Equations by First-Order System of Least-Squares Finite Element Method

Equations by First-Order System of Least-Squares Finite Element Method

Sang Dong Kim and EunJung Lee Department of Mathematics,

Kyungpook National University, Taegu, Korea 702-701

Received 15 November 1999; accepted 13 April 2001

This article applies the first-order system least-squares (fosls) finite element method developed by Cai, Manteuffel and McCormick to the compressible Stokes equations. By introducing a new dependent velocity flux variable, we recast the compressible Stokes equations as a first-order system. Then it is shown that the ellipticity and continuity hold for the least-squares functionals employing the mixture ofH⁻¹andL², so that thefoslsfinite element methods yield best approximations for the velocity flux and velocity. c 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 689–699, 2001

Keywords: Compressible Stokes equations; first-order system least-squares

I. INTRODUCTION

LetΩbe a bounded, open domain in<ⁿ(n = 2,3)with Lipschitz boundary∂Ω. Consider the system of equations of the stationary compressible Stokes equations withzeroboundary conditions (cf. [1]) for thevelocityu= (u₁,· · ·, u_n)^tandpressurepas follows:





−µ∆u+ρ(β· ∇)u+∇p = f, in Ω,

∇ ·u+κβ· ∇p = g, in Ω,

u = 0, on ∂Ω, (1.1)

where the symbols ∆,∇, and∇·stand for the Laplacian, gradient, and divergence operators, respectively (∆uis the vector of components∆ui);β = (β1,· · · , βn)^tandP are givenC¹ functions describing the given ambient flow;ρ=ρ(P)is a given positive increasing function as a function of pressure;ρ⁰ = _dP^dρ ,κ= ^ρ_ρ⁰;the numberµ >0is a viscous constant;f is a given

Correspondence to:Sang Dong Kim, Department of Mathematics, Kyungpook National University, Taegu, Korea 702-701 (e-mail: skim@sobolev.knu.ac.kr); EunJung Lee, Department of Mathematics, Kyungpook National University, Taegu, Korea 702-701 (e-mail: ejlee@sobolev.knu.ac.kr)

c

2001 John Wiley & Sons, Inc.

vector function. We further assume that Z

Ωp dΩ = 0 andβ=0on∂Ω,ρ, andκare bounded.

The equations (1.1) are obtained by linearizing the barotropic Navier-Stokes equations around the ambient flowsβandP, neglecting the bulk viscosity constant term and the zero-order terms (see [1]). The traditional numerical approach is to use a mixed finite element method for these equations (1.1)(see [1]). The least-squares approach for second and higher order problems was developed in [2], [3], and [4], for example, and it was developed for the Stokes and Navier- Stokes equations in [5]. The analysis for thefoslsmethods applied to the incompressible Stokes equations by introducing a new dependent variable had been carried out by several authors (see, for example, [6] and [7] and [8]-[10]). These methods are well known as possessing the freedom of choosing finite element spaces, incorporating additional equations, and imposing additional boundary conditions as long as the system is consistent. The motivation of this article is to adopt such benefits to the compressible Stokes equations (1.1). For these, we reformulate the compressible Stokes equations as a first-order system derived in terms of an additionalvelocity flux. For the systems proposed in this article, the least-squares functionals are defined usingL² andH⁻¹norms or a mixture ofL²,H⁻¹norms and integrals. Recently, a least-squares approach for the compressible Stokes equations reported in [11] did not focus on the auxiliary variable velocity flux, in which one may lose some accuracy to approximate the auxiliaryfluxvariable from the velocityvariable by numerical differentiation. Compared with the works in [11], the foslsformulations in this article give the way to approximate thefluxvariable directly with the best accuracy as the velocity variable has. With the assumption that the viscosity constant µ is sufficiently large, we show that the resulting functionals are equivalent to a product norm kUk²+kuk²₁+kpk²_βββfollowing ideas in [10] (see the notationk · k_βββin section 2). TheH⁻¹fosls approaches enable us to provide a possible optimal convergence rate, which can be a contrast to a mixed finite element method, from the convergence estimate (see Theorem 4.1). This may be one of the benefits for employing thefoslsapproach to the compressible Stokes equations (1.1).

