2. Preliminaries
Let Ω be an open bounded domain of ℝ2 with Lipschitz continuous boundary ∂Ω and satisfy a further condition stated in (A1) below. The incompressible stationary Navier-Stokes equations with the homogeneous Dirichlet boundary condition are
(2.1)-νΔu+(u·∇)u+∇p=f in Ω,div u=0, in Ω,u|∂Ω=0 on ∂Ω,
where u=(u1(x),u2(x))T represents the velocity, p=p(x) the pressure, f=f(x)∈L2(Ω)2 the prescribed body force, ν>0 the viscosity coefficient.
In order to introduce the variational formulation for problem (2.1), we set
(2.2)X=H01(Ω)2, Y=L2(Ω)2, D(A)=H2(Ω)2∩X,M=L02(Ω)={q∈L2(Ω):∫Ωq dx=0}.
The standard notations of Sobolev space Wm,r(Ω) are used. To simplify, we use Hm(Ω) instead of Wm,r(Ω) as r=2 and ∥·∥m for ∥·∥m,2. The spaces L2(Ω)m (m=1,2) are endowed with the usual L2-scalar product (·,·) and L2-norm ∥·∥0. The spaces H01(Ω) and X are equipped with the scalar product (∇u,∇v) and the norm |u|1,Ω2, u, v∈ H01(Ω) (or X).
Define Au=-Δu is the operator associated with the Navier-Stokes problem, it is positive self-adjoint operator from D(A) onto Y.
Introducing the bilinear operator
(2.3)B(u,v)=(u·∇)v+12(div u)v ∀u,v∈X,
and defining a trilinear form on X×X×X as follows:
(2.4)b(u,v,w)=〈B(u,v),w〉X′×X=((u·∇)v,w)+12((div u)v,w)=12((u·∇)v,w)-12((u·∇)w,v).
The variational formulation of problem (2.1) reads as: find (u,p)∈(X,M) such that for all (v,q)∈(X,M)(2.5)a(u,v)-d(v,p)+d(u,q)+b(u,u,v)=(f,v),
where
(2.6)a(u,v)=ν(∇u,∇v), d(v,q)=-(v,∇q)=(q,div v),B0((u,p);(v,q))=a(u,v)-d(v,p)+d(u,q).
Clearly, the bilinear forms a(·,·) and d(·,·) are continuous on X×X and X×M, respectively. Moreover, d(·,·) also satisfies (see [20]):
(2.7)sup 0≠v∈X|d(v,q)||v|1,Ω≥β∥q∥0,Ω,
where β is a positive constant depending only on Ω.
It is easy to verify that B0 satisfies the following important properties for all (u,p), (v,q)∈(X,M) (see [1]):
(2.8)B0((u,p);(u,p))=ν∥u∥1,Ω2,(2.9)|B0((u,p);(v,q))|≤C(∥u∥1,Ω+∥p∥0,Ω)(∥v∥1,Ω+∥q∥0,Ω),(2.10)β0(∥u∥1,Ω+∥p∥0,Ω)≤sup (v,q)∈(X,M)|B0((u,p);(v,q))|∥v∥1,Ω+∥q∥0,Ω,
where β0>0 is a constant. Here and below, the letter C (with or without subscript) denotes a generic positive constant, depending at most on the data ν, Ω and f. Furthermore, the following estimates about b(·,·,·) are hold [1, 20]:
(2.11)b(u,v,w)=-b(u,w,v),(2.12)|b(u,v,w)|≤12c0∥u∥0,Ω1/2|u|1,Ω1/2(|v|1,Ω∥w∥0,Ω1/2|w|1,Ω1/2+∥v∥0,Ω1/2|v|1,Ω1/2|w|1,Ω),
for all u, v, w∈X and
(2.13)|b(u,v,w)|+|b(v,u,w)|+|b(w,u,v)|≤C|u|1,Ω∥Av∥0,Ω∥w∥0,Ω,
for all u∈X, v∈D(A), w∈Y.
As mentioned above, a further assumption about Ω is needed (see [1]).
(
A
1
)
Assume that Ω is regular so that the unique solution (v,q)∈(X,M) of the steady Stokes equations
(2.14)-νΔv+∇q=g, div v=0 in Ω, v|∂Ω=0,
for a prescribed g∈Y exists and satisfies
(2.15)∥Av∥0,Ω+|q|1,Ω≤C∥g∥0,Ω.
Under the assumption of (A1), if ∂Ω is of C2 or Ω is a two-dimensional convex polygon, it has been shown that (see [20])
(2.16)∥v∥0,Ω≤γ0|v|1,Ω, ∀v∈X, |v|1,Ω≤γ0∥Av∥0,Ω, ∀v∈D(A),
where γ0 is a positive constant only depending on Ω.
The following existence and uniqueness results about problem (2.5) are classical (see [1, 20]).
Theorem 2.1.
Assume that ν and f∈Y satisfy the following uniqueness condition:
(2.17)1-c0γ02ν2∥f∥0,Ω>0.
Then problem (2.5) admits a unique solution (u,p)∈(D(A),H1(Ω)∩M) with div u=0 such that
(2.18)|u|1,Ω≤γ0ν∥f∥0,Ω, ∥Au∥0,Ω+|p|1,Ω≤C∥f∥0,Ω.
