# Two-Level Stabilized Finite Volume Methods for Stationary Navier-Stokes Equations

Authors: Anas Rachid; Mohamed Bahaj; Noureddine Ayoub
Journal: Advances in Numerical Analysis
 (2012)

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## Abstract

We propose two algorithms of two-level methods for resolving the nonlinearity in the stabilized finite volume approximation of the Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. A macroelement condition is introduced for constructing the local stabilized finite volume element formulation. Moreover the two-level methods consist of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. The error analysis shows that the two-level stabilized finite volume element method provides an approximate solution with the convergence rate of the same order as the usual stabilized finite volume element solution solving the Navier-Stokes equations on a fine mesh for a related choice of mesh widths.


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## Body

## 1. Introduction

We consider a two-level method for the resolution of the nonlinear system arising from finite volume discretizations of the equilibrium, incompressible Navier-Stokes equations:(1.1)-νΔu+(u⋅∇)u+∇p=finΩ,(1.2)∇⋅u=0inΩ,(1.3)u=0in∂Ω,
where u=(u1(x),u2(x)) is the velocity vector, p=p(x) is the pressure, f=f(x) is the body force, ν>0 is the viscosity of the fluid, and Ω⊂ℝ2, the flow domain, is assumed to be bounded, to have a Lipschitz-continuous boundary ∂Ω, and to satisfy a further condition stated in (H1).Finite volume method is an important numerical tool for solving partial differential equations. It has been widely used in several engineering fields, such as fluid mechanics, heat and mass transfer, and petroleum engineering. The method can be formulated in the finite difference framework or in the Petrov-Galerkin framework. Usually, the former one is called finite volume method [1, 2], MAC (marker and cell) method [3], or cell-centered method [4], and the latter one is called finite volume element method (FVE) [5–7], covolume method [8], or vertex-centered method [9, 10]. We refer to the monographs [11, 12] for general presentations of these methods. The most important property of FVE is that it can preserve the conservation laws (mass, momentum, and heat flux) on each control volume. This important property, combined with adequate accuracy and ease of implementation, has attracted more people to do research in this field.On the other hand, the two-level finite element strategy based on two finite element spaces on one coarse and one fine mesh has been widely studied for steady semilinear elliptic equations [13, 14] and the Navier-Stokes equations [15–22]. For the finite volume element method, Bi and Ginting [23] have studied two-grid finite volume element method for linear and nonlinear elliptic problems; Chen et al. [24] have applied two-grid methods for solving a two-dimensional nonlinear parabolic equation using finite volume element method. Chen and Liu [25] have also studied this method for semilinear parabolic problems. However, to the best of our knowledge, there is no two-level finite volume convergence analysis for the Navier-Stokes equations in the literature.In this paper we aim to combine FVE method based onP1-P0 macroelement with two-level strategy to solve the two-dimensional Navier-Stokes (1.1)–(1.3). The heart of the analysis is the use of a transfer operator to connect finite volume and finite element estimations which will lead to more difficult term to estimate. We choose the two-grid spaces as two conforming finite element spaces XH and Xh on one coarse grid with mesh size H and one fine grid with mesh size h≪H. We propose two algorithms of two-level method for resolving the nonlinearity in the stabilized finite volume approximation of the problem (1.1)–(1.3): the simple and Newton algorithms. First we prove that the simple two-level stabilized finite volume solution (uh,ph) is the following error estimate:(1.4)‖u-uh‖1+‖p-ph‖0≤C(h+H2).
Second we prove that the Newton two-level stabilized finite volume solution (uh,ph) is the following error estimate:(1.5)‖u-uh‖1+‖p-ph‖0≤C(h+H3|log⁡⁡h|1/2),
where C denotes some generic constant which may stand for different values at its different occurrences.Hence, the two-level algorithms achieve asymptotically optimal approximation as long as the mesh sizes satisfyh=O(H2) for the simple two-level stabilized finite volume solution and h=O(H3|logh|1/2) for the Newton two-level stabilized finite volume solution. As a result, solving the nonlinear Navier-Stokes equations will not be much more difficult than solving one single linearized equation.The rest of this paper is organized as follows. In the next section, we introduce some notations and construct a FVE scheme. In Section3 we recall same preliminary estimates of the stabilized finite volume approximations. Finally the two-level FVE algorithms and the improved error estimates are presented and established in Section 4.