We also discuss the computable functional and provide its error estimate.

This article proceeds as follows. In section 2, the notations, definitions, and first-order system formulations are introduced. In section 3, the least-squares functionals are provided, and coercivity results are derived. In section 4, using these coercivity results, we note thatfoslsformulation can lead to the optimal convergence rate, and we establish discretization error estimates with respect to order of approximation.

II. FOSLSFORMULATION

For convenience, we will adopt the notations introduced in [12] and put the necessary definitions in this section. A new independent variable related to then²-vector function of gradients of the displacement vectors,u_i,i= 1,· · ·nwill be given. For a givenu= (u₁,· · ·, u_n)^t, the gradient is∇u= (∂1u1, ∂2u1, ∂3u1,· · · , ∂2u3, ∂3u3)^t.

For a function with vector componentsU= (U₁,U₂,· · ·,U_n)^t, the divergence is defined as

∇ ·U= (∇ ·U1,∇ ·U2,· · · ,∇ ·Un)^t.

The inner products and norms on the block column vector functions are defined in the natural componentwise way, for example:

kUk²=Xⁿ

i=1

kU_ik² and Z

ΩUdΩ²=Xⁿ

i=1

Z

ΩU_idΩ².

We use standard notations and definitions for the Sobolev spacesH^s(Ω)ⁿ, associated inner prod- ucts(·,·)_s, and respective normsk · k_s,s≥0. Whens = 0,H⁰(Ω)ⁿis the usualL²(Ω)ⁿ, in which case the norm and inner product will be denoted byk · kand(·,·), respectively.L²₀(Ω)will denote the space ofL²(Ω)functionspsuch thatR

Ωpdx= 0. The spaceH₀^s(Ω)is the closure of the linear space of infinitely differentiable functions with compact support inΩwith respect to the normk · ks. From now on, we will omit the superscriptnandΩif the dependence of vector norms on dimension is clear by context. We useH₀⁻¹(Ω) to denote the dual spaces ofH₀¹(Ω) with norm defined by

|φ|−1,0= sup

ψ∈H₀¹(Ω)

(φ, ψ)

|ψ|₁ . We will use

|u|∞ = sup

x∈Ω|u(x)|.

Define the product spacesH₀¹(Ω)ⁿ andL²(Ω)ⁿin the usual way with standard product norms and define a space

Q={q∈L²(Ω) : (kqk²+µ²kκβ· ∇qk²)¹² <∞ }, which is a Hilbert space with norm

kqk_βββ= (kqk²+µ²kκβ· ∇qk²)¹². (2.1) Let

H(div; Ω) ={v∈L²(Ω)ⁿ:∇ ·v∈L²(Ω)}, which is a Hilbert space (see [13]) under the norm

kvk_H(div;Ω):= kvk²+k∇ ·vk²¹₂ .

Following [10] for Stokes equations, introducing thevelocity fluxvariableU=∇u, that is, U= (U₁₁, U₁₂, U₁₃,· · ·, U₃₂, U₃₃)^t=∇u,

the compressible Stokes equations (1.1), with scaledpressurep, may be written as the following equivalent first-order system:









U− ∇u = 0, in Ω,

−∇ ·U+_µ^ρβ· ∇u+∇p = _µ¹f, in Ω,

∇ ·u+µκβ· ∇p = g, in Ω, u = 0, on ∂Ω,

(2.2)

where

β· ∇u= (β· ∇u1,· · ·,β· ∇un)^t.

Note that the first and last equations in (2.2) implies Z

ΩUdΩ = Z

Ω∇udΩ =0.

Hence we may add these two conditions to (2.2), so that an extended equivalent first-order system is given as :

















U− ∇u = 0, in Ω,

−∇ ·U+_µ^ρβ· ∇u+∇p = _µ¹f, in Ω,

∇ ·u+µκβ· ∇p = g, in Ω, u = 0, on ∂Ω, R

ΩUdΩ = 0, R

Ω∇udΩ = 0.

