# Two-Level Brezzi-Pitkäranta Stabilized Finite Element Methods for the Incompressible Flows

Authors: Rong An; Xian Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We present a new stabilized finite element method for incompressible flows based on Brezzi-Pitkäranta stabilized method. The stability and error estimates of finite element solutions are derived for classical one-level method. Combining the techniques of two-level discretizations, we propose two-level Stokes/Oseen/Newton iteration methods corresponding to three different linearization methods and show the stability and error estimates of these three methods. We also propose a new Newton correction scheme based on the above two-level iteration methods. Finally, some numerical experiments are given to support the theoretical results and to check the efficiency of these two-level iteration methods.


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## Body

## 1. Introduction

In this paper, we consider steady Navier-Stokes equations with homogeneous Dirichlet boundary conditions:(1)-μΔu+(u·∇)u-∇p=f,inΩ,div⁡u=0,inΩ,u=0,on⁡∂Ω,
where Ω⊂ℝ2 is a bounded convex domain with boundary ∂Ω. μ>0 represents the viscous coefficient. u=(u1(x),u2(x)) denotes the velocity vector, p=p(x) the pressure, and f=(f1(x),f2(x)) the prescribed body force vector. The solenoidal condition div⁡u=0 means that the flows are incompressible.In computational fluid dynamics, it is very important in searching the appropriate mixed finite element approximation to solve the numerical solutions of the problem (1) quickly and efficiently. Roughly speaking, the selected finite element spaces are required to satisfy the inf-sup condition, such as the finite element space constructed by the P2-P1 pair. However, from the computational cost point of view, the P1-P1 pair is of practical importance in scientific computation with the lower computational cost. Therefore, much attention has been attracted by the P1-P1 pair for simulating the incompressible flow. But, in this case, the inf-sup condition is not satisfied. A usual technique is to introduce the stabilized term in the finite element variational equation such that the inf-sup condition is enforced. There exist many stabilized methods, such as Brezzi-Pitkäranta stabilized method [1], locally stabilized method [2, 3], pressure stabilized method [4], stream upwind Petrov-Galerkin method [5], Douglas-Wang absolutely stabilized method [6], and pressure projection stabilized method [7, 8] and the references cited therein. Most of these stabilized methods necessarily introduce the stabilized parameters. Moreover, some of these methods are conditionally stable; that is, the stabilized parameters must satisfy some stable condition. Therefore, the development of stabilized methods free from stabilized parameters has become increasingly important.In this paper, we combine the Brezzi-Pitkäranta stabilized method, which is unconditionally stable [9], with techniques of two-level discretizations to solve the numerical solution of the problem (1) under the uniqueness condition. Two-level discretization method has become a powerful tool in solving nonlinear partial differential equations. The basic idea is to capture “large eddies" by computing the initial approximation on the coarse mesh and then to obtain the fine approximation by solving a linearized problem corresponding to nonlinear partial differential equations on the fine mesh. More details can be referred to in the works of Xu [10, 11]. There exists a large amount of references about two-level finite element method for Navier-Stokes equations. For details, please see the works of An and Qiu [12], Ervin et al. [13], Franca and Nesliturk [14], de Frutos et al. [15, 16], Girault and Lions [17], Goswami and Damázio [18], He [19], He and Li [20], He and Wang [21], He et al. [22], Huang et al. [23], Layton [24], Layton and Tobiska [25], Li [26], Li and An [27, 28], Liu and Hou [29], and Zhu and Chen [30] and the references cited therein.Based on the Brezzi-Pitkäranta stabilized finite element method, in this paper, we solve the nonlinear Navier-Stokes equations on the coarse mesh with mesh sizeH in Step I and then solve a linear system according to Stokes/Oseen/Newton iterative method on the fine mesh with mesh size h in Step II. Denote by (uh,ph) the finite element approximation solution on the fine mesh. If we suppose (u,p)∈(H2(Ω)2,H1(Ω)), then the error estimate derived is
(2)∥u-uh∥V+∥p-ph∥≤c(h+H2),
where c>0 is independent of h and H and the norms ∥·∥V and ∥·∥ are defined in the next section. It is obvious that if we choose H=O(h1/2), then two-level method discussed in this paper provides the same convergence order as the classical one-level method. Finally, we propose a Newton correction scheme on the fine mesh. The numerical solution (uh,ph) in Step II is as the iterative initial value. Then the finite element approximation solution (u⋆h,p⋆h) is solved in terms of Newton iterative scheme on the fine mesh in Step III. The error estimate derived for this Newton correction scheme is
(3)∥u-u⋆h∥V+∥p-p⋆h∥≤c(h+H4).
Thus, if H=O(h1/4), then this new two-level method also is of the same convergence order as the classical one-level method.This paper is organized as follows. In Section2, we introduce some function spaces and some classical results about Navier-Stokes equations. In Section 3, the Brezzi-Pitkäranta stabilized finite element approximation will be applied and the error estimates about the velocity in H1-norm and L2-norm and the pressure in L2-norm are derived. In Section 4, the two-level discretization finite element methods are proposed and the error estimates (2) and (3) are shown. In the final section, the numerical experiments are displaced to support the theoretical results.


## 2. Navier-Stokes Equations

In what follows, we employ the standard notationHl(Ω) (or Hl(Ω)2), l≥0, for the Sobolev spaces of all functions having square integrable derivatives up to order l in Ω. Denote the standard Sobolev norm by ∥·∥l. If l=0, we write L2(Ω) (or L2(Ω)2) and ∥·∥ instead of H0(Ω) (or H0(Ω)2) and ∥·∥0, respectively. The symbol c always denotes some positive constant which is independent of the mesh parameters h and H and can be a different constant even in the same formulation.Introduce the following spaces usually used in this paper:(4)V=H01(Ω)2,M=L02(Ω)={q∈L2(Ω);∫Ω‍qdx=0}.
The space V is equipped with the norm
(5)∥v∥V=(∫Ω‍|∇v|2dx)1/2,∀v∈V.
It is well known that ∥v∥V is equivalent to ∥v∥1 due to Poincaré inequality. Introduce the following bilinear and trilinear forms:
(6)a(u,v)=μ∫Ω‍∇u·∇vdx,∀u,v∈V,d(v,q)=∫Ω‍qdiv⁡vdx,∀v∈V,q∈M,b(u,v,w)=∫Ω‍(u·∇)v·wdx-12∫Ω‍div⁡uv·wdx=12∫Ω‍(u·∇)v·wdx-12∫Ω‍(u·∇)w·vdx.
It is easy to check that this trilinear form satisfies the following important properties [20, 31]:
(7)b(u,v,w)=-b(u,w,v),(8)b(u,v,w)≤N∥u∥V∥v∥V∥w∥V,(9)b(u,v,w)≤N2∥u∥1/2∥u∥V1/2×(∥v∥V∥w∥1/2∥w∥V1/2+∥w∥V∥v∥1/2∥v∥V1/2),
for all u,v,w∈V and
(10)|b(u,v,w)|+|b(v,u,w)|+|b(w,u,v)|≤N∥u∥V∥v∥2∥w∥,
for all u∈V, v∈H2(Ω)2, and w∈L2(Ω)2, where N>0 depends only on Ω.Givenf∈L2(Ω)2, under the above notations, the variational formulation of the problem (1) reads as follows: find (u,p)∈(V,M) such that for all (v,q)∈(V,M)(11)a(u,v)+b(u,u,v)-d(v,p)=(f,v),d(u,q)=0.
Define a generalized bilinear form on (V,M)×(V,M) by
(12)ℬ(u,p;v,q)=a(u,v)-d(v,p)+d(u,q);
then the problem (11) also takes the following form:
(13)ℬ(u,p;v,q)+b(u,u,v)=(f,v),∀(v,q)∈(V,M).The following existence, uniqueness, and regularity results concerning the solution(u,p) to the problem (13) are classical [32–34].Theorem 1.
Assuming thatμ and f satisfy the following uniqueness condition:
(14)2μ-2N∥f∥<1,
then the problem (13) exists a unique solution (u,p)∈(V,M) satisfying
(15)∥u∥V≤1μ∥f∥<μ2N.
Furthermore, if ∂Ω is of class C2, then the solution (u,p) to the problem (13) satisfies the following regularity property:
(16)∥u∥2+∥p∥1≤c∥f∥.