## 2. Finite Volume Scheme

### 2.1. Notations

We will use∥·∥m and |·|m to denote the norm and seminorm of the Sobolev space (Hm(Ω))d, d=1,2. Let H01(Ω) be the standard Sobolev subspace of H1(Ω) of functions vanishing on ∂Ω. We introduce the following notations:(2.1)X=(H01(Ω))2,Y=L02(Ω)={q:q∈L2(Ω),∫Ωq=0}.
The scalar product and norm in Y are denoted by the usual L2(Ω) inner product (·,·) and ∥·∥0, respectively. As mentioned above, we need a further assumption on Ω.H1
Assume thatΩ is regular so that the unique solution (v,q)∈X×Y of the steady Stokes problem
(2.2)-Δv+∇q=g,∇⋅v=0;v|∂Ω=0
for a prescribed g∈(L2(Ω))2 exists and satisfies
(2.3)‖v‖2+‖q‖1≤C‖g‖0,
where C>0 is a constant depending on Ω.The weak formulation of the problem (1.1)–(1.3) is to find (u,p)∈X×Y such that(2.4)a(u,v)-d(v,p)+b(u,u,v)=(f,v),∀v∈X,d(u,q)=0,∀q∈Y,
where the bilinear forms a(·,·), d(·,·) and the trilinear form b(·,·,·) are given by(2.5)a(u,v)=ν(∇u,∇v)=ν∫Ω∇u:∇vdx,∀u,v∈X,d(v,q)=(∇⋅v,q)=∫Ωq∇⋅vdx,∀u,q∈X×Y,b(u,w,v)=((u⋅∇w),v)+12((∇⋅u)w,v)=12((u⋅∇w),v)-12((u⋅∇v),w)∀u,v,w∈X.
Introducing the generalized bilinear form on (X×Y)2 by(2.6)B((u,p);(v,q))=a(u,v)-d(v,p)+d(u,q),
we can rewrite (2.4) in a compact form: find (u,p)∈X×Y such that(2.7)B((u,p);(v,q))+b(u,u,v)=(f,v),∀(v,q)∈X×Y.Let𝒯h be a quasi-uniform triangulation of Ω with h=maxhK, where hK is the diameter of the triangle K∈𝒯h. We assume that the partition 𝒯h has been obtained from a macrotriangular partition Λh by joining the sides of each element of Λh. Every element K∈𝒯h must lie in exactly one macroelement 𝒦, which implies that macroelements do not overlap. For each 𝒦, the set of interelement edges which are strictly in the interior of 𝒦 will be denoted by Γ𝒦, and the length of an edge e∈Γ𝒦 is denoted by he.Based on this triangulation, letXh be the standard conforming finite element subspace of piecewise linear velocity,(2.8)Xh={v∈C(Ω)∩X:v∣Kislinear,∀K∈Th;v∣∂Ω=0},
and let Yh be the piecewise constant pressure subspace(2.9)Yh={q∈Y:q∣Kisconstant,∀K∈Th}.
It is well known that the standard P1-P0 element does not satisfy the inf-sup condition and cannot be applied to problem (1.1)–(1.3) directly. But a locally stabilized method based on the macroelement can be used to yield adequate approximations [6].In order to describe the FVE method for solving problem (1.1)–(1.3), we will introduce a dual partition 𝒯h* based upon the original partition 𝒯h whose elements are called control volumes. We construct the control volumes in the same way as in [5, 26]. Let zK be the barycenter of K∈𝒯h. We connect zK with line segments to the midpoints of the edges of K, thus partitioning K into three quadrilaterals Kz, z∈Zh(K), where Zh(K) are the vertices of K. Then with each vertex z∈Zh=∪K∈𝒯hZh(K), we associate a control volume Vz, which consists of the union of the subregions Kz, sharing the vertex z. Thus we finally obtain a group of control volumes covering the domain Ω, which is called the dual partition 𝒯h* of the triangulation 𝒯h. We denote by Zh0 the set of interior vertices.We call the partition𝒯h* regular or quasi-uniform if there exists a positive constant C>0 such that(2.10)C-1h2≤meas⁡(Vz)≤Ch2,Vz∈Th*.
If the finite element triangulation 𝒯h is quasi-uniform, then the dual partition 𝒯h* is also quasi-uniform [23].