(2.3)

Some existence, uniqueness, and regularity results for compressible linearized Stokes equations are discussed in [14], [15], [16], and [17]. It is well known that the solutionsu∈ H₀¹(Ω)and p∈L²₀(Ω)to (1.1) satisfy

kuk²₁+kpk² ≤ C(kfk²_−1,0+kgk²), (2.4) whereCdepends onµ, ρ, κandβ.

III. FIRST-ORDER SYSTEM OF LEAST-SQUARES FUNCTIONALS

The main objective in this section is to establish ellipticity and continuity of least-squares func- tionals based on (2.2) and (2.3) in appropriate Sobolev spaces.

The least-squares functionals considered in this article are G1(U,u, p;f, g) = k −∇ ·U+ ρ

µ(β· ∇)u+∇p− 1

µf k²_−1,0 (3.1) + k∇ ·u+µκβ· ∇p−gk²+kU− ∇uk².

and

G2(U,u, p;f, g) =G1(U,u, p;f, g) + Z

ΩUdΩ ₂

+ Z

Ω∇udΩ ₂

. (3.2)

Define

M(U,u, p) =kUk²+kuk²₁+kpk_βββ², and let

V:=L²(Ω)ⁿ²×H₀¹(Ω)ⁿ× Q∩L²₀(Ω) .

The first-order system least-squares variational problem for the compressible Stokes equations corresponding (2.2) and (2.3) is to minimize the quadratic functionalG_i(i= 1,2)overV: find (U,u, p)∈ V such that, fori= 1,2,

G_i(U,u, p;f, g) = inf

(V,v,q)∈VG_i(V,v, q;f, g). (3.3) The following lemma was proved by Neˇcas [18] and also can be found in [13].

Lemma 3.1. There is a constantCΩ>0, which depends onΩsuch that kpk ≤ CΩk∇pk−1, for p∈L²₀(Ω).

Theorem 3.1. Assume that there areµandβsuch that ¹_µkρβk_∞+µk∇ ·(κβ)k_∞ is small enough. Then there are two positive constantscandCsuch that for all(U,u, p)∈ V,

cM(U,u, p)≤G1(U,u, p;0,0)≤CM(U,u, p). (3.4) Proof. Upper bound in (3.4) is a simple consequence from the triangle inequality, Cauchy- Schwarz inequality and the bound of _µ¹kρβk_∞. Because of limiting arguments, it is enough to show that lower bound in (3.4) holds forVe=H(div; Ω)ⁿ×H₀¹(Ω)ⁿ×(L²₀(Ω)∩Q). For any (U,u, p) ∈Veandφ∈H₀¹(Ω)ⁿ, from Green’s formula and Poincar´e-Friedrichs inequality we have

(∇p,φ) =

−∇ ·U+ρ

µ(β· ∇)u+∇p,φ

−(U,∇φ)− 1 µ

ρ(β· ∇)u,φ

≤G₁¹²(U,u, p;0,0)kφk1+kUkk∇φk − 1

µ(ρ(β· ∇)u,φ)

≤G₁¹²(U,u, p;0,0)kφk₁+kUkkφk₁+C₁

µ kρβk_∞k∇ukkφk₁, whereC1is a positive constant. Combining this with Lemma 3.1 yields that

kpk ≤CΩ

G₁¹²(U,u, p;0,0) +kUk+C₁

µ kρβk∞k∇uk

. (3.5)

Since

k∇uk²= (∇u−U,∇u) + (U,∇u)

≤G₁¹²(U,u, p;0,0)k∇uk+kUkk∇uk, we have, by cancellingk∇ukon both sides,

k∇uk ≤G₁¹²(U,u, p;0,0) +kUk. (3.6) From (3.5) and (3.6),

kpk ≤C_1,Ω

G₁¹²(U,u, p;0,0) +kUk

, (3.7)

whereC_1,Ω=C_Ω

1 +^C_µ¹kρβk_∞ .