## 3. Stabilized Finite Element Approximation

Let𝒯h be a family of quasiuniform triangular partitions of Ω into triangles. The corresponding ordered triangles are denoted by K1,K2,…,Kn. Let hi=diam⁡(Ki), i=1,…,n, and h=max⁡{h1,h2,…,hn}. For every K∈𝒯h, let Pr(K) denote the space of the polynomials on K of degree at most r. Consider the conforming finite element spaces Vh and Mh given by
(17)Vh={vh∈C(Ω¯)2∩V,vh∣K∈[P1(K)]2,∀K∈𝒯h},Mh={qh∈C(Ω¯)∩M,qh∣K∈P1(K),∀K∈𝒯h}.
Then the Brezzi-Pitkäranta stabilized finite element approximation of (11) is as follows: find uh∈Vh and ph∈Mh such that for all (vh,qh)∈(Vh,Mh)(18)a(uh,vh)+b(uh,uh,vh)-d(vh,ph)=(f,vh),d(uh,qh)+Ch(ph,qh)=0,
where the stabilized term is defined by
(19)Ch(ph,qh)=α∑i=1n‍hi2∫Ki‍∇ph·∇qhdx,∀ph,qh∈Mh
with some positive constant α>0. Define a mesh-dependent norm [·]h on Mh by
(20)[qh]h2=1αCh(qh,qh),∀qh∈Mh.
Then, it holds that Ch(ph,qh)≤α[ph]h[qh]h for all ph,qh∈Mh and
(21)d(v,qh)≤ch∥v∥[qh]h,∀v∈V,qh∈Mh,
which has been shown by Latché and Vola [35]. Moreover, Ch(p,q) also is defined for any couple of functions p,q∈H1(Ω) and satisfies
(22)[q]h≤ch∥q∥1,∀q∈H1(Ω).Introduce another generalized bilinear formℬh(·,·;·,·) on (Vh,Mh)×(Vh,Mh) defined by
(23)ℬh(uh,ph;vh,qh)=ℬ(uh,ph;vh,qh)+Ch(ph,qh).
Then the discrete problem (18) can be rewritten as follows:
(24)ℬh(uh,ph;vh,qh)+b(uh,uh,vh)=(f,vh).Denote byIh:H2(Ω)2∩V⟶Vh and Jh:H1(Ω)∩M⟶Mh the standard interpolation operators satisfying
(25)∥v-Ihv∥+h∥v-Ihv∥V≤ch2∥v∥2,∀v∈H2(Ω)2∩V,∥q-Jhq∥≤ch∥q∥1,∀q∈H1(Ω)∩M.
Moreover, we suppose that the inverse inequalities hold:
(26)∥∇qh∥Ki≤chi-1∥qh∥Ki,∥∇qh∥≤ch-1∥qh∥.First, we recall the following stable theorem [9].Theorem 2.
For anyα>0, there exist two positive constants β1 and β2 independent of h such that ℬh(·,·;·,·) on (Vh,Mh)×(Vh,Mh) satisfies the following continuous property:
(27)ℬh(wh,rh;vh,qh)≤β1(∥wh∥V+∥rh∥)(∥vh∥V+∥qh∥)
and the weakly coercive property:
(28)β2(∥wh∥V+∥rh∥)≤sup⁡(vh,qh)∈(Vh,Mh)ℬh(wh,rh;vh,qh)∥vh∥V+∥qh∥.A direct result of Theorem2 is that the problem (24) exists a unique solution. In order to derive the error estimate between (u,p) and (uh,ph), we introduce the following Galerkin projection operator (Rh,Qh):(V,M)⟶(Vh,Mh) defined by
(29)ℬh(Rhw,Qhr;wh,rh)=ℬ(w,r;wh,rh)
for each (w,r)∈(V,M) and all (wh,rh)∈(Vh,Mh). According to Theorem 2, it is easy to check that (Rhw,Qhr) is well defined. Moreover, there holds
(30)ℬh(Rhw,Qhr;wh,rh)=ℬh(w,r;wh,rh)-Ch(r,rh).About the Galerkin projection operator(Rh,Qh), the following approximation property has been derived in [9].Theorem 3.
For anyw∈H2(Ω)2∩V and r∈H1(Ω)∩M, there holds
(31)∥w-Rhw∥+h∥w-Rhw∥V+h∥r-Qhr∥+h[r-Qhr]h≤ch2(∥w∥2+∥r∥1).Next, we begin to show the error estimate for the one-level finite element approximation solution(uh,ph).Theorem 4.
Suppose that the uniqueness condition (14) holds. If (u,p)∈H2(Ω)2∩V×H1(Ω)∩M and (uh,ph)∈(Vh,Mh) are the solutions of (13) and (24), respectively, then, for any α>0, the following optimal error estimate holds:
(32)∥u-uh∥V+∥p-ph∥+[p-ph]h≤ch.Proof.
First, we estimate∥uh∥V. Setting vh=uh and qh=ph in (24), using (7) and Young inequality, we obtain
(33)μ∥uh∥V2≤(f,uh)≤μ2∥uh∥V2+12μ∥f∥2.
Then under the uniqueness condition (14), uh satisfies
(34)∥uh∥V≤1μ∥f∥<μ2N.
It follows from (30) that
(35)μ∥uh-Rhu∥V2+α[ph-Qhp]h2=ℬh(uh-Rhu,ph-Qhp;uh-Rhu,ph-Qhp)=ℬh(uh-u,ph-p;uh-Rhu,ph-Qhp)+ℬh(u-Rhu,p-Qhp;uh-Rhu,ph-Qhp)=b(u,u,uh-Rhu)-b(uh,uh,uh-Rhu).
According to (7), (15), (34), and Young inequality, we get
(36)b(u,u,uh-Rhu)-b(uh,uh,uh-Rhu)=b(u-uh,u,uh-Rhu)+b(uh,u-uh,uh-Rhu)=b(u-Rhu,u,uh-Rhu)+b(Rhu-uh,u,uh-Rhu)+b(uh,u-Rhu,uh-Rhu)≤N(∥u∥V+∥uh∥V)∥u-Rhu∥V∥uh-Rhu∥V+N∥u∥V∥uh-Rhu∥V2≤μ∥u-Rhu∥V∥uh-Rhu∥V+μ2∥uh-Rhu∥V2≤μ2∥uh-Rhu∥V2+μ4∥uh-Rhu∥V2+μ∥u-Rhu∥V2.
Thus, from (31) we obtain
(37)∥u-uh∥V≤∥u-Rhu∥V+∥uh-Rhu∥V≤3∥u-Rhu∥V≤ch.
Next, we estimate ∥ph-Qhp∥. It follows from (15), (28), (34), and (37) that
(38)β2∥ph-Qhp∥≤sup⁡(vh,qh)∈(Vh,Mh)ℬh(uh-Rhu,ph-Qhp;vh,qh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)((∥vh∥V+∥qh∥)-1((∥vh∥V+∥qh∥)-1ℬh(uh-u,ph-p;vh,qh)+ℬh(u-Rhu,p-Qhp;vh,qh)(∥vh∥V+∥qh∥)-1)×(∥vh∥V+∥qh∥)-1)=sup⁡(vh,qh)∈(Vh,Mh)b(u,u,vh)-b(uh,uh,vh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)b(u-uh,u,vh)+b(uh,u-uh,vh)∥vh∥V+∥qh∥≤μ∥u-uh∥V.
Thus, we obtain
(39)∥p-ph∥≤∥p-Qhp∥+∥ph-Qhp∥≤∥p-Qhp∥+c∥u-uh∥V≤ch.
Moreover,
(40)[p-ph]h≤[p-Qhp]h+[ph-Qhp]h≤[p-Qhp]h+c∥u-Rhu∥V≤ch.Next, we give theL2 error estimate ∥u-uh∥ by Aubin-Nitsche technique. This error analysis is based on the regularity assumption that the following linearized problem (41) is (H2(Ω)2,H1(Ω)) regular. Given z∈L2(Ω)2, find (w,π)∈(V,M) such that for all (v,q)∈(V,M)(41)a(w,v)+b(u,v,w)+b(v,u,w)-d(v,π)=(z,v),d(w,q)=0.
According to (7) and (15), it is easy to verify that the problem (41) exists a unique solution (w,π)∈(V,M). The assumption that (41) is (H2(Ω)2,H1(Ω)) regular means that (w,π) also belongs to (H2(Ω)2,H1(Ω)) and the following estimate holds:
(42)∥w∥2+∥π∥1≤c∥z∥.
Under the above assumption, we prove the following theorem.Theorem 5.
Suppose that the uniqueness condition (14) holds. If (u,p)∈(H2(Ω)2∩V,H1(Ω)∩M) and (uh,ph)∈(Vh,Mh) are the solutions of (13) and (24), respectively, then, for any α>0, the following optimal L2 error estimate holds:
(43)∥u-uh∥≤ch2.Proof.
Settingz=v=u-uh in the first equation of (41), it yields
(44)∥u-uh∥2=a(w,u-uh)+b(u,u-uh,w)+b(u-uh,u,w)-d(u-uh,π).
Subtracting (11) from (18) yields
(45)a(u-uh,vh)+b(u,u,vh)-b(uh,uh,vh)-d(vh,p-ph)=0,∀vh∈Vh,d(u-uh,qh)-Ch(ph,qh)=0,∀qh∈Mh.
Taking vh=Rhw and qh=Qhπ in (45) and combining them with (44), we obtain
(46)∥u-uh∥2=a(w-Rhw,u-uh)+b(u,u-uh,w)+b(u-uh,u,w)+b(uh,uh,Rhw)-b(u,u,Rhw)+d(Rhw-w,p-ph)-d(u-uh,π-Qhπ)-Ch(ph,Qhπ)=I1+⋯+I4.
Using (31), (32), and (42), I1 is estimated by
(47)I1=a(w-Rhw,u-uh)≤μ∥u-uh∥V∥w-Rhw∥V≤ch2(∥w∥2+∥π∥1)≤ch2∥u-uh∥.
Similarly, I3 is estimated by
(48)I3=d(Rhw-w,p-ph)-d(u-uh,π-Qhπ)≤∥Rhw-w∥V∥p-ph∥+∥u-uh∥V∥π-Qhπ∥≤ch2(∥w∥2+∥π∥1)≤ch2∥u-uh∥.
About I2, we rewrite it as
(49)I2=b(u,u-uh,w)+b(u-uh,u,w)+b(uh,uh,Rhw)-b(u,u,Rhw)=b(u-uh,u-uh,w)+b(u-uh,u,w-Rhw)+b(u,u-uh,w-Rhw)+b(u-uh,u-uh,Rhw-w).
Then it follows from (8), (15), (31), (32), and (42) that
(50)I2≤N∥u-uh∥V2(∥w∥2+∥w-Rhw∥V)+N∥u∥V∥u-uh∥V∥w-Rhw∥V≤ch2∥w∥2≤ch2∥u-uh∥.
Finally, using (22), (31), (32), and (42) we estimate I4 by
(51)I4=-Ch(ph,Qhπ)=Ch(p-ph,Qhπ-π)+Ch(p-ph,π)+Ch(p,π-Qhπ)-Ch(p,π)≤[p-ph]h[Qhπ-π]h+[p-ph]h[π]h+[p]h[π-Qhπ]h+[p]h[π]h≤ch2(∥w∥2+∥π∥1)≤ch2∥u-uh∥.
Combining these estimates for I1 to I4 with (46), we complete the proof of (43).


