find uhn∈Vσh such that
(34)uh0=rhu0,and for all n≥1(35)forallvh∈Vσh,uhn−uhn−1k,vh−uhnh+12auhn+uhn−1,vh−uhn+14duhn+uhn−1,uhn+uhn−1,vh−uhn+Jvh−Juhn≥fhn,vh−uhnh.Remark 2.
For existence and uniqueness of the solution of (34)-(35), one observes that the variational inequality (34)-(35) can be seen as a case of following modified variational formulation associated to the stationary Navier-Stokes equations with slip boundary condition type.
(36)Findvh∈Vσh,Tvh,wh−vh+Dvh,vh,wh−vh+jwh−jvh≥F,wh−vh,forallwh∈Vσh,where Tu,v=u,v+k/2au,v, Du,v,w=k/4du,v,w, jv=kJv and F=kfhn. Following [6, 20, 21], (36) admits a unique solution vh∈Vσh.


## 4. TheVh,·h-Stability

We start this section by performing the stability analysis of the scheme (34)-(35) in Vσh,·h and show that the solution is uniformly bounded, provided that a stability CFL-type condition is satisfied.Lemma 3.
LetM≥K122f∞/ν be arbitrarily fixed and assume that u0≤M and
(37)kS2h≤4ν15cd2M2+ν2.
Then, for any integern≥1, we have
(38)uhnh2≤1+Ck−nu02+Cf∞21−1+Ck−n,(39)uhnh≤M,(40)Juhn≤K122νf∞2,(41)k∑j=inuhj1,h2≤CM2+n−i+1kf∞2,foralli=1,⋯,n,(42)∑i=1nuhi−uhi−1h2≤CM2+nkf∞2.Proof.
We first establish the relation (52) below and next use it to handle the proof by induction. First, let vh=0 and vh=2uhn in (35); one has
(43)1kuhn−uhn−1,uhnh+12auhn+uhn−1,uhn+14duhn+uhn−1,uhn+uhn−1,uhn+Juhn=fhn,uhnh,that is
(44)2uhn−uhn−1,uhnh+kauhn+uhn−1,uhn+k2duhn+uhn−1,uhn+uhn−1,uhn+2kJuhn=2kfhn,uhnh.
Using relation (20), we have
(45)2uhn−uhn−1,uhnh=uhnh2−uhn−1h2+uhn−uhn−1h2.
Using Cauchy-Schwarz inequality, (31) and (21), we write
(46)kauhn+uhn−1,uhn=2νkuhn1,h2+kauhn−1−uhn,uhn≥2νkuhn1,h2−νkuhn1,huhn−uhn−11,h≥2νkuhn1,h2−νkShuhn1,huhn−uhn−1h≥2νkuhn1,h2−16uhn−uhn−1h2−32ν2k2S2huhn1,h2,and the right hand side of (44) is bounded as follows:
(47)2kfhn,uhnh≤2K1kfhnhuhn1,h≤νkuhn1,h2+K12νkf∞2.
To bound the nonlinear termd·,·,· in (44), we write it as
(48)k2duhn+uhn−1,uhn+uhn−1,uhn=kduhn−1,uhn+uhn−1,uhn+k2duhn−uhn−1,uhn+uhn−1,uhn.
Using (18) and (19), we have
(49)duhn−1,uhn,uhn−uhn−1=duhn−1,uhn,uhn−duhn−1,uhn,uhn−1=duhn−1,uhn,uhn+duhn−1,uhn−1,uhn=duhn−1,uhn+uhn−1,uhn.
Hence, using (16),(17), and (21) and recalling (31), we obtain the following bounds:
(50)kduhn−1,uhn+uhn−1,uhn=kduhn−1,uhn,uhn−uhn−1≤cdkShuhn−1huhn1,huhn−uhn−1h≤16uhn−uhn−1h2+32cd2k2Sh2uhn−1h2uhn1,h2,(51)k2duhn−uhn−1,uhn+uhn−1,uhn=−k2duhn−uhn−1,uhn,uhn−1≤cdkShuhn−uhn−1huhn1,huhn−1h≤16uhn−uhn−1h2+38cd2k2Sh2uhn−1h2uhn1,h2.