### 2.2. Construction of the FVE Scheme

We formulate the FVE method for the problem (1.1)–(1.3) as follows: given a z∈Zh0 and K∈𝒯h, integrating (1.1) over the associated control volume Vz and (1.1) over the element K and applying Green’s formula, we obtain an integral conservation form(2.11)-ν∫∂Vz∇unds+∫∂Vzpnds+∫Vzu⋅∇udx=∫Vzfdx,∀z∈Zh0,(2.12)∫K∇⋅udx=0,∀K∈Th,
where n denotes the unit outer normal vector to ∂Vz (Figure 1).Left-hand side: a sample region with blue lines indicating the corresponding control volumeVz. Right-hand side: a triangle K partitioned into three subregions Kz.
(a)(b)LetIh*:Xh→Xh* be the transfer operator defined by(2.13)Ih*v=∑z∈Zh0v(z)χz,∀v∈Xh,
where(2.14)Xh*={v=(v1,v2)∈(L2(Ω))2:vi∣Vzisconstant,i=1,2∀z∈Zh0},
and χz is the characteristic function of the control volume Vz. The operator Ih* satisfies [26](2.15)‖Ih*v‖0≤C‖v‖0,∀v∈Xh.
Now for an arbitrary Ih*v, we multiply (2.11) by v(z) and sum over all z∈Zh0 to get(2.16)ah(u,Ih*v)-dh(Ih*v,p)+bh(u,u,Ih*v)=(f,Ih*v),∀v∈Xh.
Here ah:X×Xh→ℝ, dh:Xh×Y→ℝ and bh:X×X×Xh→ℝ are defined by(2.17)ah(u,Ih*v)=-ν∑z∈Zh0v(z)∫∂Vz∇unds,dh(Ih*v,p)=∑z∈Zh0v(z)∫∂Vzpnds,bh(u,u,Ih*v)=∑z∈Zh0v(z)∫Vz(u⋅∇)udx.
We also define the trilinear forms b̃(·,·,·) and b¯(·,·,·) on X×X×Xh by(2.18)b̃(u,v,Ih*w)=((u⋅∇)v,Ih*w)+12((∇⋅u)v,Ih*w),b¯(u,v,w-Ih*w)=((u⋅∇)v,w-Ih*w)+12((∇⋅u)v,w-Ih*w).
To formulate the discrete problem so as to eliminate any such potential difficulties, we rewrite (2.16) as follows:(2.19)ah(u,Ih*v)-dh(Ih*v,p)+b̃(u,u,Ih*v)=(f,Ih*v),∀v∈Xh.
We multiply (2.12) by q∈Yh and sum over all K∈𝒯h: then, we obtain(2.20)b(u,q)=0,∀q∈Yh.
Now we rewrite (2.19) and (2.20) to a variational form similar to finite element problems. The locally stabilized FVE scheme is to find (uh,ph)∈Xh×Yh such that(2.21)ah(uh,Ih*vh)-dh(Ih*vh,ph)+b̃(uh,uh,Ih*vh)=(f,Ih*vh),∀vh∈Xh,d(uh,qh)-βch(ph,qh)=0,∀qh∈Yh,
where(2.22)ch(p,q)=∑K∈Λh∑e∈ΓKhe∫e[p]e[q]eds
is a stabilized form defined on (H1(Ω)+Yh)2, [·]e is the jump operator across the edge e, and β>0 is the local stabilization parameter. It is trivial that ch(p,qh)=ch(ph,q)=ch(p,q)=0, forallp,q∈H1(Ω), forallph,qh∈Yh.A general framework for analyzing the locally stabilized formulation (2.21) can be developed using the notion of equivalence class of macroelements. As in Stenberg [27] each equivalence class, denoted by ℰ𝒦̂, contains macroelements which are topologically equivalent to a reference macroelement 𝒦̂.Let(2.23)B¯h((uh,ph);(Ih*vh,qh))=ah(uh,Ih*vh)-dh(Ih*vh,ph)+d(uh,qh),(2.24)Bh((uh,ph);(Ih*vh,qh))=B¯h((uh,ph);(Ih*vh,qh))+βch(ph,qh).
We can rewrite (2.21) in a compact form: find (uh,ph)∈Xh×Yh such that(2.25)Bh((uh,ph);(Ih*vh,qh))+b̃(uh,uh,Ih*vh)=(f,Ih*vh),∀(vh,qh)∈Xh×Yh.