3. Enriched Nonconforming Finite Element Method
Let 𝒯h be a regular triangulation of Ω into element {Kj}:Ω¯=∪K¯j, that is, |Kj|≃ChKj2, where |Kj| is the area of the element Kj and hKj is the diameter of Kj; the mesh parameter h is given by h=max {hKj:Kj∈𝒯h}. Denote the boundary segment and the interior boundary by γj=∂Ω∩∂Kj and γjk=γkj=∂Kj∩∂Kk, respectively. Let Γh and Γj be the sets of γjk and γj. The centers of γj and γjk are indicated by ξj and ξjk, respectively. The finite element spaces investigated in this paper are the following mixed finite element spaces:
(3.1)NCP1={v∈Y:v|Kj∈P1(Kj)2, v(ξjk)=v(ξkj), v(ξj)=0 ∀j,k,Kj∈𝒯h},P1={q∈H1(Ω):q|Kj∈P1(Kj), ∀Kj∈𝒯h},
where P1(Kj) is the set of line polynomials on Kj, and noting that the nonconforming finite element space NCP1 is not a subspace of X. Defining the energy norm (3.2)∥v∥1,h=(∑Kj|v|1,Kj2)1/2, ∀v∈NCP1.
The finite element spaces NCP1 and P1 satisfy the following approximation property (see [4, 21]): for (v,q)∈H2(Ω)×H1(Ω), there are two approximations vI∈NCP1 and qI∈P1 such that
(3.3)∥v-vI∥0,Ω+h(∥v-vI∥1,h+∥q-qI∥0,Ω)≤Ch2(∥Av∥0,Ω+|q|1,Ω),
and the compatibility conditions hold for all j and k:
(3.4)∫γjk[v]ds=0, ∫Γjvds=0 ∀v∈NCP1,
where [v]=vγjk-vγkj denotes the jump of the function v across the boundary γjk.
Set 〈·,·〉j=(·,·)∂Kj and |·|m,j=|·|m,Kj. Then for all u, v∈H1(Kj)2, q∈L2(Ω), the discrete bilinear forms are
(3.5)ah(u,v)=∑Kjν(∇u,∇v)Kj, dh(v,p)=∑Kj(div v,p)Kj.
For the nonconforming space NCP1, we define a local operator
(3.6)Πj:H1(Kj)2→NCP1(Kj),
satisfying
(3.7)∫∂Kj(v-Πjv)ds=0.
Then the local operator Πj satisfies (see [21])
(3.8)|v-Πjv|1,Kj≤Chi|v|i+1,Kj, v∈Hi+1(Kj), i=0,1, ∥Πjv∥1,Kj≤C∥v∥1,Kj.
The global operator Πh:X→NCP1 is defined as Πhv|j=Πjv, v∈X.
As noted, the choice NCP1-P1 is an unstable pair that does not satisfy the discrete Inf-Sup condition. Therefore, we need to introduce the enrichment multiscale method to overcome this restriction.
Let Eh be a finite dimensional space, called multiscale space, such that
(3.9)Eh∈H1(𝒯h)2, Eh∩NCP1={0}, where H1(𝒯h)2={v∈Y:v|Kj∈H1(Kj)2}.
The discrete weak formulation of the Stokes equations is to find uh+ue∈NCP1⊕Eh and ph∈P1, such that
(3.10)ah(uh+ue,vh)-dh(vh,ph)+dh(uh+ue,qh)=(f,v)Ω,
for all vh∈NCP1⊕H01(𝒯h)2 and qh∈P1. Let ue|Kj=ueKj+ue∂Kj, we can solve it through the following local problem:
(3.11)-νΔueKj=f+νΔuh-∇ph in Kj, ueKj|∂Kj=0,-νΔue∂Kj=0 in Kj, ue∂Kj=ge on ∂Kj,-ν∂ssge=1he[ν∂nuh+phI·n]E, ge=0 at the nodes,
where he denotes the length of the edge E∈∂Kj; n the normal outward vector on ∂Kj; ∂s, ∂n are the tangential and normal derivative operators, respectively; I is the ℝ2×2 identity matrix. Equation (3.11) is well posed, that is, ue can be expressed by uh, ph, and f on each element Kj. For convenience, we define two local operators ℳKj:L2(Kj)2→H01(Kj)2 and ℋKj:L2(∂Kj)2→H1(Kj)2 by
(3.12)ueKj=1νℳKj(f+νΔuh-∇ph), ∀Kj∈𝒯h,ue∂K=1νℋKj([ν∂nuh+phI·n]E), ∀Kj∈𝒯h, E∈Γh.
With Green formulation and (3.12), for all (uh,ph), (vh,qh)∈NCP1×P1, (3.10) can be rewritten as
(3.13)∑Kj[ν(∇uh,∇vh)Kj-(ph,∇·vh)Kj+(qh,∇·uh)Kj] +∑Kj1ν(ℳKj(-νΔuh+∇ph)-ℋKj([ν∂nuh+phI·n]E),νΔvh+∇qh)Kj +∑E∈Γh1ν(ℋKj([ν∂nuh+phI·n]E),ν∂nvh+qhI·n)E =∑Kj[(f,vh)Kj+1ν(ℳKj(f),νΔvh+∇qh)Kj].
With the help of (3.13), the enriched nonconforming finite element method for the stationary Navier-Stokes equations (2.1) is rewritten as follows: find (uh,ph)∈NCP1×P1 such that
(3.14)B((uh,ph);(vh,qh))+b(uh,uh,vh)=F(vh,qh)
for all (vh,qh)∈NCP1×P1, where
(3.15)B((uh,ph);(vh,qh))=Bh((uh,ph);(vh,qh))+∑Kj1ν(ℳKj(∇ph),∇qh)Kj +∑E∈Γh1ν(ℋKj([ν∂nuh]),[ν∂nvh])E≜Bh((uh,ph);(vh,qh))Ω+∑KjτKj(∇ph,∇qh)Kj +∑E∈ΓhτE([ν∂nuh],[ν∂nvh])E,F(vh,qh)=∑Kj[(f,vh)Kj+1ν(ℳKj(f),∇qh)Kj],Bh((uh,ph);(vh,qh))=ah(uh,vh)-dh(vh,ph)+dh(uh,qh).
