two-level stabilized finite volume methods for stationary navier … · 2019. 7. 31. · two-level...

Hindawi Publishing CorporationAdvances in Numerical AnalysisVolume 2012, Article ID 309871, 14 pagesdoi:10.1155/2012/309871

Research ArticleTwo-Level Stabilized Finite Volume Methods forStationary Navier-Stokes Equations

Anas Rachid,1 Mohamed Bahaj,2 and Noureddine Ayoub2

1 Ecole Nationale Superieure des Arts et Metiers-Casablanca, Universite Hassan II, B.P. 150,Mohammedia, Morocco

2 Department of Mathematics and Computing Sciences, Faculty of Sciences and Technology,University Hassan 1st, B.P. 577, Settat, Morocco

Correspondence should be addressed to Anas Rachid, [email protected]

Received 17 December 2011; Accepted 17 February 2012

Academic Editor: Weimin Han

Copyright q 2012 Anas Rachid et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We propose two algorithms of two-level methods for resolving the nonlinearity in the stabilizedfinite volume approximation of the Navier-Stokes equations describing the equilibrium flow of aviscous, incompressible fluid. A macroelement condition is introduced for constructing the localstabilized finite volume element formulation. Moreover the two-level methods consist of solvinga 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 anapproximate solution with the convergence rate of the same order as the usual stabilized finitevolume element solution solving the Navier-Stokes equations on a fine mesh for a related choiceof mesh widths.

1. Introduction

We consider a two-level method for the resolution of the nonlinear system arising from finitevolume discretizations of the equilibrium, incompressible Navier-Stokes equations:

−νΔu + (u · ∇)u +∇p = f in Ω, (1.1)

∇ · u = 0 in Ω, (1.2)

u = 0 in ∂Ω, (1.3)

where u = (u1(x), u2(x)) is the velocity vector, p = p(x) is the pressure, f = f(x) is thebody force, ν > 0 is the viscosity of the fluid, and Ω ⊂ R

2, the flow domain, is assumed to

2 Advances in Numerical Analysis

be bounded, to have a Lipschitz-continuous boundary ∂Ω, and to satisfy a further conditionstated in (H1).

Finite volume method is an important numerical tool for solving partial differentialequations. 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 thefinite difference framework or in the Petrov-Galerkin framework. Usually, the former oneis called finite volume method [1, 2], MAC (marker and cell) method [3], or cell-centeredmethod [4], and the latter one is called finite volume element method (FVE) [5–7], covolumemethod [8], or vertex-centeredmethod [9, 10]. We refer to themonographs [11, 12] for generalpresentations of thesemethods. Themost important property of FVE is that it can preserve theconservation laws (mass, momentum, and heat flux) on each control volume. This importantproperty, combined with adequate accuracy and ease of implementation, has attracted morepeople to do research in this field.

On the other hand, the two-level finite element strategy based on two finite elementspaces on one coarse and one fine mesh has been widely studied for steady semilinear ellipticequations [13, 14] and the Navier-Stokes equations [15–22]. For the finite volume elementmethod, Bi and Ginting [23] have studied two-grid finite volume element method for linearand nonlinear elliptic problems; Chen et al. [24] have applied two-grid methods for solvinga two-dimensional nonlinear parabolic equation using finite volume element method. Chenand Liu [25] have also studied this method for semilinear parabolic problems. However, tothe best of our knowledge, there is no two-level finite volume convergence analysis for theNavier-Stokes equations in the literature.

In this paper we aim to combine FVE method based on P1 − P0 macroelement withtwo-level strategy to solve the two-dimensional Navier-Stokes (1.1)–(1.3). The heart ofthe analysis is the use of a transfer operator to connect finite volume and finite elementestimations which will lead to more difficult term to estimate. We choose the two-grid spacesas two conforming finite element spaces XH and Xh on one coarse grid with mesh size Hand one fine grid with mesh size h � H. We propose two algorithms of two-level method forresolving 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 stabilizedfinite volume solution (uh, ph) is the following error estimate:

∥∥∥u − uh

∥∥∥1+∥∥∥p − ph

∥∥∥0≤ C

(

h +H2)

. (1.4)

Second we prove that the Newton two-level stabilized finite volume solution (uh, ph) is thefollowing error estimate:

∥∥∥u − uh

∥∥∥1+∥∥∥p − ph

∥∥∥0≤ C

(

h +H3∣∣logh∣∣1/2

)

, (1.5)

where C denotes some generic constant which may stand for different values at its differentoccurrences.