Gathering (44)-(51), we obtain
(52)uhnh2−uhn−1h2+12uhn−uhn−1h2+νk1−158νkSh2cd2uhn−1h2+ν2uhn1,h2+2kJuhn≤K12νkf∞2.
Note that according to CFL-condition (37), if
(53)uhnh≤M,then
(54)0≤1−158νkSh2cd2uhn−1h2+ν2≤12.
We now use the induction. It is clear that (38) and (39) hold for n=0. Then assuming that (38) holds for n=0,⋯,m−1, for m≥2, we see under the assumption of Lemma 3 that (39) holds for n=0,⋯,m−1. Then (52), together with (39) and (37), implies
(55)uhnh2−uhn−1h2+12uhn−uhn−1h2+ν2kuhn1,h2+2kJuhn≤K12νkf∞2,foralln=1,⋯,m.
If we drop the last term on the left hand side and rewrite the remaining equation withn replaced by j and take the sum with j=i,⋯,n, for some 1≤i≤n, we obtain
(56)uhnh2+12∑j=inuhj−uhj−1h2+ν2k∑j=inuhj1,h2≤M2+K12νn−i+1kf∞2,and hence, (41) and (42) hold for all n=1,⋯,m−1.
Now using (30), relation (55) implies
(57)uhnh2≤11+ν/2K12kuhn−1h2+K121+ν/2K12kνkf∞2,foralln=1,⋯,m.
Using recursively (57), we obtain
(58)uhmh2≤11+ν/2K12kmuh0h2+K12νkf∞2∑i=1m11+ν/2K12ki,≤1+ν2K12k−mu02+Cf∞21−1+ν2K12k−m.Thus, (38) holds for n=m.


## 5. TheVh,·1,h-Stability

For proving the uniform bound ofuhn in Vh,·1,h for all n≥1, we first show that it is bounded on any finite interval of time. Then we extend the result to the infinite time using the discrete uniform Gronwall lemma.Lemma 4.
LetM≥K122f∞+C1AgS/ν be arbitrarily fixed and assume that u0≤M, and assume also that the CFL-condition (37) is satisfied. Assume that k also satisfies
(59)k≤4K12ν≔κ1.
Assume also that for somen the following is true:
(60)K3M2kL1uhn−11,h2+2κ1νf∞2+C1AgS2≤16,where L1=2+3cd2M2/ν2, C1=C2/λ1 is given by (65) and K3 is given by (64). Then,
(61)uhn1,h2≤uhn−11,h21+K4M2kuhn−11,h2+f∞2+AgS2+K5kf∞2+AgS2,where K4 and K5 are positive constants independent of h and n.Proof.
Letvh=uhn−Auhn+uhn−1 in (35); we obtain
(62)1kuhn−uhn−1,Auhn+uhn−1h+ν2Auhn+uhn−1h2+14duhn+uhn−1,uhn+uhn−1,Auhn+uhn−1≤Juhn−Auhn+uhn−1−Juhn+fhn,Auhn+uhn−1h,that is
(63)uhn1,h2−uhn−11,h2+ν2kAuhn+uhn−1h2+14kduhn+uhn−1,uhn+uhn−1,Auhn+uhn−1≤kJAuhn+uhn−1+kfhn,Auhn+uhn−1h.
Using relations (16) and (21) and the uniform bound (39), we majorize the trilinear form as
(64)14kduhn+uhn−1,uhn+uhn−1,Auhn+uhn−1≤14kcduhn+uhn−1h3/2uhn+uhn−11,hAuhn+uhn−1h3/2≤14kcdMuhn1,hAuhn+uhn−1h3/2+uhn−11,hAuhn+uhn−1h3/2≤ν8kAuhn+uhn−1h2+K3M2kuhn−11,h4+K3M2kuhn1,h4,where K3=27cd4/16ν3.
(65)JAuhn+uhn−1=Ag,A−1Auτhn+uτhn−1S≤AgSA−1Auτhn+uτhn−1S≤AgSuτhn+uτhn−1S≤CAgS∇uhn+uhn−1≤Cλ1AgSAuhn+uhn−1≤ν8Auhn+uhn−1h2+C12νAgS2,where C1=C2/λ1.