## 4. Two-Level Brezzi-Pitkäranta Stabilized Methods

In this section, the two-level Brezzi-Pitkäranta stabilized finite element methods for (13) are proposed in terms of Oseen/Stokes/Newton iteration method. From now on, H and h with h<H<1 are two real positive parameters. The coarse mesh triangulation 𝒯H is made as like in Section 3. And a fine mesh triangulation 𝒯h is generated by a mesh refinement process to 𝒯H. The conforming finite element space pairs (Vh,Mh) and (VH,MH)⊂(Vh,Mh) corresponding to the triangulations 𝒯h and 𝒯H, respectively, are constructed as like in Section 3. With the above notations, we propose the following two-level Brezzi-Pitkäranta stabilized finite element methods in the next subsections.


### 4.1. Two-Level Oseen Iteration Method

Step I. We solve (24) on the coarse mesh; that is, find (uH,pH)∈(VH,MH) such that for all (vH,qH)∈(VH,MH)(52)ℬH(uH,pH;vH,qH)+b(uH,uH,vH)=(f,vH).Step II. We solve a discrete Oseen problem according to Oseen iteration on the fine mesh; that is, find (uh,ph)∈(Vh,Mh) such that for all (vh,qh)∈(Vh,Mh)(53)ℬh(uh,ph;vh,qh)+b(uH,uh,vh)=(f,vh).First, we discuss the existence and uniqueness of the solution to the problem (53) under the uniqueness condition (14). In view of Theorem 2, the problem (52) exists a unique solution (uH,pH)∈(VH,MH) with
(54)∥uH∥V≤1μ∥f∥<μ2N.
Moreover, it follows from Theorems 4 and 5 that
(55)∥u-uH∥+H∥u-uH∥V+H∥p-pH∥≤cH2.
On the other hand, setting vh=uh and qh=ph in (53), it yields
(56)ℬh(uh,ph;uh,ph)+b(uH,uh,uh)=μ∥uh∥V2+[ph]h2.
Then it is easy to show that the problem (53) also exists a unique solution (uh,ph)∈(Vh,Mh) such that
(57)∥uh∥V≤1μ∥f∥<μ2N.
Next, we give the error estimate for the two-level Oseen iteration method.Theorem 6.
Suppose that the uniqueness condition (14) holds. If (u,p)∈(H2(Ω)2∩V,H1(Ω)∩M) and (uh,ph)∈(Vh,Mh) are the solutions of (13) and (53), respectively, then there holds
(58)∥u-uh∥V+∥p-ph∥≤c(h+H2).Proof.
In terms of the definitionℬh(·,·;·,·) and (30), we get
(59)μ∥uh-Rhu∥V2+[ph-Qhp]h2=ℬh(uh-Rhu,ph-Qhp;uh-Rhu,ph-Qhp)=ℬh(uh-u,ph-p;uh-Rhu,ph-Qhp)+ℬh(u-Rhu,p-Qhp;uh-Rhu,ph-Qhp)=b(u,u,uh-Rhu)-b(uH,uh,uh-Rhu).
We rewrite b(u,u,uh-Rhu)-b(uH,uh,uh-Rhu) as
(60)b(u,u,uh-Rhu)-b(uH,uh,uh-Rhu)=b(u-uH,u,uh-Rhu)-b(uH,uh-u,uh-Rhu)=b(u-uH,u,uh-Rhu)-b(uH,Rhu-u,uh-Rhu).
Then using (8), (10), and (54), we obtain
(61)b(u,u,uh-Rhu)-b(uH,uh,uh-Rhu)≤N∥u∥2∥uh-Rhu∥V∥u-uH∥+N∥uH∥V∥Rhu-u∥V∥uh-Rhu∥V≤μ2∥uh-Rhu∥V2+N2μ∥u∥22∥u-uH∥2+μ4∥Rhu-u∥V2.
Thus, there holds
(62)∥uh-Rhu∥V≤c(∥u-uH∥+∥Rhu-u∥V)≤c(h+H2),
where we use (31) and (55). A direct consequence of the above estimate is
(63)∥u-uh∥V≤c(h+H2).
From (28), (30), (55), and (63), we have
(64)β2∥ph-Qhp∥≤sup⁡(vh,qh)∈(Vh,Mh)ℬh(uh-Rhu,ph-Qhp;vh,qh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)((ℬh(uh-u,ph-p;vh,qh)+ℬh(u-Rhu,p-Qhp;vh,qh)(uh-u,ph-p;vh,qh))×(∥vh∥V+∥qh∥)-1(uh-u,ph-p;vh,qh))=sup⁡(vh,qh)∈(Vh,Mh)b(u,u,vh)-b(uH,uh,vh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)b(u-uH,u,vh)+b(uH,u-uh,vh)∥vh∥V+∥qh∥≤c(∥u-uH∥+∥u-uh∥V)≤c(h+H2),
which together with (31) yields
(65)∥p-ph∥≤c(h+H2).