Hence, the two-level algorithms achieve asymptotically optimal approximation aslong as the mesh sizes satisfy h = O(H2) for the simple two-level stabilized finite volumesolution 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 difficultthan solving one single linearized equation.

Advances in Numerical Analysis 3

The rest of this paper is organized as follows. In the next section, we introduce somenotations and construct a FVE scheme. In Section 3 we recall same preliminary estimates ofthe stabilized finite volume approximations. Finally the two-level FVE algorithms and theimproved error estimates are presented and established in Section 4.

2. Finite Volume Scheme

2.1. Notations

Wewill use ‖ · ‖m and | · |m to denote the norm and seminorm of the Sobolev space (Hm(Ω))d,d = 1, 2. LetH1

0(Ω) be the standard Sobolev subspace ofH1(Ω) of functions vanishing on ∂Ω.We introduce the following notations:

X =(

H10(Ω)

)2, Y = L2

0(Ω) ={

q : q ∈ L2(Ω),∫

Ωq = 0

}

. (2.1)

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 that Ω is regular so that the unique solution (v, q) ∈ X × Y of the steady Stokesproblem

−Δv +∇q = g, ∇ · v = 0; v|∂Ω = 0 (2.2)

for a prescribed g ∈ (L2(Ω))2 exists and satisfies

‖v‖2 +∥∥q

∥∥1 ≤ C

∥∥g

∥∥0, (2.3)

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

a(u, v) − d(

v, p)

+ b(u, u, v) =(

f, v)

, ∀v ∈ X,

d(

u, q)

= 0, ∀q ∈ Y,(2.4)


where the bilinear forms a(·, ·), d(·, ·) and the trilinear form b(·, ·, ·) are given by

a(u, v) = ν(∇u,∇v) = ν

∫

Ω∇u : ∇v dx, ∀u, v ∈ X,

d(

v, q)

=(

∇ · v, q)

=∫

Ωq∇ · v dx, ∀u, q ∈ X × Y,

b(u,w, v) = ((u · ∇w), v) +12((∇ · u)w,v)

=12((u · ∇w), v) − 1

2((u · ∇v), w) ∀u, v,w ∈ X.

(2.5)

Introducing the generalized bilinear form on (X × Y )2 by

B((

u, p)

;(

v, q))

= a(u, v) − d(

v, p)

+ d(

u, q)

, (2.6)

we can rewrite (2.4) in a compact form: find (u, p) ∈ X × Y such that

B((

u, p)

;(

v, q))

+ b(u, u, v) =(

f, v)

, ∀(

v, q)

∈ X × Y. (2.7)

Let Th be a quasi-uniform triangulation of Ω with h = maxhK, where hK is thediameter of the triangle K ∈ Th. We assume that the partition Th has been obtained froma macrotriangular partition Λh by joining the sides of each element of Λh. Every elementK ∈ Th must lie in exactly one macroelement K, which implies that macroelements do notoverlap. For each K, the set of interelement edges which are strictly in the interior of K willbe 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 subspaceof piecewise linear velocity,

Xh = {v ∈ C(Ω) ∩X : v|K is linear, ∀K ∈ Th;v|∂Ω = 0}, (2.8)

and let Yh be the piecewise constant pressure subspace

Yh ={

q ∈ Y : q|K is constant, ∀K ∈ Th

}

. (2.9)

It is well known that the standard P1 − P0 element does not satisfy the inf-sup condition andcannot be applied to problem (1.1)–(1.3) directly. But a locally stabilized method based onthe macroelement can be used to yield adequate approximations [6].

In order to describe the FVEmethod for solving problem (1.1)–(1.3), we will introducea dual partition T∗

h based upon the original partition Th whose elements are called controlvolumes. We construct the control volumes in the same way as in [5, 26]. Let zK be thebarycenter of K ∈ Th. We connect zK with line segments to the midpoints of the edgesof K, thus partitioning K into three quadrilaterals Kz, z ∈ Zh(K), where Zh(K) are thevertices ofK. Then with each vertex z ∈ Zh = ∪K∈ThZh(K), we associate a control volume Vz,which consists of the union of the subregionsKz, sharing the vertex z. Thus we finally obtain


z

Vz

(a)

K

z

Kz

zK

(b)

Figure 1: Left-hand side: a sample region with blue lines indicating the corresponding control volume Vz.Right-hand side: a triangle K partitioned into three subregionsKz.

a group of control volumes covering the domain Ω, which is called the dual partition T∗hof

the triangulation Th. We denote by Z0h the set of interior vertices.