By applying the technique to one used in [16], we can obtain that (bKj,1)Kj/|Kj|⋍C~hKj2, (aKj,1)E/he⋍he/12, τKj⋍C~hKj2 and τE⋍he/(12ν). Moreover, if f is a piecewise constant, then we have ℳKj(f)=bKjf,
(3.16)(ℳKj(f),∇qh)Kj=(bKj,1)Kj|Kj|(f,∇qh)Kj≃C~hKj2(f,∇qh)Kj.
Define the mesh-dependent norms as follows:
(3.17)∥|u|∥h2=ν∥u∥1,h2+∑E∈ΓhτE∥[ν∂nu]∥0,E2, ∥q∥h2=∑KjτKj|q|1,Kj2.
Remark 3.1.
The assumption of piecewise constant f is made simply to analyze the problem (3.14), but this assumption does not affect the precision of this method, and (3.14) may be implemented as it is presented for a general function f∈L2(Ω)2. Here, we do not give the detail proof about this fact; readers can visit Appendix B of the paper [16] for f∈H1(Ω)2.
Remark 3.2.
Generally speaking, the following linear algebra equations can be obtained from the discrete system of original problem:
(3.18)(A-DDT0)(UP)=(F0),
where the matrices A and D are deduced from the diffusion, convection, and incompressible terms; F is the variation of the source term. The norm of matrix A gets smaller as the convection increases; therefore, some unnecessary oscillations will be created. In order to eliminate these oscillations, we introduce the stabilized term, in this case, the coefficient matrix of discrete formulation transforms into
(3.19)(A-DDTG),
where G is derived from the stabilized term, that is, the term of (∇ph,∇qh). As the considered problem has strong convection, in order to obtain a good behavior of matrix (A-DDTG), we should choose a proper G. In this way, the singularly perturbed problem can be treated effectively. The reason that we treat the convection term not use enriched function technique is to simply the theoretical analysis and computation, and the discrete convection term has no influence about the stabilized term G(·,·).
Lemma 3.3.
Let (vh,qh)∈NCP1×P1, then,
(3.20)B((vh,qh);(vh,qh))=∥|vh|∥h2+∥qh∥h2.
Proof.
The results follow from the definition of (3.15) and the mesh-dependent norms in each K∈𝒯h.
Before establishing the stability of scheme (3.14), we introduce the local trace theorem (see [1]). There exists C>0, independent of h, such that
(3.21)∥u∥0,∂Kj2≤C(hKj-1∥u∥0,Kj2+hKj|u|1,Kj2) , ∀u∈H1(Kj).
Theorem 3.4.
There exist two positive constants C, β depending on ν, for all (uh,ph), (vh,qh)∈NCP1×P1 such that
(3.22)|B((uh,ph);(vh,qh))|≤C(∥uh∥1,h+∥ph∥0,Ω)(∥vh∥1,h+∥qh∥0,Ω),(3.23)sup 0≠(vh,qh)∈(NCP1,P1)|B((uh,ph);(vh,qh))|∥vh∥1,h+∥qh∥0,Ω≥β(∥uh∥1,h+∥ph∥0,Ω).
Proof.
It follows from (uh,ph), (vh,qh)∈NCP1×P1, inverse inequality, (3.15), and (3.21) that
(3.24)|B((uh,ph);(vh,qh))| ≤ν∥uh∥1,h∥vh∥1,h+∥uh∥1,h∥qh∥0,Ω+∥ph∥0,Ω∥vh∥1,h +∑KjτKj|ph|1,Kj|qh|1,Kj+∑E∈ΓhτE∥[ν∂nuh]∥0,E∥[ν∂nvh]∥0,E ≤ν∥uh∥1,h∥vh∥1,h+∥uh∥1,h∥qh∥0,Ω+∥ph∥0,Ω∥vh∥1,h+C1∥ph∥0,Ω∥qh∥0,Ω +C2∑E∈ΓhτE(hKj-1/2∥∇uh∥0,Kj+hKj1/2|∇uh|1,Kj)(hKj-1/2∥∇vh∥0,Kj+hKj1/2|∇vh|1,Kj) ≤C(ν)∥uh∥1,h∥vh∥1,h+∥uh∥1,h∥qh∥0,Ω+∥ph∥0,Ω∥vh∥1,h+C1∥ph∥0,Ω∥qh∥0,Ω ≤C(∥uh∥1,h+∥ph∥0,Ω)(∥vh∥1,h+∥qh∥0,Ω),
that is, the continuity result (3.22) holds.
From the properties of the nonconforming finite element given in [18], for all ph∈L2(Kj), there exists a function w∈H1(Kj)2, such that ∥w∥1,h=∥ph∥0 and
(3.25)(∇·w,ph)Kj=∥ph∥0,Kj2, ∥w∥1,h≤C∥ph∥0,Ω.
Using the Cauchy-Schwartz inequality and (3.25), we have
(3.26)|B((uh,ph);(-w,0))| =-ν(∇uh,∇w)+(ph,∇·w)-∑E∈ΓhτE([ν∂nuh],[ν∂nw])E ≥-ν∥uh∥1,h∥w∥1,h+C0∥w∥1,h∥ph∥0,Ω-∑E∈ΓhτE∥[ν∂nuh]∥0,E∥[ν∂nw]∥0,E ≥-(ν∥uh∥1,h2+∑E∈ΓhτE∥[ν∂nuh]0,E2∥)1/2(ν∥w∥1,h2+∑E∈ΓhτE∥[ν∂nw]∥0,E2)1/2 +C0∥w∥1,h∥ph∥0,h.
Using (3.21) and inverse inequality, we obtain that
(3.27)τE∥[ν∂nw]∥0,E2≤he12ν(hK-1∥ν∇w·n∥0,K2+hK|ν∇w·n|1,K2)≤heν12hK|w|1,K2+CKνhe12hK|w|1,K2≤ν(1+CK)12|w|1,K2.