### 4.2. Two-Level Stokes Iteration Method

Step I. We solve (24) on the coarse mesh; that is, find (uH,pH)∈(VH,MH) such that for all (vH,qH)∈(VH,MH)(66)ℬH(uH,pH;vH,qH)+b(uH,uH,vH)=(f,vH).Step II. We solve a discrete Stokes problem according to Stokes iteration on the fine mesh; that is, find (uh,ph)∈(Vh,Mh) such that for all (vh,qh)∈(Vh,Mh)(67)ℬh(uh,ph;vh,qh)+b(uH,uH,vh)=(f,vh).In this subsection, we assume that the following uniqueness conditions hold:(68)3μ-2N∥f∥<1.
Proceeding the argument as in Section 4.1, the problem (66) exists a unique solution (uH,pH)∈(VH,MH) and uH satisfies ∥uH∥V≤(1/μ)∥f∥<μ/3N. According to the definition of ℬh(·,·;·,·), the discrete Stokes problem (67) also exists a unique solution (uh,ph)∈(Vh,Mh). Moreover, uh satisfies
(69)∥uh∥V≤1μ∥f∥+Nμ∥uH∥V2≤1μ∥f∥+13∥uH∥V≤43μ∥f∥≤4μ9N<μ2N.
Then the error estimate for two-level Stokes iteration method is derived in the following theorem.Theorem 7.
Suppose that the uniqueness condition (68) holds. If (u,p)∈(H2(Ω)2∩V,H1(Ω)∩M) and (uh,ph)∈(Vh,Mh) are the solutions of (13) and (67), respectively, then one has
(70)∥u-uh∥V+∥p-ph∥≤c(h+H2).Proof.
Subtracting (13) from (67), we get
(71)ℬh(uh-u,ph-p;vh,qh)=b(u,u,vh)-b(uH,uH,vh)-Ch(p,qh).
Then, from (10), (28), (30), and (55), we have
(72)β2(∥uh-Rhu∥V+∥ph-Qhp∥)≤sup⁡(vh,qh)∈(Vh,Mh)ℬh(uh-Rhu,ph-Qhp;vh,qh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)((ℬh(uh-u,ph-p;vh,qh)+ℬh(u-Rhu,p-Qhp;vh,qh)(uh-u,ph-p;vh,qh))×(∥vh∥V+∥qh∥)-1(uh-u,ph-p;vh,qh))=sup⁡(vh,qh)∈(Vh,Mh)b(u,u,vh)-b(uH,uH,vh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)((∥vh∥V+∥qh∥)-1((∥vh∥V+∥qh∥)-1b(u-uH,u,vh)+b(u,u-uH,vh)-b(u-uH,u-uH,vh)(∥vh∥V+∥qh∥)-1)×(∥vh∥V+∥qh∥)-1)≤2N∥u∥2∥u-uH∥+N∥u-uH∥V2≤cH2,
which together with (31) completes the proof of (70).


### 4.3. Two-Level Newton Iteration Method

Step I. We solve (24) on the coarse mesh; that is, find (uH,pH)∈(VH,MH) such that for all (vH,qH)∈(VH,MH)(73)ℬH(uH,pH;vH,qH)+b(uH,uH,vH)=(f,vH).Step II. We solve discrete linearized Navier-Stokes equations according to Newton iteration on the fine mesh; that is, find (uh,ph)∈(Vh,Mh) such that for all (vh,qh)∈(Vh,Mh)(74)ℬh(uh,ph;vh,qh)+b(uH,uh,vh)+b(uh,uH,vh)=(f,vh)+b(uH,uH,vh).As in Section4.2, we modify the uniqueness condition as
(75)4μ-2N∥f∥<1.
In this case, the solution uH of the problem (73) satisfies ∥uH∥V≤(1/μ)∥f∥<μ/4N. Setting vh=uh and qh=ph in (74), we have
(76)ℬh(uh,ph;uh,ph)+b(uh,uH,uh)+b(uH,uh,uh)≥μ∥uh∥V2+[ph]h2-N∥uH∥V∥uh∥V2≥3μ4∥uh∥V2+[ph]h2.
Moreover, we can estimate uh by
(77)∥uh∥V≤43μ∥f∥+4N3μ∥uH∥V2≤43μ∥f∥+13∥uH∥V≤53μ∥f∥≤5μ12N<μ2N.
The error estimate for two-level Newton iteration method is derived in the following theorem.Theorem 8.
Suppose that the uniqueness condition (75) holds. If (u,p)∈(H2(Ω)2∩V,H1(Ω)∩M) and (uh,ph)∈(Vh,Mh) are the solutions of (13) and (74), respectively, then one has
(78)∥u-uh∥V+∥p-ph∥≤c(h+H2).Proof.
Proceeding as in the proof of Theorem7, we have
(79)μ∥uh-Rhu∥V2+[ph-Qhp]h2=ℬh(uh-Rhu,ph-Qhp;uh-Rhu,ph-Qhp)=ℬh(uh-u,ph-p;uh-Rhu,ph-Qhp)+ℬh(u-Rhu,p-Qhp;uh-Rhu,ph-Qhp)=b(u,u,uh-Rhu)-b(uH,uh,uh-Rhu)-b(uh,uH,uh-Rhu)+b(uH,uH,uh-Rhu).
We rewrite the right-hand side of the above identity as follows:
(80)b(u,u,uh-Rhu)-b(uH,uh,uh-Rhu)-b(uh,uH,uh-Rhu)+b(uH,uH,uh-Rhu)=b(u-uh,u,uh-Rhu)+b(uh,u-uh,uh-Rhu)+b(uh-uH,uh-uH,uh-Rhu)=b(u-Rhu,u,uh-Rhu)+b(Rhu-uh,u,uh-Rhu)+b(uh,u-Rhu,uh-Rhu)+b(uh-uH,Rhu-uH,uh-Rhu)=J1+⋯+J4.
Using (8) and (15), we have
(81)J1=b(u-Rhu,u,uh-Rhu)≤N∥u∥V∥u-Rhu∥V∥uh-Rhu∥V≤μ2∥u-Rhu∥V∥uh-Rhu∥V≤μ8∥uh-Rhu∥V2+8μ∥u-Rhu∥V2.
Similarly,J2 and J3 can be estimated, respectively, by
(82)J2=b(Rhu-uh,u,uh-Rhu)=N∥u∥V∥uh-Rhu∥V2≤μ2∥uh-Rhu∥V2,J3=b(uh,u-Rhu,uh-Rhu)≤N∥uh∥V∥u-Rhu∥V∥uh-Rhu∥V≤μ2∥u-Rhu∥V∥uh-Rhu∥V≤μ8∥uh-Rhu∥V2+8μ∥u-Rhu∥V2.
Finally, we estimate J4 by
(83)J4=b(uh-uH,Rhu-uH,uh-Rhu)=b(uh-Rhu,Rhu-uH,uh-Rhu)+b(Rhu-uH,Rhu-uH,uh-Rhu)≤N∥Rhu-uH∥V∥uh-Rhu∥V2+N∥Rhu-uH∥V2∥uh-Rhu∥V≤cH∥uh-Rhu∥V2+μ8∥uh-Rhu∥V2+c∥Rhu-uH∥V4.
Combining these estimates for J1 to J4 with (80), for sufficiently small H, we get
(84)∥uh-Rhu∥V≤c(∥u-Rhu∥V+∥Rhu-uH∥V2)≤c(∥u-Rhu∥V+∥Rhu-u∥V2+∥u-uH∥V2)≤c(h+H2),
which implies that
(85)∥u-uh∥V≤c(h+H2).
From (28) and (30), we have
(86)β2∥ph-Qhp∥≤sup⁡(vh,qh)∈(Vh,Mh)ℬh(uh-Rhu,ph-Qhp;vh,qh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)((ℬh(uh-u,ph-p;vh,qh)+ℬh(u-Rhu,p-Qhp;vh,qh)(uh-u,ph-p;vh,qh))×(∥vh∥V+∥qh∥)-1)=sup⁡(vh,qh)∈(Vh,Mh)(((uh,uH,vh)b(u,u,vh)-b(uH,uh,vh)-b(uh,uH,vh)+b(uH,uH,vh))×(∥vh∥V+∥qh∥)-1∑).
Since
(87)b(u,u,vh)-b(uH,uh,vh)-b(uh,uH,vh)+b(uH,uH,vh)=b(u-uh,u,vh)+b(u,u-Rhu,vh)-b(u-uh,u-Rhu,vh)+b(uh-uH,Rhu-uH,vh)-b(uH,uh-Rhu,vh)≤N∥u∥V(∥u-uh∥V+∥u-Rhu∥V)∥vh∥V+N∥u-uh∥V∥u-Rhu∥V∥vh∥V+N∥uH∥V∥uh-Rhu∥V∥vh∥V+N(∥u-uh∥V+∥u-uH∥V)×(∥u-Rhu∥V+∥u-uH∥V)∥vh∥V,
then using (31), (55), (84), and (85), we obtain
(88)∥ph-Qhp∥≤N∥u∥V(∥u-uh∥V+∥u-Rhu∥V)+N∥u-uh∥V∥u-Rhu∥V+N∥uH∥V∥uh-Rhu∥V+N(∥u-uh∥V+∥u-uH∥V)×(∥u-Rhu∥V+∥u-uH∥V)≤c(h+H2),
which together with (31) yields
(89)∥p-ph∥≤c(h+H2).