We call the partition T∗hregular or quasi-uniform if there exists a positive constant

C > 0 such that

C−1h2 ≤ meas(Vz) ≤ Ch2, Vz ∈ T∗h. (2.10)

If the finite element triangulationTh is quasi-uniform, then the dual partitionT∗his 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 ∈ Z0h and

K ∈ Th, integrating (1.1) over the associated control volume Vz and (1.1) over the elementKand applying Green’s formula, we obtain an integral conservation form

−ν∫

∂Vz

∇unds +∫

∂Vz

pn ds +∫

Vz

u · ∇udx =∫

Vz

f dx, ∀z ∈ Z0h, (2.11)

∫

K

∇ · udx = 0, ∀K ∈ Th, (2.12)

where n denotes the unit outer normal vector to ∂Vz (Figure 1).Let I∗h : Xh → X∗

h be the transfer operator defined by

I∗hv =∑

z∈Z0h

v(z)χz, ∀v ∈ Xh, (2.13)


where

X∗h =

{

v = (v1, v2) ∈(

L2(Ω))2

: vi|Vzis constant, i = 1, 2 ∀z ∈ Z0

h

}

, (2.14)

and χz is the characteristic function of the control volume Vz. The operator I∗h satisfies [26]

∥∥I∗hv

∥∥0 ≤ C‖v‖0, ∀v ∈ Xh. (2.15)

Now for an arbitrary I∗hv, we multiply (2.11) by v(z) and sum over all z ∈ Z0

hto get

ah

(

u, I∗hv)

− dh

(

I∗hv, p)

+ bh(

u, u, I∗hv)

=(

f, I∗hv)

, ∀v ∈ Xh. (2.16)

Here ah : X ×Xh → R, dh : Xh × Y → R and bh : X ×X ×Xh → R are defined by

ah

(

u, I∗hv)

= −ν∑

z∈Z0h

v(z)∫

∂Vz

∇unds,

dh

(

I∗hv, p)

=∑

z∈Z0h

v(z)∫

∂Vz

pn ds,

bh(

u, u, I∗hv)

=∑

z∈Z0h

v(z)∫

Vz

(u · ∇)udx.

(2.17)

We also define the trilinear forms b(·, ·, ·) and b(·, ·, ·) on X ×X ×Xh by

b(

u, v, I∗hw)

=(

(u · ∇)v, I∗hw)

+12(

(∇ · u)v, I∗hw)

,

b(

u, v,w − I∗hw)

=(

(u · ∇)v,w − I∗hw)

+12(

(∇ · u)v,w − I∗hw)

.

(2.18)

To formulate the discrete problem so as to eliminate any such potential difficulties, we rewrite(2.16) as follows:

ah

(

u, I∗hv)

− dh

(

I∗hv, p)

+ b(

u, u, I∗hv)

=(

f, I∗hv)

, ∀v ∈ Xh. (2.19)

We multiply (2.12) by q ∈ Yh and sum over all K ∈ Th: then, we obtain

b(

u, q)

= 0, ∀q ∈ Yh. (2.20)


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

ah

(

uh, I∗hvh

)

− dh

(

I∗hvh, ph)

+ b(

uh, uh, I∗hvh

)

=(

f, I∗hvh

)

, ∀vh ∈ Xh,

d(

uh, qh)

− βch(

ph, qh)

= 0, ∀qh ∈ Yh,(2.21)

where

ch(

p, q)

=∑

K∈Λh

∑

e∈ΓKhe

∫

e

[

p]

e

[

q]

eds (2.22)

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,for all p, q ∈ H1(Ω), for all ph, qh ∈ Yh.

A general framework for analyzing the locally stabilized formulation (2.21) can bedeveloped using the notion of equivalence class of macroelements. As in Stenberg [27] eachequivalence class, denoted by EK, containsmacroelements which are topologically equivalentto a reference macroelement K.