Combining (3.26) with (3.27) yields
(3.28)|B((uh,ph);(-w,0))| ≥-Cν∥w∥1,h(ν|uh|1,h2+∑E∈ΓhτE∥[ν∂nuh]∥0,E2)1/2+C0∥w∥1,h∥ph∥0,Ω =-Cν∥w∥1,h∥|uh|∥h+C0∥ph∥0,Ω2 ≥-Cνγ1-1∥|uh|∥h2+(C0-γ1)∥ph∥0,Ω2,
where C=(1+C0)/12 with C0=max k∈𝒯hCK, and γ1 is chosen small enough. Let
(3.29)(vh,qh)=(uh-δw,ph), δ>0.
Using (3.26) and Lemma 3.3 we have
(3.30)|B((uh,ph);(vh,qh))|=|B((uh,ph);(uh,ph))+δB((uh,ph);(-w,0))| ≥∥|uh|∥h2+∥ph∥h2+δ(-Cνγ1-1∥|uh|∥h2+(C0-γ1)∥ph∥0,Ω2) ≥(1-Cδνγ1-1)∥|uh|∥h2+∥ph∥h2+δ(C0-γ1)∥ph∥0,Ω2 ≥ν(1-Cδνγ1-1)∥uh∥1,h2+δ(C0-γ1)∥ph∥0,Ω2,
provided that 0<δ<γ1/(Cν) and 0<γ1<C0. Denote
(3.31)C(ν)≜min {ν(1-Cδνγ1-1),δ(C0-γ1)}, C(δ)≜max {2,1+2δ2}.
Then we have
(3.32)∥vh∥1,h2+∥qh∥0,Ω2=∥uh-δw∥1,h2+∥ph∥0,Ω2≤2∥uh∥1,h2+(1+2δ2)∥ph∥0,Ω2≤C(δ)(∥uh∥1,h2+∥ph∥0,Ω2).
Taking β=C(ν)/(C(δ)), we obtain the desired result (3.23).
Theorem 3.5.
Under the assumptions of Theorem 2.1 and the following condition:
(3.33) the strong uniqueness condition: 1-c0γ0γ0+ν1/2ν2∥f∥0,Ω>0.
Problem (3.14) admits a unique solution (uh,ph)∈(NCP1,P1), and satisfying
(3.34)∥uh∥1,h≤γ0+ν1/2ν∥f∥0,Ω,∥ph∥0,Ω≤β-1[(γ0+Cν)∥f∥0,Ω+c0γ0ν-2(γ0+ν1/2)2∥f∥0,Ω2].
Proof.
Let Hilbert space Hh=(NCP1,P1) be with the scalar product and norm
(3.35)((v,q);(w,r))Hh=∑Kj(∇v,∇w)Kj+(q,r),
and Kh be a nonvoid, convex, and compact subset of Hh defined by
(3.36)Kh={c0γ0(γ0+ν1/2)2βν2(v,q)∈Hh:∥v∥1,h≤γ0+ν1/2ν∥f∥0, ∥q∥0,Ω≤νγ0+Cβν∥f∥0,Ω+c0γ0(γ0+ν1/2)2βν2∥f∥0,Ω2}.
Defining a continuous mapping from Kh into Hh as follows: given (v¯,q¯)∈Kh, for all (w,r)∈Hh, find (v,q)=F(v¯,q¯) such that
(3.37)B((v,q);(w,r))+b(v¯,v,w)=(f,w)+∑KjτKj(f,∇r)Kj.
Taking (w,r)=(v,q), using (2.8)–(2.13) and inverse inequality yields
(3.38)ν∥v∥1,h2+∥q∥h2≤∥|v|∥h2+∥q∥h2≤γ0∥f∥0,Ω∥v∥1,h+∑KjτKj∥f∥0,Kj∥∇q∥0,Kj≤(γ022ν+h22)∥f∥0,Ω2+ν2∥v∥1,h2+12∑KjτKj∥∇q∥0,Kj2=(γ022ν+h22)∥f∥0,Ω2+ν2∥v∥1,h2+12∥q∥h2.
As a consequence, we have
(3.39)∥v∥1,h≤γ0+ν1/2ν∥f∥0,Ω.
Using again (2.17), (3.23), (3.37), and inverse inequality, we arrive at
(3.40)β(∥v∥1,h+∥q∥0,Ω)≤|f,w|+|∑KjτKj(f,∇r)Kj|∥w∥1,h+∥r∥0+c0γ0∥v¯∥1,h∥v∥1,h≤γ0∥f∥0,Ω+Chν∥f∥0,Ω+c0γ0ν-2(γ0+ν1/2)2∥f∥0,Ω2≤(γ0ν+Chν)∥f∥0,Ω+c0γ0ν-2(γ0+ν1/2)2∥f∥0,Ω2.
Hence, the two estimates imply (v,q)=F(v¯,q¯)∈Kh, thanks to the fixed point theorem, the mapping (v,q)=F(v¯,q¯)∈Kh has at least one fixed point (uh,ph)∈Kh; namely, (uh,ph)∈Kh is a numerical solution of problem (3.14).
Next, we shall prove that the problem (3.14) has a unique solution (uh,ph). In fact, if (vh,qh) also satisfies (3.14), then for all (w,r)∈(NCP1,P1) we have
(3.41)B((uh-vh,ph-qh);(w,r))=b(vh-uh,uh,w)+b(vh,vh-uh,w).
Taking (w,r)=(uh-vh,ph-qh) in (3.41) and using again (2.8)–(2.13), Lemma 3.3, it follows that
(3.42)ν∥uh-vh∥1,h2≤c0γ0∥uh∥1∥uh-vh∥1,h2≤c0γ0γ0+ν1/2ν∥f∥0,Ω∥uh-vh∥1,h2,
Which, together with the strong uniqueness condition
(3.43)ν-c0γ0γ0+ν1/2ν∥f∥0,Ω=ν(1-c0γ0γ0+ν1/2ν2∥f∥0,Ω)>0,
gives uh=vh. Using again (2.13), (3.23), and (3.41), we obtain β∥ph-qh∥0,Ω2≤0 which implies ph=qh.