### 4.4. Newton Correction Scheme

As a result of Theorems6–8, if we choose H=O(h1/2), then two-level Stokes/Oseen/Newton iteration methods in the above subsections provide the same convergence order as the usual one-level finite element method (24). In this subsection, we propose a new Newton correction scheme. The error estimate for this scheme implies that if H=O(h1/4), then this correction scheme also provides the same convergence order as the usual one-level finite element method (24).Step I. Solve (uH,pH)∈(VH,MH) on the coarse mesh by the problem (52).Step II. Solve (uh,ph)∈(Vh,Mh) on the fine mesh by the problem (53) or (67) or (74).Step III. Solve a Newton correction solution (u⋆h,p⋆h) on the fine mesh; that is, find (u⋆h,p⋆h)∈(Vh,Mh) such that for all (vh,qh)∈(Vh,Mh)(90)ℬh(u⋆h,p⋆h;vh,qh)+b(uh,u⋆h,vh)+b(u⋆h,uh,vh)=(f,vh)+b(uh,uh,vh).First, we discuss the existence and uniqueness of the solution(u⋆h,p⋆h) to the problem (90). In terms of (57), (84), and (77), the solution uh to the problem (53) or (67) or (74) satisfies ∥uh∥V≤μ/2N. Then taking vh=u⋆h and qh=p⋆h in (90), we get
(91)ℬh(u⋆h,p⋆h;u⋆h,p⋆h)+b(uh,u⋆h,u⋆h)+b(u⋆h,uh,u⋆h)≥μ∥u⋆h∥V2+[p⋆h]h2-N∥uh∥V∥u⋆h∥V2≥μ2∥u⋆h∥V2+[p⋆h]h2.
Thus, we conclude that the problem (90) exists a unique solution (u⋆h,p⋆h)∈(Vh,Mh). Moreover, it is easy to check that u⋆h satisfies
(92)∥u⋆h∥V≤2μ∥f∥+μ2N≤3μ2N.Theorem 9.
Suppose that the uniqueness condition (14) or (68) or (75) holds. If (uh,ph)∈(Vh,Mh) and (u⋆h,p⋆h)∈(Vh,Mh) are the solutions of (24) and (90), respectively, then one has
(93)∥uh-u⋆h∥V+∥ph-p⋆h∥≤c∥uh-uh∥V2,
where uh is the solution to the problem (53) or (67) or (74).Proof.
Subtracting (24) from (90), we get
(94)ℬh(uh-u⋆h,ph-p⋆h;vh,qh)=b(uh,u⋆h,vh)+b(u⋆h,uh,vh)-b(uh,uh,vh)-b(uh,uh,vh)=b(u⋆h-uh,uh,vh)+b(uh,u⋆h-uh,vh)-b(uh-uh,uh-uh,vh).
Setting vh=uh-u⋆h and qh=ph-p⋆h in (94) and using (57), it yields
(95)μ∥uh-u⋆h∥V2≤b(u⋆h-uh,uh,uh-u⋆h)-b(uh-uh,uh-uh,uh-u⋆h)≤N∥uh∥V∥uh-u⋆h∥V2+N∥uh-uh∥V2∥uh-u⋆h∥V≤μ2∥uh-u⋆h∥V2+μ4∥uh-u⋆h∥V2+N2μ∥uh-uh∥V4.
Thus, we obtain
(96)∥uh-u⋆h∥V≤2Nμ∥uh-uh∥V2.
It follows from (28), (94), and (96) that
(97)β2∥ph-p⋆h∥≤sup⁡(vh,qh)∈(Vh,Mh)ℬh(uh-u⋆h,ph-p⋆h;vh,qh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)((b(u⋆h-uh,uh,vh)+b(uh,u⋆h-uh,vh)-b(uh-uh,uh-uh,vh))×(∥vh∥V+∥qh∥)-1(u⋆h-uh,uh,vh))≤2N∥uh∥V∥uh-u⋆h∥V+N∥uh-uh∥V2≤μ∥uh-u⋆h∥V+N∥uh-uh∥V2≤3N∥uh-uh∥V2.Combining Theorem9 with Theorems 6–8 and Theorem 4, we obtain the following error estimate between the solutions (u,p) and (u⋆h,p⋆h) to the problems (13) and (90), respectively.Theorem 10.
Under the assumption in Theorem9, if (u,p)∈(H2(Ω)2∩V,H1(Ω)∩M) and (u⋆h,p⋆h)∈(Vh,Mh) are the solutions of (13) and (90), respectively, then one has
(98)∥u-u⋆h∥V+∥p-p⋆h∥≤c(h+H4).


## 4.1. Two-Level Oseen Iteration Method

Step I. We solve (24) on the coarse mesh; that is, find (uH,pH)∈(VH,MH) such that for all (vH,qH)∈(VH,MH)(52)ℬH(uH,pH;vH,qH)+b(uH,uH,vH)=(f,vH).Step II. We solve a discrete Oseen problem according to Oseen iteration on the fine mesh; that is, find (uh,ph)∈(Vh,Mh) such that for all (vh,qh)∈(Vh,Mh)(53)ℬh(uh,ph;vh,qh)+b(uH,uh,vh)=(f,vh).First, we discuss the existence and uniqueness of the solution to the problem (53) under the uniqueness condition (14). In view of Theorem 2, the problem (52) exists a unique solution (uH,pH)∈(VH,MH) with
(54)∥uH∥V≤1μ∥f∥<μ2N.
Moreover, it follows from Theorems 4 and 5 that
(55)∥u-uH∥+H∥u-uH∥V+H∥p-pH∥≤cH2.
On the other hand, setting vh=uh and qh=ph in (53), it yields
(56)ℬh(uh,ph;uh,ph)+b(uH,uh,uh)=μ∥uh∥V2+[ph]h2.
Then it is easy to show that the problem (53) also exists a unique solution (uh,ph)∈(Vh,Mh) such that
(57)∥uh∥V≤1μ∥f∥<μ2N.
Next, we give the error estimate for the two-level Oseen iteration method.Theorem 6.
Suppose that the uniqueness condition (14) holds. If (u,p)∈(H2(Ω)2∩V,H1(Ω)∩M) and (uh,ph)∈(Vh,Mh) are the solutions of (13) and (53), respectively, then there holds
(58)∥u-uh∥V+∥p-ph∥≤c(h+H2).Proof.
In terms of the definitionℬh(·,·;·,·) and (30), we get
(59)μ∥uh-Rhu∥V2+[ph-Qhp]h2=ℬh(uh-Rhu,ph-Qhp;uh-Rhu,ph-Qhp)=ℬh(uh-u,ph-p;uh-Rhu,ph-Qhp)+ℬh(u-Rhu,p-Qhp;uh-Rhu,ph-Qhp)=b(u,u,uh-Rhu)-b(uH,uh,uh-Rhu).
We rewrite b(u,u,uh-Rhu)-b(uH,uh,uh-Rhu) as
(60)b(u,u,uh-Rhu)-b(uH,uh,uh-Rhu)=b(u-uH,u,uh-Rhu)-b(uH,uh-u,uh-Rhu)=b(u-uH,u,uh-Rhu)-b(uH,Rhu-u,uh-Rhu).
Then using (8), (10), and (54), we obtain
(61)b(u,u,uh-Rhu)-b(uH,uh,uh-Rhu)≤N∥u∥2∥uh-Rhu∥V∥u-uH∥+N∥uH∥V∥Rhu-u∥V∥uh-Rhu∥V≤μ2∥uh-Rhu∥V2+N2μ∥u∥22∥u-uH∥2+μ4∥Rhu-u∥V2.
Thus, there holds
(62)∥uh-Rhu∥V≤c(∥u-uH∥+∥Rhu-u∥V)≤c(h+H2),
where we use (31) and (55). A direct consequence of the above estimate is
(63)∥u-uh∥V≤c(h+H2).
From (28), (30), (55), and (63), we have
(64)β2∥ph-Qhp∥≤sup⁡(vh,qh)∈(Vh,Mh)ℬh(uh-Rhu,ph-Qhp;vh,qh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)((ℬh(uh-u,ph-p;vh,qh)+ℬh(u-Rhu,p-Qhp;vh,qh)(uh-u,ph-p;vh,qh))×(∥vh∥V+∥qh∥)-1(uh-u,ph-p;vh,qh))=sup⁡(vh,qh)∈(Vh,Mh)b(u,u,vh)-b(uH,uh,vh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)b(u-uH,u,vh)+b(uH,u-uh,vh)∥vh∥V+∥qh∥≤c(∥u-uH∥+∥u-uh∥V)≤c(h+H2),
which together with (31) yields
(65)∥p-ph∥≤c(h+H2).