Let

Bh

((

uh, ph)

;(

I∗hvh, qh

))

= ah

(

uh, I∗hvh

)

− dh

(

I∗hvh, ph

)

+ d(

uh, qh)

, (2.23)

Bh

((

uh, ph)

;(

I∗hvh, qh

))

= Bh

((

uh, ph)

;(

I∗hvh, qh

))

+ βch(

ph, qh)

. (2.24)

We can rewrite (2.21) in a compact form: find (uh, ph) ∈ Xh × Yh such that

Bh

((

uh, ph)

;(

I∗hvh, qh))

+ b(

uh, uh, I∗hvh

)

=(

f, I∗hvh

)

, ∀(

vh, qh)

∈ Xh × Yh. (2.25)

3. Technical Preliminaries

This section considers preliminary estimates which will be very useful in the error estimatesof two-level finite volume solution (uh, ph).

The following lemma gives the boundedness of the trilinear form b(·, ·, ·).


Lemma 3.1 (see [21]). The following estimates hold:

b(u,w, v) = −b(u, v,w),

|b(u,w, v)| ≤ 12C0‖u‖1/20 |u|1/21

(

|w|1‖v‖1/20 |v|1/21 + ‖w‖1/20 ‖w‖1/21 ‖v‖1/21

)

, ∀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∣∣logh

∣∣1/2|u|1|v|1‖w‖0, ∀u, v,w ∈ Xh.

(3.1)

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 that ν > 0 and f ∈ Y satisfy the following uniqueness condition:

1 − N1

ν2∥∥f

∥∥−1 > 0, (3.2)

where

N1 = supu,v,w∈X

b(u, v,w)|u|1|v|1|w|1

. (3.3)

Then the problem (2.7) admits a unique solution (u, p) ∈ (H10(Ω)2 ∩X,H1(Ω) ∩ Y ) such that

|u|1 ≤1ν

∥∥f

∥∥−1, ‖u‖2 +

∥∥p

∥∥1 ≤ C

∥∥f

∥∥0. (3.4)

In [30] the following lemma was proved, which shows that the finite volume elementbilinear forms ah(·, I∗h·) and dh(I∗h·, ·) are equal to the finite element ones, respectively.

Lemma 3.3. For any uh, vh ∈ Xh, and qh ∈ Yh, one has

ah

(

uh, I∗hvh

)

= a(uh, vh),

dh

(

I∗hvh, qh)

= d(

vh, qh)

.(3.5)

The following theorem establishes the weak coercivity of (2.24) [6, 31].

Theorem 3.4. Given a a stabilization parameter β ≥ β0, suppose that every macroelement K ∈Λh belongs to one of the equivalence classes EK 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

supp v ⊂ K1 ∩K2

∫

K1∩K2

v · nds/= 0. (3.6)

Then

α1(

|uh|1 +∥∥ph

∥∥0

)

≤ sup(vh,qh)∈Xh×Yh

Bh

((

u, p)

;(

I∗hvh, qh

))

|vh|1 +∥∥qh

∥∥0

, (3.7)

for all (uh, ph) ∈ Xh × Yh, and

∣∣Bh

((

u, p)

;(

I∗hvh, qh))∣∣ ≤ α2

(

|uh|1 +∥∥ph

∥∥0

)(

|vh|1 +∥∥qh

∥∥0

) (

vh, qh)

∈ Xh × Yh, (3.8)

where α1, α2 > 0 are constants independent of h and β, β0 is some fixed positive constant, and n is theout-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 Theorems 3.2 and 3.4 hold, and the body force fsatisfies the following uniqueness condition

1 − 4Nν2

∥∥f

∥∥−1 > 0. (3.9)

Then the variation problem (2.25) admits a unique solution (uh, ph) ∈ (Xh × Yh) such that

|uh|1 ≤1ν

∥∥f

∥∥−1,

∥∥ph

∥∥0 ≤

α2

α1

∥∥f

∥∥−1 +

4α2N

α1ν2∥∥f

∥∥−1, (3.10)

where

N = max{CN1,N2}, N2 = supu,v,w∈X

b(

u, v, I∗hw)

|u|1|v|1|w|1. (3.11)

For the error estimate, we introduce the Galerkin projection (Rh,Qh) : X×Y → Xh×Yh

defined by

Bh

((

Rh

(

v, q)

, Qh

(

v, q))

;(

I∗hvh, qh))

= Bh

((

v, q)

;(

I∗hvh, qh))

(3.12)

for each (v, q) ∈ X × Y and all (vh, qh) ∈ Xh × Yh. We obtain the following results by using thestandard Galerkin finite element [6, 17].