Using Cauchy-Schwarz inequality and relation (21), we have that
(66)kfhn,Auhn+uhn−1h≤kf∞Auhn+uhn−1h≤ν8kAuhn+uhn−1h2+2νkf∞2.
Gathering relations (63)-(66), we find
(67)uhn1,h2−uhn−11,h2+ν4kAuhn+uhn−1h2≤K3M2kuhn−11,h4+K3M2kuhn1,h4+2νkf∞2+C12νkAgS2,from which we obtain
(68)K3M2kuhn1,h4−uhn1,h2+K3M2kuhn−11,h4+uhn−11,h2+2νkf∞2+C12νkAgS2≥0,foralln≥1.
From (68), we have either
(69)uhn1,h2≤1−Δhn−12K3M2kor
(70)uhn1,h2≥1+Δhn−12K3M2k,where
(71)Δhn−1=1−4K3M2kK3M2kuhn−11,h4+uhn−11,h2+2νkf∞2+C12νkAgS2≥13by59and60.
Let us show that with our assumption, (70) is impossible. Taking vh=uhn−1 in (35), we find
(72)uhn−uhn−1h2+ν2kuhn1,h2−ν2kuhn−11,h2+14kduhn+uhn−1,uhn+uhn−1,uhn−uhn−1+kJuhn−Juhn−1≤kfhn,uhn−uhn−1h.
Using (30) and (21), we bound the right hand side of (72) by
(73)K1kfuhn1,h+K1kfuhn−11,h≤ν12kuhn1,h2+ν2kuhn−11,h2+7K122νkf∞2.
Sinced·,·,· is a trilinear form, we can rewrite the nonlinear term in (72) as
(74)14kduhn+uhn−1,uhn+uhn−1,uhn−uhn−1=12kduhn,uhn−1,uhn−12kduhn−1,uhn,uhn−1,and using property (17), we obtain the following bounds:
(75)12kduhn,uhn−1,uhn≤12cdkuhnhuhn1,huhn−11,h≤ν12kuhn1,h2+34νcd2kuhnh2uhn−11,h2,12kduhn−1,uhn,uhn−1≤12cdkuhn−1huhn−11,huhn1,h≤ν12kuhn1,h2+34νcd2kuhn−1h2uhn−11,h2.
Employing (40), we bound the last term of the left hand side of (72) by
(76)−K122νkf∞2≤kJuhn−Juhn−1≤K122νkf∞2.
Gathering (72)-(76) and recalling (39), we obtain
(77)uhn−uhn−1h2+ν4kuhn1,h2−ν+32νcd2M2kuhn−1h2≤8K122νkf∞2,and hence,
(78)kuhn1,h2≤22+3ν2cd2M2kuhn−1h2+16K12ν2kf∞2,from which we find, using (60),
(79)2K3M2kuhn1,h2≤23<1.
(79) contradicts (70), and therefore, we obtain
(80)uhn1,h2≤1−Δhn−12K3M2k=2K3M2kuhn−11,h4+uhn−11,h2+2/νkf∞2+C12/νkAgS21+1−x,where x=4K3M2kK3M2kuhn−11,h4+uhn−11,h2+2/νkf∞2+C12/νkAgS2.
Sincex≤4/5 (by (60)) and
(81)21+1−x≤1+x2if0≤x≤45,we obtain, using (59) and (60) and the fact that M≥K122f∞+C1AgS/ν,
(82)uhn1,h2≤K3M2kuhn−11,h4+uhn−11,h2+2νkf∞2+C12νkAgS2×1+2K3M2kK3M2kuhn−11,h4+uhn−11,h2+2νkf∞2+C12νkAgS2≤K3M2kuhn−11,h4+uhn−11,h2+2νkf∞2+C12νkAgS2+2K3M2kL1uhn−11,h2+κ1νf∞2+C1AgS22≤uhn−11,h21+K4M2kuhn−11,h2+f∞2+AgS2+K5kf∞2+AgS2,with appropriate choice of constants K4 and K5.To prove that scheme (35) is conditionally stable on a finite interval of time, we need the following discrete Gronwall lemma [22].Lemma 5.