## 2.1. Notations

We will use∥·∥m and |·|m to denote the norm and seminorm of the Sobolev space (Hm(Ω))d, d=1,2. Let H01(Ω) be the standard Sobolev subspace of H1(Ω) of functions vanishing on ∂Ω. We introduce the following notations:(2.1)X=(H01(Ω))2,Y=L02(Ω)={q:q∈L2(Ω),∫Ωq=0}.
The scalar product and norm in Y are denoted by the usual L2(Ω) inner product (·,·) and ∥·∥0, respectively. As mentioned above, we need a further assumption on Ω.H1
Assume thatΩ is regular so that the unique solution (v,q)∈X×Y of the steady Stokes problem
(2.2)-Δv+∇q=g,∇⋅v=0;v|∂Ω=0
for a prescribed g∈(L2(Ω))2 exists and satisfies
(2.3)‖v‖2+‖q‖1≤C‖g‖0,
where C>0 is a constant depending on Ω.The weak formulation of the problem (1.1)–(1.3) is to find (u,p)∈X×Y such that(2.4)a(u,v)-d(v,p)+b(u,u,v)=(f,v),∀v∈X,d(u,q)=0,∀q∈Y,
where the bilinear forms a(·,·), d(·,·) and the trilinear form b(·,·,·) are given by(2.5)a(u,v)=ν(∇u,∇v)=ν∫Ω∇u:∇vdx,∀u,v∈X,d(v,q)=(∇⋅v,q)=∫Ωq∇⋅vdx,∀u,q∈X×Y,b(u,w,v)=((u⋅∇w),v)+12((∇⋅u)w,v)=12((u⋅∇w),v)-12((u⋅∇v),w)∀u,v,w∈X.
Introducing the generalized bilinear form on (X×Y)2 by(2.6)B((u,p);(v,q))=a(u,v)-d(v,p)+d(u,q),
we can rewrite (2.4) in a compact form: find (u,p)∈X×Y such that(2.7)B((u,p);(v,q))+b(u,u,v)=(f,v),∀(v,q)∈X×Y.Let𝒯h be a quasi-uniform triangulation of Ω with h=maxhK, where hK is the diameter of the triangle K∈𝒯h. We assume that the partition 𝒯h has been obtained from a macrotriangular partition Λh by joining the sides of each element of Λh. Every element K∈𝒯h must lie in exactly one macroelement 𝒦, which implies that macroelements do not overlap. For each 𝒦, the set of interelement edges which are strictly in the interior of 𝒦 will be denoted by Γ𝒦, and the length of an edge e∈Γ𝒦 is denoted by he.Based on this triangulation, letXh be the standard conforming finite element subspace of piecewise linear velocity,(2.8)Xh={v∈C(Ω)∩X:v∣Kislinear,∀K∈Th;v∣∂Ω=0},
and let Yh be the piecewise constant pressure subspace(2.9)Yh={q∈Y:q∣Kisconstant,∀K∈Th}.
It is well known that the standard P1-P0 element does not satisfy the inf-sup condition and cannot be applied to problem (1.1)–(1.3) directly. But a locally stabilized method based on the macroelement can be used to yield adequate approximations [6].In order to describe the FVE method for solving problem (1.1)–(1.3), we will introduce a dual partition 𝒯h* based upon the original partition 𝒯h whose elements are called control volumes. We construct the control volumes in the same way as in [5, 26]. Let zK be the barycenter of K∈𝒯h. We connect zK with line segments to the midpoints of the edges of K, thus partitioning K into three quadrilaterals Kz, z∈Zh(K), where Zh(K) are the vertices of K. Then with each vertex z∈Zh=∪K∈𝒯hZh(K), we associate a control volume Vz, which consists of the union of the subregions Kz, sharing the vertex z. Thus we finally obtain a group of control volumes covering the domain Ω, which is called the dual partition 𝒯h* of the triangulation 𝒯h. We denote by Zh0 the set of interior vertices.We call the partition𝒯h* regular or quasi-uniform if there exists a positive constant C>0 such that(2.10)C-1h2≤meas⁡(Vz)≤Ch2,Vz∈Th*.
If the finite element triangulation 𝒯h is quasi-uniform, then the dual partition 𝒯h* is also quasi-uniform [23].