Theorem 3.6. Under the assumptions of Theorems 3.2 and 3.4, the projection (Rh,Qh) satisfies

∣∣v − Rh

(

v, q)∣∣1 +

∥∥q −Qh

(

v, q)∥∥0 ≤ C

(

|v|1 +∥∥q

∥∥0

)

, (3.13)

for all (v, q) ∈ X × Y and

∥∥v − Rh

(

v, q)∥∥0 + h

(∣∣v − Rh

(

v, q)∣∣1 +

∥∥q −Qh

(

v, q)∥∥0

)

≤ Ch2(‖v‖2 +∥∥q

∥∥1

)

, (3.14)

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 Theorems 3.2, 3.4, 3.5, and 3.6, the solution(uh, ph) of (2.25) satisfies

‖u − uh‖0 + h(

|u − uh|1 +∥∥p − ph

∥∥0

)

≤ Ch2. (3.15)

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 methodis to reduce the nonlinear problem on a fine mesh into a linear system on a fine mesh bysolving a nonlinear problem on a coarse mesh. The basic mechanisms are two quasi-uniformtriangulations of Ω, TH , and Th, with two different mesh sizes H and h(h � H), and thecorresponding 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 mesh TH , solve the stabilized Navier-Stokes problem.Find (uH, pH) ∈ XH × YH such that, for all (vH, qH) ∈ XH × YH ,

BH

((

uH, pH)

;(

I∗HvH, qH))

+ b(

uH, uH, I∗HvH

)

=(

f, I∗HvH

)

. (4.1)

Step 2. On the fine mesh Th, solve the stabilized linear Stokes problem.Find (uh, ph) ∈ Xh × Yh such that, for all (vh, qh) ∈ Xh × Yh,

Bh

((

uh, ph)

;(

I∗hvh, qh))

+ b(

uH, uH, I∗hvh

)

=(

f, I∗hvh

)

. (4.2)

Next, we study the convergence of (uh, ph) to (u, p) in some norms. For convenience, we sete = Rh(u, p) − uh and η = Qh(u, p) − ph.

Theorem 4.2. Under the assumptions of Theorems 3.2, 3.4, 3.5, and 3.6 for H and h, the simpletwo-level stabilized FVE solution (uh, ph) satisfies the following error estimates:

∣∣∣u − uh

∣∣∣1+∥∥∥p − ph

∥∥∥0≤ C

(

h +H2)

. (4.3)


Proof. Subtracting (4.2) from (2.25) and using the Galerkin projection (3.12), it is easy to seethat

Bh

((

e, η)

;(

I∗hvh, qh))

+ b(u, u − uH, vh) + b(u − uH, u, vh) − b(u − uH, u − uH, vh)

+ b(

u − uH, u − uH, vh − I∗hvh

)

− b(

u − uH, u, vh − I∗hvh

)

− b(

u, u − uH, vh − I∗hvh

)

+ b(

u, u, vh − I∗hvh

)

=(

f, vh − I∗hvh

)

,

(4.4)

for all (vh, qh) ∈ Xh × Yh. Due to (3.7), Lemma 3.1, (3.15), and (4.4), we have

α1(

|e|1 +∥∥η

∥∥0

)

≤ sup(vh,qh)∈Xh×Yh

Bh

((

u, p)

;(

I∗hvh, qh

))

|vh|1 +∥∥qh

∥∥0

≤ C‖u‖2‖u − uH‖0 + C|u − uH |21 + Ch‖u‖2|u − uH |1 + Ch

≤ C(

H2 + h)

,

(4.5)

which, along with (3.14), yields

∣∣∣u − uh

∣∣∣1+∥∥∥p − ph

∥∥∥0≤∣∣u − Rh

(

u, p)∣∣1 +

∥∥p −Qh

(

u, p)∥∥0 + |e|1 +

∥∥η

∥∥0

≤ C(

h +H2)

.