## 4.2. Two-Level Stokes Iteration Method

Step I. We solve (24) on the coarse mesh; that is, find (uH,pH)∈(VH,MH) such that for all (vH,qH)∈(VH,MH)(66)ℬH(uH,pH;vH,qH)+b(uH,uH,vH)=(f,vH).Step II. We solve a discrete Stokes problem according to Stokes iteration on the fine mesh; that is, find (uh,ph)∈(Vh,Mh) such that for all (vh,qh)∈(Vh,Mh)(67)ℬh(uh,ph;vh,qh)+b(uH,uH,vh)=(f,vh).In this subsection, we assume that the following uniqueness conditions hold:(68)3μ-2N∥f∥<1.
Proceeding the argument as in Section 4.1, the problem (66) exists a unique solution (uH,pH)∈(VH,MH) and uH satisfies ∥uH∥V≤(1/μ)∥f∥<μ/3N. According to the definition of ℬh(·,·;·,·), the discrete Stokes problem (67) also exists a unique solution (uh,ph)∈(Vh,Mh). Moreover, uh satisfies
(69)∥uh∥V≤1μ∥f∥+Nμ∥uH∥V2≤1μ∥f∥+13∥uH∥V≤43μ∥f∥≤4μ9N<μ2N.
Then the error estimate for two-level Stokes iteration method is derived in the following theorem.Theorem 7.
Suppose that the uniqueness condition (68) holds. If (u,p)∈(H2(Ω)2∩V,H1(Ω)∩M) and (uh,ph)∈(Vh,Mh) are the solutions of (13) and (67), respectively, then one has
(70)∥u-uh∥V+∥p-ph∥≤c(h+H2).Proof.
Subtracting (13) from (67), we get
(71)ℬh(uh-u,ph-p;vh,qh)=b(u,u,vh)-b(uH,uH,vh)-Ch(p,qh).
Then, from (10), (28), (30), and (55), we have
(72)β2(∥uh-Rhu∥V+∥ph-Qhp∥)≤sup⁡(vh,qh)∈(Vh,Mh)ℬh(uh-Rhu,ph-Qhp;vh,qh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)((ℬh(uh-u,ph-p;vh,qh)+ℬh(u-Rhu,p-Qhp;vh,qh)(uh-u,ph-p;vh,qh))×(∥vh∥V+∥qh∥)-1(uh-u,ph-p;vh,qh))=sup⁡(vh,qh)∈(Vh,Mh)b(u,u,vh)-b(uH,uH,vh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)((∥vh∥V+∥qh∥)-1((∥vh∥V+∥qh∥)-1b(u-uH,u,vh)+b(u,u-uH,vh)-b(u-uH,u-uH,vh)(∥vh∥V+∥qh∥)-1)×(∥vh∥V+∥qh∥)-1)≤2N∥u∥2∥u-uH∥+N∥u-uH∥V2≤cH2,
which together with (31) completes the proof of (70).


## 4.3. Two-Level Newton Iteration Method

Step I. We solve (24) on the coarse mesh; that is, find (uH,pH)∈(VH,MH) such that for all (vH,qH)∈(VH,MH)(73)ℬH(uH,pH;vH,qH)+b(uH,uH,vH)=(f,vH).Step II. We solve discrete linearized Navier-Stokes equations according to Newton iteration on the fine mesh; that is, find (uh,ph)∈(Vh,Mh) such that for all (vh,qh)∈(Vh,Mh)(74)ℬh(uh,ph;vh,qh)+b(uH,uh,vh)+b(uh,uH,vh)=(f,vh)+b(uH,uH,vh).As in Section4.2, we modify the uniqueness condition as
(75)4μ-2N∥f∥<1.
In this case, the solution uH of the problem (73) satisfies ∥uH∥V≤(1/μ)∥f∥<μ/4N. Setting vh=uh and qh=ph in (74), we have
(76)ℬh(uh,ph;uh,ph)+b(uh,uH,uh)+b(uH,uh,uh)≥μ∥uh∥V2+[ph]h2-N∥uH∥V∥uh∥V2≥3μ4∥uh∥V2+[ph]h2.
Moreover, we can estimate uh by
(77)∥uh∥V≤43μ∥f∥+4N3μ∥uH∥V2≤43μ∥f∥+13∥uH∥V≤53μ∥f∥≤5μ12N<μ2N.
The error estimate for two-level Newton iteration method is derived in the following theorem.Theorem 8.
Suppose that the uniqueness condition (75) holds. If (u,p)∈(H2(Ω)2∩V,H1(Ω)∩M) and (uh,ph)∈(Vh,Mh) are the solutions of (13) and (74), respectively, then one has
(78)∥u-uh∥V+∥p-ph∥≤c(h+H2).Proof.
Proceeding as in the proof of Theorem7, we have
(79)μ∥uh-Rhu∥V2+[ph-Qhp]h2=ℬh(uh-Rhu,ph-Qhp;uh-Rhu,ph-Qhp)=ℬh(uh-u,ph-p;uh-Rhu,ph-Qhp)+ℬh(u-Rhu,p-Qhp;uh-Rhu,ph-Qhp)=b(u,u,uh-Rhu)-b(uH,uh,uh-Rhu)-b(uh,uH,uh-Rhu)+b(uH,uH,uh-Rhu).
We rewrite the right-hand side of the above identity as follows:
(80)b(u,u,uh-Rhu)-b(uH,uh,uh-Rhu)-b(uh,uH,uh-Rhu)+b(uH,uH,uh-Rhu)=b(u-uh,u,uh-Rhu)+b(uh,u-uh,uh-Rhu)+b(uh-uH,uh-uH,uh-Rhu)=b(u-Rhu,u,uh-Rhu)+b(Rhu-uh,u,uh-Rhu)+b(uh,u-Rhu,uh-Rhu)+b(uh-uH,Rhu-uH,uh-Rhu)=J1+⋯+J4.
Using (8) and (15), we have
(81)J1=b(u-Rhu,u,uh-Rhu)≤N∥u∥V∥u-Rhu∥V∥uh-Rhu∥V≤μ2∥u-Rhu∥V∥uh-Rhu∥V≤μ8∥uh-Rhu∥V2+8μ∥u-Rhu∥V2.
Similarly,J2 and J3 can be estimated, respectively, by
(82)J2=b(Rhu-uh,u,uh-Rhu)=N∥u∥V∥uh-Rhu∥V2≤μ2∥uh-Rhu∥V2,J3=b(uh,u-Rhu,uh-Rhu)≤N∥uh∥V∥u-Rhu∥V∥uh-Rhu∥V≤μ2∥u-Rhu∥V∥uh-Rhu∥V≤μ8∥uh-Rhu∥V2+8μ∥u-Rhu∥V2.
Finally, we estimate J4 by
(83)J4=b(uh-uH,Rhu-uH,uh-Rhu)=b(uh-Rhu,Rhu-uH,uh-Rhu)+b(Rhu-uH,Rhu-uH,uh-Rhu)≤N∥Rhu-uH∥V∥uh-Rhu∥V2+N∥Rhu-uH∥V2∥uh-Rhu∥V≤cH∥uh-Rhu∥V2+μ8∥uh-Rhu∥V2+c∥Rhu-uH∥V4.
Combining these estimates for J1 to J4 with (80), for sufficiently small H, we get
(84)∥uh-Rhu∥V≤c(∥u-Rhu∥V+∥Rhu-uH∥V2)≤c(∥u-Rhu∥V+∥Rhu-u∥V2+∥u-uH∥V2)≤c(h+H2),
which implies that
(85)∥u-uh∥V≤c(h+H2).
From (28) and (30), we have
(86)β2∥ph-Qhp∥≤sup⁡(vh,qh)∈(Vh,Mh)ℬh(uh-Rhu,ph-Qhp;vh,qh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)((ℬh(uh-u,ph-p;vh,qh)+ℬh(u-Rhu,p-Qhp;vh,qh)(uh-u,ph-p;vh,qh))×(∥vh∥V+∥qh∥)-1)=sup⁡(vh,qh)∈(Vh,Mh)(((uh,uH,vh)b(u,u,vh)-b(uH,uh,vh)-b(uh,uH,vh)+b(uH,uH,vh))×(∥vh∥V+∥qh∥)-1∑).
Since
(87)b(u,u,vh)-b(uH,uh,vh)-b(uh,uH,vh)+b(uH,uH,vh)=b(u-uh,u,vh)+b(u,u-Rhu,vh)-b(u-uh,u-Rhu,vh)+b(uh-uH,Rhu-uH,vh)-b(uH,uh-Rhu,vh)≤N∥u∥V(∥u-uh∥V+∥u-Rhu∥V)∥vh∥V+N∥u-uh∥V∥u-Rhu∥V∥vh∥V+N∥uH∥V∥uh-Rhu∥V∥vh∥V+N(∥u-uh∥V+∥u-uH∥V)×(∥u-Rhu∥V+∥u-uH∥V)∥vh∥V,
then using (31), (55), (84), and (85), we obtain
(88)∥ph-Qhp∥≤N∥u∥V(∥u-uh∥V+∥u-Rhu∥V)+N∥u-uh∥V∥u-Rhu∥V+N∥uH∥V∥uh-Rhu∥V+N(∥u-uh∥V+∥u-uH∥V)×(∥u-Rhu∥V+∥u-uH∥V)≤c(h+H2),
which together with (31) yields
(89)∥p-ph∥≤c(h+H2).