## 2.2. Construction of the FVE Scheme

We formulate the FVE method for the problem (1.1)–(1.3) as follows: given a z∈Zh0 and K∈𝒯h, integrating (1.1) over the associated control volume Vz and (1.1) over the element K and applying Green’s formula, we obtain an integral conservation form(2.11)-ν∫∂Vz∇unds+∫∂Vzpnds+∫Vzu⋅∇udx=∫Vzfdx,∀z∈Zh0,(2.12)∫K∇⋅udx=0,∀K∈Th,
where n denotes the unit outer normal vector to ∂Vz (Figure 1).Left-hand side: a sample region with blue lines indicating the corresponding control volumeVz. Right-hand side: a triangle K partitioned into three subregions Kz.
(a)(b)LetIh*:Xh→Xh* be the transfer operator defined by(2.13)Ih*v=∑z∈Zh0v(z)χz,∀v∈Xh,
where(2.14)Xh*={v=(v1,v2)∈(L2(Ω))2:vi∣Vzisconstant,i=1,2∀z∈Zh0},
and χz is the characteristic function of the control volume Vz. The operator Ih* satisfies [26](2.15)‖Ih*v‖0≤C‖v‖0,∀v∈Xh.
Now for an arbitrary Ih*v, we multiply (2.11) by v(z) and sum over all z∈Zh0 to get(2.16)ah(u,Ih*v)-dh(Ih*v,p)+bh(u,u,Ih*v)=(f,Ih*v),∀v∈Xh.
Here ah:X×Xh→ℝ, dh:Xh×Y→ℝ and bh:X×X×Xh→ℝ are defined by(2.17)ah(u,Ih*v)=-ν∑z∈Zh0v(z)∫∂Vz∇unds,dh(Ih*v,p)=∑z∈Zh0v(z)∫∂Vzpnds,bh(u,u,Ih*v)=∑z∈Zh0v(z)∫Vz(u⋅∇)udx.
We also define the trilinear forms b̃(·,·,·) and b¯(·,·,·) on X×X×Xh by(2.18)b̃(u,v,Ih*w)=((u⋅∇)v,Ih*w)+12((∇⋅u)v,Ih*w),b¯(u,v,w-Ih*w)=((u⋅∇)v,w-Ih*w)+12((∇⋅u)v,w-Ih*w).
To formulate the discrete problem so as to eliminate any such potential difficulties, we rewrite (2.16) as follows:(2.19)ah(u,Ih*v)-dh(Ih*v,p)+b̃(u,u,Ih*v)=(f,Ih*v),∀v∈Xh.
We multiply (2.12) by q∈Yh and sum over all K∈𝒯h: then, we obtain(2.20)b(u,q)=0,∀q∈Yh.
Now we rewrite (2.19) and (2.20) to a variational form similar to finite element problems. The locally stabilized FVE scheme is to find (uh,ph)∈Xh×Yh such that(2.21)ah(uh,Ih*vh)-dh(Ih*vh,ph)+b̃(uh,uh,Ih*vh)=(f,Ih*vh),∀vh∈Xh,d(uh,qh)-βch(ph,qh)=0,∀qh∈Yh,
where(2.22)ch(p,q)=∑K∈Λh∑e∈ΓKhe∫e[p]e[q]eds
is a stabilized form defined on (H1(Ω)+Yh)2, [·]e is the jump operator across the edge e, and β>0 is the local stabilization parameter. It is trivial that ch(p,qh)=ch(ph,q)=ch(p,q)=0, forallp,q∈H1(Ω), forallph,qh∈Yh.A general framework for analyzing the locally stabilized formulation (2.21) can be developed using the notion of equivalence class of macroelements. As in Stenberg [27] each equivalence class, denoted by ℰ𝒦̂, contains macroelements which are topologically equivalent to a reference macroelement 𝒦̂.Let(2.23)B¯h((uh,ph);(Ih*vh,qh))=ah(uh,Ih*vh)-dh(Ih*vh,ph)+d(uh,qh),(2.24)Bh((uh,ph);(Ih*vh,qh))=B¯h((uh,ph);(Ih*vh,qh))+βch(ph,qh).
We can rewrite (2.21) in a compact form: find (uh,ph)∈Xh×Yh such that(2.25)Bh((uh,ph);(Ih*vh,qh))+b̃(uh,uh,Ih*vh)=(f,Ih*vh),∀(vh,qh)∈Xh×Yh.