(4.6)

Algorithm 4.3 (The Newton two-level stabilized FVE approximation). We have the followingsteps:

Step 1. On the coarse mesh TH , solve the stabilized Navier-Stokes problem.Find (uH, pH) ∈ XH × YH by (4.1).

Step 2. On the fine mesh Th, solve the stabilized linear Stokes problem.Find (uh, ph) ∈ Xh × Yh such that, for all (vh, qh) ∈ Xh × Yh,

Bh

((

uh, ph)

;(

I∗hvh, qh))

+ bh(

uh, uH, I∗hvh

)

+ bh(

uH, uh, I∗hvh

)

=(

f, I∗hvh

)

+ bh(

uH, uH, I∗hvh

)

.

(4.7)

Now, we will study the convergence of the Newton two-level stabilized finite elementsolution (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 Theorems 3.2, 3.4, 3.5, and 3.6 for H and h, the Newtontwo-level stabilized FVE solution (uh, ph) satisfies the following error estimates:

∣∣∣u − uh

∣∣∣1+∥∥∥p − ph

∥∥∥0≤ C

(

h +∣∣logh

∣∣1/2

H3)

. (4.8)


Proof. Subtracting (4.7) from (2.25), using the Galerkin projection (3.12) and taking (vh, qh) =(e, η), we get

Bh

((

e, η)

;(

I∗he, η))

+ b(E, u, e) + b(Rh − uH, u − uH, e) + b(uH, E, e)

− b(

Rh − uH,Rh − uH, e − I∗he)

− b(

Rh, Rh − u, e − I∗he)

− b(

Rh, u, e − I∗he)

− b(e, uH, e) + b(

e, uH, e − I∗he)

+ b(

uH, e, e − I∗he)

=(

f, e − I∗he)

.

(4.9)

Using Lemma 3.1 and Theorems 3.2, 3.5, and 3.7, we obtain

∣∣b(E, u, e) + b(uH, E, e) +

(

f, e − I∗he)∣∣ ≤ C(|u|1 + |uH |1)|E|1|e|1 +

∥∥f

∥∥0

∥∥e − I∗he

∥∥0

≤ Ch|e|1,(4.10)

|b(Rh − uH, u − uH, e)| = |b(Rh − uH, e, u − uH)|

≤ C∣∣logh

∣∣1/2|Rh − uH |1|e|1|u − uH |0

≤ C∣∣logh

∣∣1/2|Rh − u|1 + |u − uH |1|e|1|u − uH |0

≤ C∣∣logh

∣∣1/2

H3|e|1.

(4.11)

∣∣∣b(

Rh − uH,Rh − uH, e − I∗he)∣∣∣ ≤ C

∣∣logh

∣∣1/2|Rh − uH |1|Rh − uH |1

∥∥e − I∗he

∥∥0

≤ C∣∣logh

∣∣1/2

H3|e|1.(4.12)

∣∣∣b(

Rh, Rh − u, e − I∗he)∣∣∣ ≤ N|Rh|1|Rh − u|1|e|1 ≤ Ch|e|1, (4.13)

∣∣∣b(

Rh − u, e − I∗he)∣∣∣ ≤ C|Rh|1‖u‖2

∥∥e − I∗he

∥∥0 ≤ Ch|e|1 ≤ Ch|e|1, (4.14)

ν|e|21 − |b(e, uH, e)| −∣∣∣b(

e, uH, e − I∗he)∣∣∣ −

∣∣∣b(

uH, e, e − I∗he)∣∣∣

≥ ν|e|21 − 3N|uH |1|e|21

≥ ν

(

1 − 3Nν2

∥∥f

∥∥−1

)

|e|21.

(4.15)

Combining (4.10)–(4.15)with (4.9) yields

|e|1 ≤ C(

h +∣∣logh

∣∣1/2

H3)

. (4.16)


Thanks to (3.7), (4.9), Theorems 3.2 and 3.5, and estimates (4.10)–(4.14) and (4.16), we have

∥∥η

∥∥0 ≤ (α1)−1 sup

(vh,qh)∈Xh×Yh

Bh

((

e, η)

;(

I∗hvh, qh))

|vh|1 +∥∥qh

∥∥0

≤ C(

h + log |h|1/2H3 + |e|1)

≤ C(

h + log |h|1/2H3)

.

(4.17)

Combining (4.16) and (4.17)with (3.14) and Theorem 3.2 yields (2.13).

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