## 4.4. Newton Correction Scheme

As a result of Theorems6–8, if we choose H=O(h1/2), then two-level Stokes/Oseen/Newton iteration methods in the above subsections provide the same convergence order as the usual one-level finite element method (24). In this subsection, we propose a new Newton correction scheme. The error estimate for this scheme implies that if H=O(h1/4), then this correction scheme also provides the same convergence order as the usual one-level finite element method (24).Step I. Solve (uH,pH)∈(VH,MH) on the coarse mesh by the problem (52).Step II. Solve (uh,ph)∈(Vh,Mh) on the fine mesh by the problem (53) or (67) or (74).Step III. Solve a Newton correction solution (u⋆h,p⋆h) on the fine mesh; that is, find (u⋆h,p⋆h)∈(Vh,Mh) such that for all (vh,qh)∈(Vh,Mh)(90)ℬh(u⋆h,p⋆h;vh,qh)+b(uh,u⋆h,vh)+b(u⋆h,uh,vh)=(f,vh)+b(uh,uh,vh).First, we discuss the existence and uniqueness of the solution(u⋆h,p⋆h) to the problem (90). In terms of (57), (84), and (77), the solution uh to the problem (53) or (67) or (74) satisfies ∥uh∥V≤μ/2N. Then taking vh=u⋆h and qh=p⋆h in (90), we get
(91)ℬh(u⋆h,p⋆h;u⋆h,p⋆h)+b(uh,u⋆h,u⋆h)+b(u⋆h,uh,u⋆h)≥μ∥u⋆h∥V2+[p⋆h]h2-N∥uh∥V∥u⋆h∥V2≥μ2∥u⋆h∥V2+[p⋆h]h2.
Thus, we conclude that the problem (90) exists a unique solution (u⋆h,p⋆h)∈(Vh,Mh). Moreover, it is easy to check that u⋆h satisfies
(92)∥u⋆h∥V≤2μ∥f∥+μ2N≤3μ2N.Theorem 9.
Suppose that the uniqueness condition (14) or (68) or (75) holds. If (uh,ph)∈(Vh,Mh) and (u⋆h,p⋆h)∈(Vh,Mh) are the solutions of (24) and (90), respectively, then one has
(93)∥uh-u⋆h∥V+∥ph-p⋆h∥≤c∥uh-uh∥V2,
where uh is the solution to the problem (53) or (67) or (74).Proof.
Subtracting (24) from (90), we get
(94)ℬh(uh-u⋆h,ph-p⋆h;vh,qh)=b(uh,u⋆h,vh)+b(u⋆h,uh,vh)-b(uh,uh,vh)-b(uh,uh,vh)=b(u⋆h-uh,uh,vh)+b(uh,u⋆h-uh,vh)-b(uh-uh,uh-uh,vh).
Setting vh=uh-u⋆h and qh=ph-p⋆h in (94) and using (57), it yields
(95)μ∥uh-u⋆h∥V2≤b(u⋆h-uh,uh,uh-u⋆h)-b(uh-uh,uh-uh,uh-u⋆h)≤N∥uh∥V∥uh-u⋆h∥V2+N∥uh-uh∥V2∥uh-u⋆h∥V≤μ2∥uh-u⋆h∥V2+μ4∥uh-u⋆h∥V2+N2μ∥uh-uh∥V4.
Thus, we obtain
(96)∥uh-u⋆h∥V≤2Nμ∥uh-uh∥V2.
It follows from (28), (94), and (96) that
(97)β2∥ph-p⋆h∥≤sup⁡(vh,qh)∈(Vh,Mh)ℬh(uh-u⋆h,ph-p⋆h;vh,qh)∥vh∥V+∥qh∥=sup⁡(vh,qh)∈(Vh,Mh)((b(u⋆h-uh,uh,vh)+b(uh,u⋆h-uh,vh)-b(uh-uh,uh-uh,vh))×(∥vh∥V+∥qh∥)-1(u⋆h-uh,uh,vh))≤2N∥uh∥V∥uh-u⋆h∥V+N∥uh-uh∥V2≤μ∥uh-u⋆h∥V+N∥uh-uh∥V2≤3N∥uh-uh∥V2.Combining Theorem9 with Theorems 6–8 and Theorem 4, we obtain the following error estimate between the solutions (u,p) and (u⋆h,p⋆h) to the problems (13) and (90), respectively.Theorem 10.
Under the assumption in Theorem9, if (u,p)∈(H2(Ω)2∩V,H1(Ω)∩M) and (u⋆h,p⋆h)∈(Vh,Mh) are the solutions of (13) and (90), respectively, then one has
(98)∥u-u⋆h∥V+∥p-p⋆h∥≤c(h+H4).