## 3. Technical Preliminaries

This section considers preliminary estimates which will be very useful in the error estimates of two-level finite volume solution(uh,ph).The following lemma gives the boundedness of the trilinear formb(·,·,·).Lemma 3.1 (see [21]).
The following estimates hold:(3.1)b(u,w,v)=-b(u,v,w),|b(u,w,v)|≤12C0‖u‖01/2|u|11/2(|w|1‖v‖01/2|v|11/2+‖w‖01/2‖w‖11/2‖v‖11/2),∀u,v,w∈X,|b(u,v,w)|+|b(v,u,w)|+|b(w,u,v)|≤C1|u|1‖v‖2‖w‖0,∀u∈X,v∈(H2(Ω))2∩X,w∈Y,|b(u,v,w)|≤C|log⁡⁡h|1/2|u|1|v|1‖w‖0,∀u,v,w∈Xh.
Here and after Ci, i=1,2, and C are positive constants depending only on the data (ν,f,Ω).The existence and uniqueness results of (2.7) can be found in [28, 29].Theorem 3.2.
Assume thatν>0 and f∈Y satisfy the following uniqueness condition:
(3.2)1-N1ν2‖f‖-1>0,
where
(3.3)N1=sup⁡u,v,w∈Xb(u,v,w)|u|1|v|1|w|1.
Then the problem (2.7) admits a unique solution (u,p)∈(H01(Ω)2∩X,H1(Ω)∩Y) such that
(3.4)|u|1≤1ν‖f‖-1,‖u‖2+‖p‖1≤C‖f‖0.In [30] the following lemma was proved, which shows that the finite volume element bilinear forms ah(·,Ih*·) and dh(Ih*·,·) are equal to the finite element ones, respectively.Lemma 3.3.
For anyuh,vh∈Xh, and qh∈Yh, one has
(3.5)ah(uh,Ih*vh)=a(uh,vh),dh(Ih*vh,qh)=d(vh,qh).The following theorem establishes the weak coercivity of (2.24) [6, 31].Theorem 3.4.
Given a a stabilization parameterβ≥β0, suppose that every macroelement 𝒦∈Λh belongs to one of the equivalence classes ℰ𝒦̂ and that the following macroelement connectivity condition is valid: for any two neighboring macroelements 𝒦1 and 𝒦2 with ∫𝒦1∩𝒦2≠0, there exists v∈Xh such that
(3.6)supp⁡v⊂K1∩K2∫K1∩K2v⋅nds≠0.
Then
(3.7)α1(|uh|1+‖ph‖0)≤sup⁡(vh,qh)∈Xh×YhBh((u,p);(Ih*vh,qh))|vh|1+‖qh‖0,
for all (uh,ph)∈Xh×Yh, and
(3.8)|Bh((u,p);(Ih*vh,qh))|≤α2(|uh|1+‖ph‖0)(|vh|1+‖qh‖0)(vh,qh)∈Xh×Yh,
where α1,α2>0 are constants independent of h and β, β0 is some fixed positive constant, and n is the out-normal vector.Next, we establish the existence and the uniqueness of FVE scheme (2.25), by the fixed-point theorem, in the following.Theorem 3.5 (see [6]).
Suppose the assumptions of Theorems3.2 and 3.4 hold, and the body force f satisfies the following uniqueness condition
(3.9)1-4Nν2‖f‖-1>0.
Then the variation problem (2.25) admits a unique solution (uh,ph)∈(Xh×Yh) such that
(3.10)|uh|1≤1ν‖f‖-1,‖ph‖0≤α2α1‖f‖-1+4α2Nα1ν2‖f‖-1,
where
(3.11)N=max⁡⁡{CN1,N2},N2=sup⁡u,v,w∈Xb̃(u,v,Ih*w)|u|1|v|1|w|1.For the error estimate, we introduce the Galerkin projection(Rh,Qh):X×Y→Xh×Yh defined by(3.12)Bh((Rh(v,q),Qh(v,q));(Ih*vh,qh))=B¯h((v,q);(Ih*vh,qh))
for each (v,q)∈X×Y and all (vh,qh)∈Xh×Yh. We obtain the following results by using the standard Galerkin finite element [6, 17].Theorem 3.6.
Under the assumptions of Theorems3.2 and 3.4, the projection (Rh,Qh) satisfies
(3.13)|v-Rh(v,q)|1+‖q-Qh(v,q)‖0≤C(|v|1+‖q‖0),
for all (v,q)∈X×Y and
(3.14)‖v-Rh(v,q)‖0+h(|v-Rh(v,q)|1+‖q-Qh(v,q)‖0)≤Ch2(‖v‖2+‖q‖1),
for all (v,q)∈(D(A)∩X)×(H1(Ω)∩Y).Then the optimal error estimates can be obtained as follows.Theorem 3.7 (see [6, 32]).
Under the assumptions of Theorems3.2, 3.4, 3.5, and 3.6, the solution (uh,ph) of (2.25) satisfies
(3.15)‖u-uh‖0+h(|u-uh|1+‖p-ph‖0)≤Ch2.