From the Poincar´e-Friedrichs inequality, Green’s formula, (3.6), and (3.7), we have

(U,U) = (U− ∇u,U) + (∇u,U) (3.8)

= (U− ∇u,U) +

−∇ ·U+1

µ(ρβ· ∇)u+∇p,u

−1 µ

(ρβ· ∇)u,u

+ (p,∇ ·u+µκβ· ∇p)−µ(p, κβ· ∇p)

≤ G₁¹²(U,u, p;0,0)kUk+CFG₁¹²(U,u, p;0,0)k∇uk +CF

µ kρβk∞k∇uk²+G₁¹²(U,u, p;0,0)kpk+µ

2k∇ ·(κβ)k∞kpk²,

≤ C2G1(U,u, p;0,0) +C3G₁¹²(U,u, p;0,0)kUk+C4kUk² where

C₂=C_F+ 2

µC_Fkρβk_∞+C_1,Ω+C_1,Ω² µk∇ ·(κβ)k_∞, and C₃= 1 +C_F +C_1,Ω and

C4= 2

µCFkρβk∞+µ

2C_1,Ω² k∇ ·(κβ)k∞, and(p,(β· ∇p)) =−¹₂R

Ω(∇ ·β)p²dΩis used in the first inequality. Recall the-inequality 2ab≤ 1

a²+b². Now using the assumption thatµandβare chosen so that

2

µCFkρβk∞+µ

2C_1,Ω² k∇ ·(κβ)k∞

is small enough, we may have bounds ofC₂andC₃independent ofµandβand with an absolute constantδ <1

C₄≤1−δ, so that, using- inequality in (3.8), we have

kUk² ≤ C₅G₁(U,u, p;0,0) (3.9) where the constantC5depends onΩ. We now then have, using (3.9) in (3.7) and (3.6),

kpk²≤C₆G₁(U,u, p;0,0) (3.10) and

k∇uk²≤C6G1(U,u, p;0,0), (3.11) whereC₆is a positive constant dependent onΩ. Finally, we have

µ²kκβ· ∇pk² = (µκβ· ∇p+∇ ·u, µκβ· ∇p)−(∇ ·u, µκβ· ∇p)

≤

G₁¹²(U,u, p;0,0) + 2k∇uk

µkκβ· ∇pk.

Then, cancellingµkκβ· ∇pkand squaring the remaining, we have from (3.11),

µ²kκβ· ∇pk²≤C6G1(U,u, p;0,0) (3.12) The theorem now follows from (3.9), (3.10), (3.12) and the usual continuity arguments.

As a result of this theorem, the following theorem comes easily, from which the existence of the unique minimizer of the functionalG₂can be shown.

Theorem 3.2. Under the same assumption of Theorem 3.1, there are two positive constantsc andCsuch that for all(U,u, p)∈ V,

cM(U,u, p)≤G2(U,u, p;0,0)≤CM(U,u, p).

IV. FINITE ELEMENT APPROXIMATIONS

In this section, we provide the finite element approximation on the minimization of the least- squares functionals G₁ andG₂ defined in (3.1) and (3.2), respectively. LetT_h be a family of triangulations ofΩby a standard finite element subdivisions ofΩinto quasi-uniform triangles with h= max{diam(K) :K∈ T_h}. Near a curved portion of the boundary∂Ω, we use isoparametric triangles with one curved side. It is assumed that the curved elements in the triangulations of the domain ofΩnot to be ‘‘too curved,’’ so that the standard interpolation error on a straight triangle will hold for a curved triangle (see [19], [20], and [21]). LetV_h:=U_h× S_0,h× W_hbe the finite dimensional subspace ofVwith the following properties: there exist a constantCand integersr, s, andk, for any(U,u, p)∈ H^r(Ω)ⁿ²×H₀^s+1(Ω)ⁿ×H^k+1(Ω)