4. Error Estimates
In order to derive the error estimates of the numerical solution (uh,ph), we introduce the Galerkin projection (Rh,Qh):(X,M)→(NCP1,P1) defined as follows: for all (vh,qh)∈(NCP1,P1)(4.1)B((Rh(v,q),Qh(v,q));(vh,qh))=B0((v,q);(vh,qh)).
Noting the Theorem 3.4, (Rh(v,q),Qh(v,q)) is well defined.
By using a similar argument to the one used in [14, 22], we have the following lemma.
Lemma 4.1.
Let (u,p)∈D(A)×(H1(Ω)∩M); under the assumptions of Theorems 3.4 and 3.5, the projection operator (Rh,Qh) satisfies
(4.2)∥u-Rh(u,p)∥0,Ω+h(∥u-Rh(u,p)∥1,h+∥p-Qh(u,p)∥0,Ω)≤Ch2(∥Au∥0,Ω+|p|1,Ω).
Proof.
From (u,p)∈[H2(Ω)2∩X]×[H1(Ω)∩M], we have [ν∂nu]E=0. For all (vh,qh)∈NCP1×P1, using (4.1) yields
(4.3)B((Rh(u,p),Qh(u,p))(vh,qh))=B0((u,p);(vh,qh))=B((u,p);(vh,qh))-∑KjτKj(∇p,∇qh)Kj.
From the definition of (Rh(u,p),Qh(u,p)), (3.3), combining Theorem 3.4, (4.3), the triangular with inverse inequalities, we arrive at
(4.4)∥u-Rh(u,p)∥1,h+∥p-Qh(u,p)∥0,Ω≤∥u-uI∥1,h+∥p-pI∥0,Ω+∥uI-Rh(u,p)∥1,h+∥pI-Qh(u,p)∥0,Ω≤∥u-uI∥1,h+∥p-pI∥0,Ω+β-1sup (vh,qh)∈NCP1×P1|B((uI-Rh(u,p),pI-Qh(u,p));(vh,qh))|∥vh∥1,h+∥qh∥0,Ω≤β-1sup (vh,qh)∈NCP1×P1|B((uI-u,pI-p);(vh,qh))|+|∑KjτKj(∇p,∇qh)Kj|∥vh∥1,h+∥qh∥0,Ω +∥u-uI∥1,h+∥p-pI∥0,Ω.
It is easy to check that
(4.5)|B((uI-u,pI-p);(vh,qh))|≤C(∥u-uI∥1,h+∥p-pI∥0,Ω+h|p|1)(∥vh∥1,h+∥qh∥0,Ω).
Combining (4.4), (4.5), and inverse inequality yields
(4.6)∥u-Rh(u,p)∥1,h+∥p-Qh(u,p)∥0,Ω ≤C(∥u-uI∥1,h+∥p-pI∥0,Ω+h|p|1)+β-1sup (vh,qh)∈NCP1×P1|∑KjτKj(∇p,∇qh)Kj|∥vh∥1,h+∥qh∥0,Ω ≤C1h(∥Au∥0,Ω+|p|1,Ω)+C2hsup (vh,qh)∈NCP1×P1∑KjhKj∥∇p∥0,Kj∥∇qh∥0,Kj∥vh∥1,h+∥qh∥0,Ω ≤Ch(∥Au∥0,Ω+∥p∥1,Ω).
In order to derive the estimate in the L2-norm, we consider the following dual problem with (e,η)=(u-Rh(u,p),p-Qh(u,p)):
(4.7)-ΔΦ+∇Ψ=e in Ω,(4.8)div Φ=0 in Ω,(4.9)Φ|∂Ω=0 on ∂Ω.
Based on the assumption of (A1), (4.7)–(4.9) have a unique solution and satisfy
(4.10)∥AΦ∥0,Ω+|Ψ|1,Ω≤C∥u-Rh(u,p)∥0,Ω.
Multiplying (4.7) and (4.8) by e and η, respectively, integrating over Ω, and using (4.3) with (vh,qh)=(ΦI,ΨI), we see that
(4.11)∥u-Rh(u,p)∥02 =B0((u-Rh(u,p),p-Qh(u,p));(Φ,Ψ)) -∑Kj〈∂Φ∂n,u-Rh(u,p)〉j+∑Kj〈(u-Rh(u,p))·n,Ψ〉j =B((u-Rh(u,p),p-Qh(u,p));(Φ,Ψ))-∑KjτKj(∇(p-Qh(u,p)),∇Ψ)Kj -∑Kj〈∂Φ∂n,u-Rh(u,p)〉j+∑Kj〈(u-Rh(u,p))·n,Ψ〉j =B((u-Rh(u,p),p-Qh(u,p));(Φ-ΦI,Ψ-ΨI)) -∑KjτKj(∇(p-Qh(u,p)),∇Ψ)Kj+∑KjτKj(∇p,∇ΨI)Kj -∑Kj〈∂Φ∂n,u-Rh(u,p)〉j+∑Kj〈(u-Rh(u,p))·n,Ψ〉j,
where (ΦI,ΨI) is the finite element interpolation of (Φ,Ψ) in (NCP1,P1) and satisfies (3.3). For each E∈∂Kj, we define the mean value of u-Rh(u,p) and Ψ on E(4.12)u-Rh(u,p)¯=1he∫E(u-Rh(u,p))|Kjds; Ψ¯=1he∫EΨ|Kjds.