## 4. Two-Level FVE Algorithms and Its Error Analysis

In this section, we will present two-level stabilized finite volume element algorithm for (1.1)–(1.3) and derive some optimal bounds for errors. The idea of the two-level method is to reduce the nonlinear problem on a fine mesh into a linear system on a fine mesh by solving a nonlinear problem on a coarse mesh. The basic mechanisms are two quasi-uniform triangulations of Ω, 𝒯H, and 𝒯h, with two different mesh sizes H and h(h≪H), and the corresponding solutions spaces (XH,YH) and (Xh,Yh), which satisfy (XH,YH)⊂(Xh,Yh) and will be called the coarse and the fine spaces, respectively. Now find (uh,ph) as follows.Algorithm 4.1 (Simple two-level stabilized FVE approximation).
We have the following steps:Step 1.
On the coarse mesh𝒯H, solve the stabilized Navier-Stokes problem.
Find(uH,pH)∈XH×YH such that, forall(vH,qH)∈XH×YH,
(4.1)BH((uH,pH);(IH*vH,qH))+b̃(uH,uH,IH*vH)=(f,IH*vH).Step 2.
On the fine mesh𝒯h, solve the stabilized linear Stokes problem.
Find(uh,ph)∈Xh×Yh such that, forall(vh,qh)∈Xh×Yh,
(4.2)Bh((uh,ph);(Ih*vh,qh))+b̃(uH,uH,Ih*vh)=(f,Ih*vh).Next, we study the convergence of(uh,ph) to (u,p) in some norms. For convenience, we set e=Rh(u,p)-uh and η=Qh(u,p)-ph.Theorem 4.2.
Under the assumptions of Theorems3.2, 3.4, 3.5, and 3.6 for H  and h, the simple two-level stabilized FVE solution (uh,ph) satisfies the following error estimates:
(4.3)|u-uh|1+‖p-ph‖0≤C(h+H2).Proof.
Subtracting (4.2) from (2.25) and using the Galerkin projection (3.12), it is easy to see that
(4.4)Bh((e,η);(Ih*vh,qh))+b(u,u-uH,vh)+b(u-uH,u,vh)-b(u-uH,u-uH,vh)+b̅(u-uH,u-uH,vh-Ih*vh)-b̅(u-uH,u,vh-Ih*vh)-b̅(u,u-uH,vh-Ih*vh)+b̅(u,u,vh-Ih*vh)=(f,vh-Ih*vh),
for all (vh,qh)∈Xh×Yh. Due to (3.7), Lemma 3.1, (3.15), and (4.4), we have
(4.5)α1(|e|1+‖η‖0)≤sup⁡(vh,qh)∈Xh×YhBh((u,p);(Ih*vh,qh))|vh|1+‖qh‖0≤C‖u‖2‖u-uH‖0+C|u-uH|12+Ch‖u‖2|u-uH|1+Ch≤C(H2+h),
which, along with (3.14), yields
(4.6)|u-uh|1+‖p-ph‖0≤|u-Rh(u,p)|1+‖p-Qh(u,p)‖0+|e|1+‖η‖0≤C(h+H2).Algorithm 4.3 (The Newton two-level stabilized FVE approximation).