Discrete Gronwall Lemma.
Givenk>0, an integer n⋆>0, and positive sequences αn, βn, and γn such that
(83)αn≤αn−11+kβn−1+kγn,foralln=1,⋯,n⋆,we have
(84)αn≤α0expk∑i=0n−1βi+∑i=1n−1kγiexpk∑j=in−1βj+kγn,foralln=2,⋯,n⋆.Proof.
Using recursively (83), we derive
(85)αn≤α0∏i=0n−11+kβi+∑i=1n−1kγi∏j=in−11+kβi+kγn,and since 1+x≤expx, for all x∈ℝ, the conclusion of the lemma follows.Proposition 6.
Estimates on a finite interval of time.
LetT>0 and M≥K122f∞+C1AgS/ν be fixed, and let u0≤M. Assume that, besides the CFL-condition (37), k also satisfies
(86)k≤minκ1,κ2M,f∞,AgS,κ3M,uh01,h,f∞,AgS,T,where
(87)κ2M,f∞=112K3K6M2f∞2+AgS2,κ3M,uh01,h,f∞,T=112K3M2L1L2M,uh01,h,f∞,AgS,T.
L2·,·,·,· is a monotonically increasing function in all its arguments and is given in (95) below and K6=8K12/ν2.
Then,(a)
Relation (58) holds for all n=1,⋯,N=T/k (integer part of T/k)(b)
uhn1,h2≤L2M,uh01,h,f∞,AgS,nk,foralln=1,⋯,N=T/kProof.
LetT>0 and let h,k be such that (37) and (86) are satisfied.
We will use induction onn. If n=1, assumption (86) implies
(88)K3M2kL1uh01,h2+2κ1νf∞2+C1AgS2≤16.
Thus, the conclusion (61) of Lemma 4 holds for n=1. Now assume that (60) holds for n=1,⋯,m, for some m≤N. Hence, (61) holds for n=1,⋯,m. If we rewrite (61) as (83) with
(89)αn=uhn1,h2,βn=K4M2uhn1,h2+f∞2+AgS2andγn=K5f∞2+AgS2and noting that, using (41), we have
(90)k∑j=im−1βj=K4M2k∑j=im−1uhj1,h2+f∞2+AgS2≤2K7M2M2+m−ikf∞2+AgS2,and therefore,
(91)∑i=1m−1kγiexpk∑j=im−1βj≤K5kf∞2+AgS2×∑i=1m−1exp2K7M2M2+m−ikf∞2+AgS2≤K5f∞2+AgS2mkexp2K7M4exp2K7M2mkf∞2+AgS2.
Similarly, fori=0, we have
(92)k∑j=0m−1βj=K4M2k∑j=0m−1uhj1,h2+f∞2+AgS2≤2K7M2M2+mkf∞2+AgS2+K4M2kuh01,h2.
Using (86) and recalling that L1≥2, the last term of (83) can be bounded as
(93)K4M2kuh01,h2≤K4uh01,h212K3L1L2M,uh01,h,f∞,AgS,T≤K424K3.
Then, Lemma5 and relations (90)-(93) imply
(94)uhm1,h2≤L2M,uh01,h,f∞,AgS,mk,where
(95)L2M,uh01,h,f∞,AgS,mk=uh01,h2exp2K7M4+K424K3×exp2K7M2mkf∞2+AgS2+2K5f∞2+AgS2mkexp2K7M2exp2K7M2mkf∞2+AgS2.
Using (94) and recalling assumption (86), it is easily checked that condition (60) holds for n−1=m, and by the same Lemma 5, we have (61) that holds for n=m+1.To prove the uniform bound ofuhn1,h for all n≥1, we will repeatedly apply Proposition 6 on different intervals of time, considering different initial values, and we will need the following discrete uniform Gronwall lemma, a generalized version of the discrete uniform Gronwall lemma of Shen [22], whose proof can be found in [7].Lemma 7.
Discrete uniform Gronwall lemma.