Note that each interior edge appears twice in the sum of (4.11); u-Rh(u,p)¯ and Ψ¯ are constants. Then it follows from (4.11) that
(4.13)∥u-Rh(u,p)∥0,Ω2 =B((u-Rh(u,p),p-Qh(u,p));(Φ-ΦI,Ψ-ΨI))+∑KjτKj(∇p,∇ΨI)Kj -∑KjτKj(∇(p-Qh(u,p)),∇Ψ)Kj+∑Kj ∑E∈∂Kj((u-Rh(u,p))·n,Ψ-Ψ¯)E -∑Kj ∑E∈∂Kj(∂Φ∂n,u-Rh(u,p)-u-Rh(u,p)¯)E.
Combining (3.3) with Lemma 4.1, we deduce that
(4.14)B((u-Rh(u,p),p-Qh(u,p));(Φ-ΦI,Ψ-ΨI)) ≤Ch2(∥Au∥0,Ω+|p|1,Ω)(∥AΦ∥0,Ω+|Ψ|1,Ω),(4.15) ∑KjτKj(∇p,∇ΨI)Kj-∑KjτKj(∇(p-Qh(u,p)),∇Ψ)Kj ≤∑KjτKj|p|1,Kj|ΨI|1,Kj+∑KjτKj|p-Qh(u,p)|1,Kj|Ψ|1,Kj.
With the help of (4.12), we have
(4.16)∫E[(u-Rh(u,p))-u-Rh(u,p)¯]ds=0.
Combining the definition of Πj, (4.16), and local trace theorem (3.21) with the standard argument for the nonconforming element (see [21]), we see that
(4.17)∑Kj ∑E∈∂Kj(∂Φ∂n,u-Rh(u,p)-u-Rh(u,p)¯)E =∑Kj ∑E∈∂Kj(∂Φ∂n-∂(ΠjΦ)∂n,u-Rh(u,p)-u-Rh(u,p)¯)E ≤∑Kj ∑E∈∂Kj∥∇(Φ-ΠjΦ)∥L2(E)∥u-Rh(u,p)-u-Rh(u,p)¯∥L2(E) ≤Ch2(∥Au∥0,Ω+|p|1,Ω)∥AΦ∥0,Ω.
In a similar way, we have
(4.18)∑Kj ∑E∈∂Kj((u-Rh(u,p))·n,Ψ-Ψ¯)E≤Ch2(∥Au∥0,Ω+|p|1)|Ψ|1,Ω.
By combining (4.13)–(4.15) with (4.17)-(4.18), we deduce that
(4.19)∥u-Rh(u,p)∥0,Ω≤Ch2(∥Au∥0,Ω+|p|1,Ω),
which, together with (4.6). We finish the proof.
Theorem 4.2.
Assume that the conditions of Theorems 3.4 and 3.5 are valid; let (u,p), (uh,ph) be the solutions of (2.1) and (3.14), respectively, then
(4.20)∥u-uh∥1,h+∥p-ph∥0,Ω≤Ch.
Proof.
We get the following error equation by combining (2.1) with (3.14), for all (vh,qh)∈(NCP1,P1)(4.21)B((u-uh,p-ph);(vh,qh))+b(u-uh,u,vh)+b(uh,u-uh,vh)-∑Kj〈∂u∂n,vh〉j +∑Kj〈vh·n,p〉j=∑KjτKj(∇p,∇qh)Kj-∑KjτKj(f,∇qh)Kj.
With (4.3), (4.21) can be rewritten as
(4.22)B((eh,ηh);(vh,qh))+b(u-Rh(u,p)+eh,u,vh)+b(uh,u-Rh(u,p)+eh,vh) -∑Kj〈∂u∂n,vh〉j+∑Kj〈vh·n,p〉j=-∑KjτKj(f,∇qh)Kj,
where eh=Rh(u,p)-uh and ηh=Qh(u,p)-ph.
From Theorem 3.5 and (4.22), we get that
(4.23)∥ηh∥0,Ω≤β-1sup 0≠(vh,qh)∈(NCP1,P1)|B((eh,ηh);(vh,qh))|∥vh∥1,h+∥qh∥0,Ω≤β-1sup 0≠(vh,qh)∈(NCP1,P1)1∥vh∥1,h+∥qh∥0,Ω ×(|b(u-Rh(u,p)+eh,u,vh)|+|∑KjτKj(f,∇qh)Kj| +|b(uh,u-Rh(u,p)+eh,vh)|+|∑Kj〈∂u∂n,vh〉j|+|∑Kj〈vh·n,p〉j|).
Again, with (2.13), Theorem 2.1, inverse inequality, and Lemma 4.1, we have
(4.24)|b(u-Rh(u,p),u,vh)|+|b(uh,u-Rh(u,p),vh)| ≤c0γ0(|u|1,Ω+∥uh∥1,h)∥u-Rh(u,p)∥1,h∥vh∥1,h≤Ch∥vh∥1,h,(4.25)|b(eh,u,vh)|+|b(uh,eh,vh)|≤c0γ0(|u|1,Ω+∥uh∥1,h)∥eh∥1,h∥vh∥1,h,(4.26)∑KjτKj|(f,∇qh)Kj|≤∑KjτKj∥f∥0,Kj∥∇qh∥0,Kj≤∑KjC~hKj2∥f∥0,Kj∥∇qh∥0,Kj≤C(ν)h∥f∥0,Ω∥qh∥0,Ω,
and using the similar arguments as for (4.17)-(4.18) yields
(4.27)|∑Kj〈∂u∂n,vh〉j|+|∑Kj〈vh·n,p〉j|≤Ch2(∥Au∥0,Ω+|p|1,Ω)∥vh∥1,h.
Combining (4.23)–(4.27) with Theorem 3.5, we arrive at
(4.28)∥ηh∥0,Ω≤Ch+2c0γ0γ0+ν1/2ν∥f∥0,Ω∥eh∥1,h.