## 5. Numerical Experiments

In this section, we make some numerical experiments to support the theoretical results derived in Section4. The body force f is appropriately selected such that the exact solution of the problem (1) is given by
(99)u(x,y)=(u1(x,y),u2(x,y)),p(x,y)=x2-y2,u1(x,y)=x2(x-1)2y(y-1)(2y-1),u2(x,y)=-x(x-1)(2x-1)y2(y-1)2
in the unit square Ω=(0,1)×(0,1).In all experiments, we choose the viscous coefficientμ=0.1 and stabilized parameter α=0.01 in (18). According to Theorems 6–8, we choose H=h1/2; then two-level finite element approximation solution is of the following optimal error estimate:
(100)∥u-uh∥V+∥p-ph∥≤ch.
Here we select eight fine mesh values h=1/42,1/62,…,1/182. Then the corresponding coarse mesh values are obtained. These fine mesh values also are used in the numerical experiment for one-level finite element method. The numerical results are displayed in Tables 1, 2, 3, and 4, from which the observations and conclusions are presented as follows.(i)
Based on Table1, the numerical convergence orders reach the optimal orders which coincide with the theoretical results derived in Theorems 4 and 5, namely, O(h) for the velocity in H1-norm and the pressure in L2-norm and O(h2) for the velocity in L2-norm. We also observe that if h=1/182, in this case, the standard one-level method can not work and does not obtain the predicted numerical results.(ii)
From Tables2–4, we can see that if H=h1/2, all three two-level Stokes/Oseen/Newton iteration methods can reach the optimal convergence orders of O(h) for both velocity and pressure, in H1-norm and L2-norm, respectively, as proven in Theorems 6–8. Besides, we find that these methods can achieve the optimal convergence orders of O(h2) for velocity in the sense of L2-norm as expected.(iii)
From the view of computational cost, we can obviously observe by comparing Table1 and Tables 2–4 that these two-level iteration methods significantly save CPU time compared with the one-level method and, meanwhile, obtain nearly the same approximation results.Table 1
Convergence of one-level method.
1
/
h
∥
u
-
u
h
∥
/
∥
u
∥
Rate
∥
u
-
u
h
∥
V
/
∥
u
∥
V
Rate
∥
p
-
p
h
∥
/
∥
p
∥
Rate
CPU (s)
4
2
5.05728
e
-
02
/
2.04342
e
-
01
/
7.00342
e
-
03
/
0.271
6
2
9.71970
e
-
03
2.0338
8.42186
e
-
02
1.0930
1.74965
e
-
03
1.7104
1.211
8
2
3.03424
e
-
03
2.0234
4.57602
e
-
02
1.0602
6.91717
e
-
04
1.6129
4.526
1
0
2
1.23515
e
-
03
2.0139
2.87904
e
-
02
1.0383
3.42162
e
-
04
1.5772
10.509
1
2
2
5.94367
e
-
04
2.0059
1.98039
e
-
02
1.0261
1.93897
e
-
04
1.5576
22.588
1
4
2
3.21053
e
-
04
1.9977
1.44653
e
-
02
1.0189
1.20442
e
-
04
1.5445
44.181
1
6
2
1.88826
e
-
04
1.9875
1.10329
e
-
02
1.0143
7.99308
e
-
05
1.5352
80.286
1
8
2
OUT
OF
MEMORYTable 2
Convergence of two-level Oseen iteration method.
1
/
H
1
/
h
∥
u
-
u
h
∥
/
∥
u
∥
Rate
∥
u
-
u
h
∥
V
/
∥
u
∥
V
Rate
∥
p
-
p
h
∥
/
∥
p
∥
Rate
CPU (s)
4
4
2
5.05858
e
-
02
/
2.04349
e
-
01
/
7.00341
e
-
03
/
0.179
6
6
2
9.73019
e
-
03
2.0328
8.42217
e
-
02
1.0930
1.74965
e
-
03
1.7104
0.738
8
8
2
3.04358
e
-
03
2.0199
4.57618
e
-
02
1.0602
6.91725
e
-
04
1.6129
2.105
10
1
0
2
1.24408
e
-
03
2.0046
2.87914
e
-
02
1.0383
3.42169
e
-
04
1.5772
5.081
12
1
2
2
6.03073
e
-
04
1.9858
1.98046
e
-
02
1.0261
1.93904
e
-
04
1.5575
10.546
14
1
4
2
3.29574
e
-
04
1.9599
1.44658
e
-
02
1.0189
1.20448
e
-
04
1.5444
19.846
16
1
6
2
1.97162
e
-
04
1.9238
1.10332
e
-
02
1.0143
7.99367
e
-
05
1.5352
36.305
18
1
8
2
1.26739
e
-
04
1.8759
8.69472
e
-
03
1.0112
5.57676
e
-
05
1.5284
56.658Table 3
Convergence of two-level Stokes iteration method.
1
/
H
1
/
h
∥
u
-
u
h
∥
/
∥
u
∥
Rate
∥
u
-
u
h
∥
V
/
∥
u
∥
V
Rate
∥
p
-
p
h
∥
/
∥
p
∥
Rate
CPU (s)
4
4
2
5.05773
e
-
02
/
2.04346
e
-
01
/
7.00344
e
-
03
/
0.163
6
6
2
9.72434
e
-
03
2.0333
8.42205
e
-
02
1.0930
1.74968
e
-
03
1.7103
0.606
8
8
2
3.03897
e
-
03
2.0215
4.57613
e
-
02
1.0602
6.91743
e
-
04
1.6128
1.808
10
1
0
2
1.23981
e
-
03
2.0089
2.87911
e
-
02
1.0383
3.42185
e
-
04
1.5772
4.451
12
1
2
2
5.98991
e
-
04
1.9950
1.98044
e
-
02
1.0261
1.93918
e
-
04
1.5575
9.314
14
1
4
2
3.25639
e
-
04
1.9768
1.44657
e
-
02
1.0189
1.20460
e
-
04
1.5443
17.481
16
1
6
2
1.93359
e
-
04
1.9518
1.10331
e
-
02
1.0143
7.99470
e
-
05
1.5350
30.246
18
1
8
2
1.23069
e
-
04
1.9179
8.69461
e
-
03
1.0112
5.57768
e
-
05
1.5283
49.980Table 4
Convergence of two-level Newton iteration method.
1
/
H
1
/
h
∥
u
-
u
h
∥
/
∥
u
∥
Rate
∥
u
-
u
h
∥
V
/
∥
u
∥
V
Rate
∥
p
-
p
h
∥
/
∥
p
∥
Rate
CPU (s)
4
4
2
5.05710
e
-
02
/
2.04342
e
-
01
/
7.00340
e
-
03
/
0.204
6
6
2
9.71757
e
-
03
2.0340
8.42186
e
-
02
1.0930
1.74964
e
-
03
1.7104
0.848
8
8
2
3.03203
e
-
03
2.0243
4.57601
e
-
02
1.0602
6.91711
e
-
04
1.6129
2.768
10
1
0
2
1.23292
e
-
03
2.0163
2.87903
e
-
02
1.0383
3.42158
e
-
04
1.5772
6.316
12
1
2
2
5.92128
e
-
04
2.0113
1.98039
e
-
02
1.0261
1.93895
e
-
04
1.5576
13.269
14
1
4
2
3.18804
e
-
04
2.0083
1.44653
e
-
02
1.0189
1.20440
e
-
04
1.5445
24.717
16
1
6
2
1.86566
e
-
04
2.0062
1.10328
e
-
02
1.0143
7.99295
e
-
05
1.5352
42.030
18
1
8
2
1.16338
e
-
04
2.0049
8.69440
e
-
03
1.0112
5.57612
e
-
05
1.5285
69.469The numerical results for two-level Newton correction method also are displayed in Table5. Based on Theorem 10, the optimal convergence order O(h) for the velocity in H1-norm and the pressure in L2-norm can be reached as H≈h1/4, which has been reflected in Table 5. However, this Newton correction method only can save about 85% CPU time compared with the one-level method. The reason is that the Newton correction method needs two-step computation in the fine mesh.Table 5
Convergence of two-level Newton correction scheme.
1
/
H
1
/
h
∥
u
-
u
⋆
h
∥
/
∥
u
∥
Rate
∥
u
-
u
⋆
h
∥
V
/
∥
u
∥
V
Rate
∥
p
-
p
⋆
h
∥
/
∥
p
∥
Rate
CPU (s)
2
4
2
5.05706
e
-
02
/
2.04341
e
-
01
/
7.00308
e
-
03
/
0.263
2
6
2
9.71752
e
-
03
2.0340
8.42185
e
-
02
1.0930
1.74964
e
-
03
1.7104
1.206
3
8
2
3.03204
e
-
03
2.0243
4.57601
e
-
02
1.0602
6.91710
e
-
04
1.6129
3.729
3
1
0
2
1.23293
e
-
03
2.0163
2.87903
e
-
02
1.0383
3.42158
e
-
04
1.5772
9.294
3
1
2
2
5.92131
e
-
04
2.0113
1.98039
e
-
02
1.0261
1.93895
e
-
04
1.5576
19.497
4
1
4
2
3.18806
e
-
04
2.0082
1.44653
e
-
02
1.0189
1.20440
e
-
04
1.5445
37.133
4
1
6
2
1.86567
e
-
04
2.0062
1.10328
e
-
02
1.0143
7.99295
e
-
05
1.5352
65.678
4
1
8
2
OUT
OF
MEMORYFinally, we show the contour plots of the exact solution and the numerical solution to exhibit the approximation profiles. Figures1 and 2 display the exact solution and the numerical solution by one-level stabilized method. Besides, as to the two-level method, here only the numerical solution by Newton iteration method is displayed in Figure 3. From these three groups of contour plots, we can observe the good coincidence with each other to illustrate the stability of the present stabilized methods.Contour plots of exact solution. From left to right: two components of velocity and pressure.
(a)
(b)
(c)Contour plots of numerical solution by one-level stabilized method. From left to right: two components of velocity and pressure.
(a)
(b)
(c)Contour plots of numerical solution by two-level Newton iteration method. From left to right: two components of velocity and pressure.
(a)
(b)
(c)



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*Source: 698354-2014-05-08.xml*