Givenk>0, positive integers n1,n2,n⋆ such that n1≤n⋆, n1+n2+1≤n⋆, and positive sequences αn, βn and γn such that
(96)αn≤αn−11+kβn−1+kγn,foralln=1,⋯,n⋆.
Assume also that for anyn′ satisfying n1≤n′≤n⋆−n2(97)∑n=n′n′+n2kβn≤C1n1,n⋆,∑n=n′n′+n2kαn≤C2n1,n⋆,∑n=n′n′+n2kγn≤C3n1,n⋆,then we have
(98)αn≤C3n1,n⋆kn2+C2n1,n⋆expC1n1,n⋆,foranyn1+n2+1≤n≤n⋆.Theorem 8. Uniform bound ofuhn1,h for all n≥1.
Letu0∈Vσ∩C1Ω¯2, f∈L∞ℝ+;H, and assume that u0≤M, where M≥K122f∞+C1AgS/ν. Also let r≥4κ1 be arbitrarily fixed and assume that, besides the CFL-condition (37), k also satisfies
(99)k≤minκ1,κ2M,f∞,AgS,κ3M,K2uh0C1Ω¯2,f∞,AgS,r,κ3M,ρ1,f∞,AgS,r,where κ1,κ2,κ3 are defined above and ρ1 is given in (107) below.
Then, we have(100)uhn1,h2≤L3uh0C1Ω¯2,f∞,AgS,foralln≥1,where L3·,·,· is a continuous function defined on ℝ+3, increasing.
Moreover, there exists anN>0 such that
(101)uhn1,h2≤L4f∞,AgS,foralln≥N.Proof.
In order to derive uniform boundsuhn1,h for all n≥1, we apply Proposition 6 on successive intervals of time, with different initial values. On each interval considered, we obtain a bound L2·,·,·,· which depends on the norm uh01,h and on the length of the interval. Using the discrete uniform Gronwall lemma, we majorize the norm of the initial values uh01,h by a constant ρ1, and recalling the fact that L2 is an increasing function of its arguments, we obtain a bound independent on the initial value considered.
First using (33) and (99) and since κ3 is a decreasing function of its arguments, we can apply Proposition 6 with T=r to obtain
(102)uhn1,h2≤uhn−11,h21+K4M2kuhn−11,h2+f∞2+AgS2+K5kf∞2+AgS2,(103)uhn1,h2≤L2M,uh01,h,f∞,AgS,r,foralln=1,⋯,Nr≔r/k.
To extend the bound (103) to n=Nr+1,⋯,2Nr, we apply again Proposition 6, namely, L2M,uhNr1,h,f∞,AgS,r depends on the discrete initial value; we want to bound uhNr1,h independent of h and k.
Rewrite (102) in the form of (96) with αn=uhn1,h2, γn=K5f∞2+AgS2 and βn=K4M2uhn−11,h2+f∞2+AgS2. Then, we apply Lemma 7 with n1=1, n2=Nr−2, n⋆=Nr to obtain the bound of uhNr1,h. For n′=1,2, using (41), we have
(104)k∑n=n′n′+n2βn=K4M2k∑n=n′n′+n2uhn1,h2+f∞2+AgS2≤K8M2M2+rf∞2+AgS2,(105)k∑n=n′n′+n2γn=K5k∑n=n′n′+n2f∞2+AgS2≤K5rf∞2+AgS2,(106)k∑n=n′n′+n2αn=k∑n=n′n′+n2uhn1,h2≤K9M2+rf∞2.
Then, Lemma7, together with the assumption r≥4κ1, yields
(107)uhNr1,h2≤2K9M2/r+f∞2+K5rf∞2+AgS2×expK8M2M2+rf∞2+AgS2≔ρ1M,f∞,AgS,r.
Taking into account assumption (99) on the time step k, relation (107), and the fact that L2·,·,· is an increasing function of its arguments, we apply Proposition 6 with T=r and initial data uhNr. We obtain that the relation (61) holds for all n=Nr+1,⋯,2Nr, and
(108)uhn1,h2≤L2M,uhNr1,h,f∞,AgS,r≤L2M,ρ1,f∞,AgS,r,foralln=Nr+1,⋯,2Nr.