Choosing (vh,qh)=(eh,ηh) in (4.22), we obtain that
(4.29)B((eh,ηh);(eh,ηh))+b(eh,u,eh)+∑KjτKj(f,∇ηh)Kj =-b(u-Rh(u,p),u,eh)-b(uh,u-Rh(u,p),eh) +∑Kj〈∂u∂n,vh〉j-∑Kj〈eh·n,p〉j.
Using (2.13), (2.17), Theorem 2.1, and Lemma 3.3, we get
(4.30)B((eh,ηh);(eh,ηh))-b(eh,u,eh)=∥|eh|∥h2+∥ηh∥h2-c0γ0|u|1∥eh∥1,h2≥ν∥eh∥1,h2-c0γ0|u|1∥eh∥1,h2≥ν(1-c0γ02ν-2∥f∥0)∥eh∥1,h2>0.
Combining (4.24)–(4.28) with (4.29) yields:
(4.31)∥eh∥1,h≤Ch.
From (4.28) and (4.31), we obtain that ∥ηh∥0,Ω≤Ch. Furthermore, we finish the proof by combining triangles inequality with Lemma 4.1, (4.28), and (4.31).
Theorem 4.3.
Let (u,p) and (uh,ph) be the solutions of (2.1) and (3.14), respectively, then we have
(4.32)∥u-uh∥0,Ω≤Ch2(∥Au∥0,Ω+|p|1,Ω).
Proof.
Using the duality argument for a linearized stationary Navier-Stokes problem; for some given g∈Y and the solution (u,p) of (2.1), defining (Φ,Ψ)∈(X,M) by
(4.33)-νΔΦ+∇Ψ+B~(u,Φ)-B(u,Φ)=g, in Ω,(4.34)div Φ=0 in Ω,(4.35)u|∂Ω=0 on ∂Ω,
where B~(u,Φ) is defined as 〈v,B~(u,Φ)〉X×X′=b(v,u,Φ), for all v∈X; multiplying (4.33) and (4.34) by v∈X and q∈M, respectively; integrating over Ω, from (2.8)–(2.11), it is easily to see that the bilinear form a(Φ,v)-d(v,Ψ)+d(Φ,q) is continuity and X×M coercive, by using the Lax-Milgram’s Lemma, (4.33)–(4.35) have a unique solution (Φ,Ψ).
Multiplying (4.33) and (4.34) by Φ and Ψ, respectively, using (2.13) and Theorem 2.1, we have
(4.36)ν|Φ|1,Ω2-b(Φ,u,Φ)≥ν|Φ|1,Ω2-c0γ0|u|1,Ω|Φ|1,Ω2≥ν(1-c0γ02ν-2∥f∥0,Ω)|Φ|1,Ω2>0.
On the other hand, estimating the right term yields
(4.37)(Φ,g)≤∥Φ∥0,Ω∥g∥0,Ω≤γ0|Φ|1,Ω∥g∥0,Ω.
By using (4.36) and (4.37), we arrive at
(4.38)|Φ|1,Ω≤C∥g∥0,Ω.
Setting Ψ=0 and taking the scalar product of (4.33) with AΦ in Y yields
(4.39)ν∥AΦ∥02+b(AΦ,u,Φ)-b(u,Φ,AΦ)=(g,AΦ).
Using the Gagliardo-Nirenberg inequality yields
(4.40)∥v∥L42≤C∥v∥0,Ω|v|1,Ω, ∀v∈X,∥∇v∥L42≤C∥Av∥0,Ω|v|1,Ω, ∀v∈H2(Ω)2∩X.
With the help of the Agmon’s inequality, we have
(4.41)∥v∥L∞2≤C∥v∥0,Ω∥Av∥0,Ω, ∀v∈H2(Ω)2∩X.
Furthermore, the following estimates are hold:
(4.42)b(AΦ,u,Φ)≤C∥AΦ∥0,Ω(∥∇u∥L4∥Φ∥L4+∥u∥L∞∥∇Φ∥0,Ω)≤ν4∥AΦ∥0,Ω2+Cν-1∥Au∥0,Ω2∥∇Φ∥0,Ω2),b(u,Φ,AΦ)≤ν4∥AΦ∥0,Ω2+Cν-1∥Au∥0,Ω2∥∇Φ∥0,Ω2),(g,AΦ)≤ν4∥AΦ∥0,Ω2+Cν-1∥g∥0,Ω.
Combining above inequalities with (4.38) and (4.39), we arrive at
(4.43)∥AΦ∥0,Ω2≤cν-2(1+∥Au∥0,Ω2)∥g∥0,Ω2.
Applying the continuous Inf-Sup condition (2.8) yields
(4.44)|Ψ|1,Ω≤cν∥AΦ∥0,Ω+c∥Au∥0,Ω|Φ|1,Ω+c∥g∥0,Ω.
Combining (4.43)-(4.44) with (2.18), (4.38), we arrive at
(4.45)∥AΦ∥0,Ω+|Ψ|1,Ω≤C∥g∥0,Ω.
Taking g=Rh(u,p)-uh, multiplying (4.33) and (4.34) by Rh(u,p)-uh and Qh(u,p)-ph, respectively, using (4.1) yields
(4.46)∥Rh(u,p)-uh∥0,Ω2 =B((Rh(u,p)-uh,Qh(u,p)-ph);(Φ,Ψ))+b(u,Rh(u,p)-uh,Φ) -∑KjτKj(∇(Qh(u,p)-ph),∇Ψ)Kj+∑Kj〈Ψ,(Rh(u,p)-uh)·n,〉j +b(Rh(u,p)-uh,u,Φ)-∑Kj〈∂Φ∂n,Rh(u,p)-uh〉j.