We have the following steps:Step 1.
On the coarse mesh𝒯H, solve the stabilized Navier-Stokes problem.
Find(uH,pH)∈XH×YH by (4.1).Step 2.
On the fine mesh𝒯h, solve the stabilized linear Stokes problem.
Find(uh,ph)∈Xh×Yh such that, forall(vh,qh)∈Xh×Yh,
(4.7)Bh((uh,ph);(Ih*vh,qh))+bh(uh,uH,Ih*vh)+bh(uH,uh,Ih*vh)=(f,Ih*vh)+bh(uH,uH,Ih*vh).Now, we will study the convergence of the Newton two-level stabilized finite element solution(uh,ph) to (u,p) in some norms. To do this, let us set e=Rh(u,p)-uh, E=u-Rh(u,p), and η=Qh(u,p)-ph.Theorem 4.4.
Under the assumptions of Theorems3.2, 3.4, 3.5, and 3.6 for H and h, the Newton two-level stabilized FVE solution (uh,ph) satisfies the following error estimates:
(4.8)|u-uh|1+‖p-ph‖0≤C(h+|log⁡⁡h|1/2H3).Proof.
Subtracting (4.7) from (2.25), using the Galerkin projection (3.12) and taking (vh,qh)=(e,η), we get
(4.9)Bh((e,η);(Ih*e,η))+b(E,u,e)+b(Rh-uH,u-uH,e)+b(uH,E,e)-b̅(Rh-uH,Rh-uH,e-Ih*e)-b̅(Rh,Rh-u,e-Ih*e)-b̅(Rh,u,e-Ih*e)-b(e,uH,e)+b̅(e,uH,e-Ih*e)+b̅(uH,e,e-Ih*e)=(f,e-Ih*e).
Using Lemma 3.1 and Theorems 3.2, 3.5, and 3.7, we obtain
(4.10)|b(E,u,e)+b(uH,E,e)+(f,e-Ih*e)|≤C(|u|1+|uH|1)|E|1|e|1+‖f‖0‖e-Ih*e‖0≤Ch|e|1,(4.11)|b(Rh-uH,u-uH,e)|=|b(Rh-uH,e,u-uH)|≤C|log⁡⁡h|1/2|Rh-uH|1|e|1|u-uH|0≤C|log⁡⁡h|1/2|Rh-u|1+|u-uH|1|e|1|u-uH|0≤C|log⁡⁡h|1/2H3|e|1.(4.12)|b̅(Rh-uH,Rh-uH,e-Ih*e)|≤C|log⁡⁡h|1/2|Rh-uH|1|Rh-uH|1‖e-Ih*e‖0≤C|log⁡⁡h|1/2H3|e|1.(4.13)|b̅(Rh,Rh-u,e-Ih*e)|≤N|Rh|1|Rh-u|1|e|1≤Ch|e|1,(4.14)|b̅(Rh-u,e-Ih*e)|≤C|Rh|1‖u‖2‖e-Ih*e‖0≤Ch|e|1≤Ch|e|1,(4.15)ν|e|12-|b(e,uH,e)|-|b¯(e,uH,e-Ih*e)|-|b¯(uH,e,e-Ih*e)|≥ν|e|12-3N|uH|1|e|12≥ν(1-3Nν2‖f‖-1)|e|12.
Combining (4.10)–(4.15) with (4.9) yields
(4.16)|e|1≤C(h+|log⁡⁡h|1/2H3).
Thanks to (3.7), (4.9), Theorems 3.2 and 3.5, and estimates (4.10)–(4.14) and (4.16), we have
(4.17)‖η‖0≤(α1)-1sup⁡(vh,qh)∈Xh×YhBh((e,η);(Ih*vh,qh))|vh|1+‖qh‖0≤C(h+log⁡|h|1/2H3+|e|1)≤C(h+log⁡|h|1/2H3).
Combining (4.16) and (4.17) with (3.14) and Theorem 3.2 yields (2.13).



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*Source: 309871-2012-05-07.xml*