Applying again Lemma7 with n1=Nr+1, n2=Nr−2 and n⋆=2Nr, we obtain
(109)uh2Nr1,h2≤ρ1.
Iterating the above procedure, we find(110)uhn1,h2≤L2M,ρ1,f∞,AgS,r≔L3f∞,AgS,foralln≥Nr,and recalling (103), we conclude
(111)uhn1,h2≤maxL2M,uh01,h,f∞,AgS,r,L3f∞,AgS≤maxL2M,K2u0C1Ω¯2,f∞,AgS,r,L3f∞,AgSby32≔L4K2u0C1Ω¯2,f∞,AgS,foralln≥1.
As for theN beyond which uhn1,h is bounded independent of u0, we can evidently take N=Nr (see (110)). This completes the proof of the theorem.


## 6. Numerical Experiments

Let us explain our numerical experiments. We assumeΩ=0,12, the boundary of which consists of two portions Γ and S given by
(112)Γ=0,y,0<y<1Ux,0,0<x<1U1,y,0<y1Ux,1,0<x<1,S=x,10<x<1.The time interval is given by0,T with T=1. For the triangulation Th of Ω¯, we employ a uniform N×N mesh, where N denotes the division number of each side of the domain. The implementation is done by extending the Matlab code developed in [23, 24]. In all the examples presented, the velocity and pressure will be approximated by P2−P1 element. Let us consider
(113)u1t,x,y=20x21−x2y1−2yexp−t,u2t,x,y=−20x1−x1−2x1−y2y2exp−t,pt,x,y=2x−12y−1exp−t.The initial condition is given byu0x,y=u10,x,y,u2x,y.The functionsf and g are chosen such that u,p defined above is the solution of (1)-(5).It is easy to verify that the solutionu satisfies u=0 on Γ, u·n=u2=0,u1≠0 on S. By direct computations, we have
(114)στ=−60x21−x2exp−tonS×0,T,uτ=20x21−x2exp−tonS×0,T,maxSστ=3.75exp−t,∀t∈0,T.On the other hand, from the slip boundary conditions (5), we have
(115)στ≤gonS×0,T,then we find from (104) that for the given function g:
(116)g≥3.75exp−t⇒113remainsasolution,g<3.75exp−t⇒6.3isnolongerasolutionandanon−trivialslipoccurs.Indeed, it is observable in Figure1, slip and non-slip condition on the boundary. In fact in Figure 1(a), g<3.75exp−t and we see the manifestation of the slip due to the adherence of the flow at the boundary, whereas in Figure 1(b), g≥3.75exp−t and no slip occurs.Figure 1
Velocity field, respectively, forg=1 and g=4.
(a)(b)To analyze the convergence rate, we simulated the same problem. Since we do not know the explicit exact solution wheng=1, we employ the approximate solutions with N=60 as the reference solutions uref,pref, and we compute the L2-norm for velocity of the difference of the reference solution and the approximate solution uh,ph.For the convergence with respect to the mesh sizeh, we choose k=h2 and we solve problem (35) with different values of h (h=1/5;1/10;1/15;1/20;1/25). In Figure 2(a), we plot the log of L2-errors against logh.Figure 2
L2-error estimate, respectively, for mesh size h and time step k.
(a)(b)For the convergence with respect to the time stepk, h is fixed (h=0.01) and we solve problem (35) with different time steps k=0.1;0.05;0.025;0.0125. Figure 2(b) shows the plots of logL2-error norm against logk.


## 7. Conclusions

In this paper, we have proposed and analyzed the Crank-Nicolson scheme for the two-dimensional Navier-Stokes equations driven by slip boundary conditions of friction type. We established the well-posedness and stability of the numerical scheme inL2-norm and H1-norm for all positive time using the Crank-Nicholson scheme in time and the finite element method in space. We have proven that the numerical scheme is stable in L2 and H1-norms with the aid of different versions of discrete Grownwall lemmas, under a CFL-type condition.



---

*Source: 7742867-2022-03-20.xml*