Setting (vh,qh)=(ΦI,ΨI) in (4.22), and using (4.46), we obtain that
(4.47)∥Rh(u,p)-uh∥0,Ω2=B((Rh(u,p)-uh,Qh(u,p)-ph);(Φ-ΦI,Ψ-ΨI))-∑KjτKj(∇(Qh(u,p)-ph),∇Ψ)Kj+b(u-uh,Rh(u,p)-uh,Φ)+∑Kj〈∂u∂n,vh〉j+b(u-Rh(u,p),u,ΦI)+b(Rh(u,p)-uh,u,Φ-ΦI)+b(uh,u-Rh(u,p),Φ-ΦI)+b(uh,Rh(u,p)-uh,Φ-ΦI)-b(uh,u-Rh(u,p),Φ)+∑Kj〈Ψ,(Rh(u,p)-uh)·n,〉j-∑KjτKj(f,∇ΨI)Kj-∑Kj〈∂Φ∂n,Rh(u,p)-uh〉j-∑Kj〈vh·n,p〉j.
We now estimate the right terms of (4.47). Thanks to (3.3); Theorems 2.1, 3.4-3.5, and 4.2; inverse inequality; Lemma 4.1; we know that
(4.48)B((Rh(u,p)-uh,Qh(u,p)-ph);(Φ-ΦI,Ψ-ΨI))≤C(∥Rh(u,p)-uh∥1,h+∥Qh(u,p)-ph∥0,Ω)(∥Φ-ΦI∥1,h+∥Ψ-ΨI∥0,Ω+h|Ψ|1,Ω)≤Ch2(∥Au∥0,Ω+|p|1,Ω)(∥AΦ∥0,Ω+|Ψ|1,Ω),b(Rh(u,p)-uh,u,Φ-ΦI)+b(uh,Rh(u,p)-uh,Φ-ΦI)≤2c0(|u|1,Ω+∥uh∥1,h)∥Rh(u,p)-uh∥1,h∥Φ-ΦI∥1,h≤Ch2(∥Au∥0,Ω+|p|1,Ω)∥AΦ∥0,Ω,∑KjτKj(∇(Qh(u,p)-ph),∇Ψ)Kj≤∑KjC~hK2∥∇(Qh(u,p)-ph)∥0,Kj∥∇Ψ∥0,Kj≤Ch2(∥Au∥0,Ω+|p|1)|Ψ|1,Ω,b(u-Rh(u,p),u,ΦI)≤C∥u-Rh(u,p)∥0,Ω∥Au∥0,Ω|ΦI|1,h≤Ch2(∥Au∥0,Ω+|p|1,Ω)|Φ|1,Ω,b(u-uh,Rh(u,p)-uh,Φ)≤c0∥u-uh∥1,h∥Rh(u,p)-uh∥1,h|Φ|1,Ω≤Ch2(∥Au∥0,Ω+|p|1)|Φ|1,Ω,b(uh,u-Rh(u,p),Φ)≤C∥uh∥1,h∥u-Rh(u,p)∥0,Ω∥AΦ∥0,Ω,∑KjτKj(f,∇ΨI)Kj≤∑KjC~hKj2∥f∥0,Kj∥∇ΨI∥0,Kj≤Ch2∥f∥0,Ω|Ψ|1,Ω,b(uh,u-Rh(u,p),Φ-ΦI)≤c0∥uh∥1,h∥u-Rh(u,p)∥1,h∥Φ-ΦI∥1,h≤Ch2(∥Au∥0,Ω+|p|1,Ω)∥AΦ∥0,Ω.
Applying the argument used for (4.18)-(4.21), (4.28) gives
(4.49)∑Kj〈∂Φ∂n,Rh(u,p)-uh〉j+∑Kj〈Ψ,(Rh(u,p)-uh)·n,〉j ≤Ch2(∥Au∥0,Ω+|p|1,Ω)(∥AΦ∥0,Ω+|Ψ|1,Ω),∑Kj〈∂u∂n,ΦI〉+∑Kj〈ΦI·n,p〉j≤Ch2(∥Au∥0,Ω+∥p∥1,Ω)∥ΦI∥1,h≤Ch2(∥Au∥0,Ω+∥p∥1,Ω)∥ΦI∥1,Ω.
Combining above inequalities, Theorems 2.1, 3.5, (4.46), and (4.51), we get that
(4.50)∥Rh(u,p)-uh∥0,Ω2 ≤Ch2(∥Au∥0,Ω+|p|1,Ω)(∥AΦ∥0,Ω+|Ψ|1,Ω)+C∥uh∥1,h∥u-Rh(u,p)∥0,Ω∥AΦ∥0,Ω ≤Ch2(∥Au∥0,Ω+|p|1,Ω)∥Rh(u,p)-uh∥1,Ω+Cγ0+ν1/2ν∥f∥0,Ω∥u-Rh(u,p)∥0,Ω2.
Choosing the appropriate ν,Ω and f such that 1-C((γ0+ν1/2)/ν)∥f∥0>0, then we have
(4.51)∥Rh(u,p)-uh∥0,Ω≤Ch2(∥Au∥0,Ω+∥p∥1,Ω).
By applying the triangles inequality, Lemma 4.1 and (4.51), we finish the proof.
Lemma 4.4.
Under the assumptions of Theorem 4.2, the following estimate about u-uh in mesh-dependent norm
(4.52)∥|u-uh|∥h≤Ch,
holds, where u and uh are the solution of problem (2.1) and (3.14), respectively.
Proof.
According to the definition of ∥|·|∥h, uh∈Xh, with Theorems 2.1 and 4.2, inverse and local trace inequalities (3.21), we have
(4.53)∥|u-uh|∥h2=ν∥|u-uh|∥1,h2+∑E∈ΓhτE∥ν∂n(u-uh)∥0,E2=ν∥|u-uh|∥1,h2+∑E∈Γhhe12ν∥ν∂n(u-uh)∥0,E2≤Ch2+C∑E∈Γhhe[hKj-1∥∇(u-uh)·n∥0,Kj2+hKj|∇(u-uh)·n|1,Kj2]≤Ch2.