∩ V,1≤r,1≤sand1≤k, there exists a pair(U_h,u_h, p_h)∈ V_hsuch that

Uhinf∈Uh{kU−U_hk+hkU−U_hk₁} ≤Ch^rkUk_r, (4.1)

uhinf∈S0,h{ku−uhk+hku−uhk1} ≤Ch^s+1kuks+1, (4.2)

phinf∈Wh{kp−p_hk+hkp−p_hk₁} ≤Ch^k+1kpk_k+1. (4.3) The finite element approximation to (3.3) becomes: find(U_h,u_h, p_h)∈ V_hthat satisfies

Gi(Uh,uh, ph;f, g) = inf

(Vh,vh,qh)∈VhGi(Vh,vh, qh;f, g), (i= 1,2).

Since the functionalGiinvolves theH⁻¹norm, we need a solution operator of boundary value problem for its evaluation, for which we adopt the well-knownH⁻¹norm technique proposed in [22].

LetT : H₀⁻¹(Ω)ⁿ −→H₀¹(Ω)ⁿbe the solution operator(u=Tf) for the following elliptic boundary value problem:

−∆u+u = f, in Ω, u = 0, on ∂Ω.

The following well-known result can be found in Lemma 2.1 in [22].

(f, Tf) =kfk²_−1,0= sup φ∈H¹₀(Ω)ⁿ

(f,φ)²

kφk²₁ for all f ∈H₀⁻¹(Ω)ⁿ. (4.4)

Then, from (4.4), we have G₁(U,u, p;0,0) =

T(−∇ ·U+ρ

µβ· ∇u+∇p), −∇ ·U+ρ

µβ· ∇u+∇p

+ (∇ ·u+µκβ· ∇p , ∇ ·u+µκβ· ∇p) + (U− ∇u, U− ∇u) and

G₂(U,u, p;0,0) =

T(−∇ ·U+ρ

µβ· ∇u+∇p), −∇ ·U+ρ

µβ· ∇u+∇p

+ (∇ ·u+µκβ· ∇p , ∇ ·u+µκβ· ∇p) + (U− ∇u, U− ∇u) + 1

|Ω|

Z

ΩUdΩ, Z

ΩUdΩ

+ 1

|Ω|

Z

Ω∇udΩ, Z

Ω∇udΩ

,

where|Ω|denotes the measure of domainΩ.

Theorem 4.1. Suppose that the assumption of Theorem 3.1 holds. Let(U,u, p) ∈ V be the solution of the minimization ofG₁overVand(U_h,u_h, p_h)be the unique minimizer ofG₁over V_h. Then

kU − U_hk²+ku−u_hk²₁+kp−p_hk_βββ² (4.5)

≤ C inf

(Vh,vh,qh)∈Vh kU−V_hk²+ku−v_hk²₁+kp−q_hk²_βββ .

Proof. For convenience, let [U,u, p; V,v, q] =

T(−∇ ·U+ ρ

µβ· ∇u+∇p), −∇ ·V+ ρ

µβ· ∇v+∇q

+(∇ ·u+µκβββ· ∇p , ∇ ·v+µκβ· ∇q) + (U− ∇u, V− ∇v).

(4.4), Theorem 3.1, the orthogonality of the error(U−Uh,u−uh, p−ph)toVh, with respect to the above inner product and Cauchy-Schwarz inequality imply that, for all(Vh,vh, qh)∈ Vh,

kU−Uhk² + ku−uhk²+kp−phk_βββ²

≤ C[U−U_h,u−u_h, p−p_h; U−U_h,u−u_h, p−p_h]

= C[U−Uh,u−uh, p−ph; U−Vh,u−vh, p−qh]

= T(−∇ ·(U−Uh) + ρ

µβ· ∇(u−uh) +∇(p−ph)),

−∇ ·(U−Vh) +ρ

µβ· ∇(u−vh) +∇(p−qh)

+(∇ ·(u−uh) +µκβ· ∇(p−ph), ∇ ·(u−vh) +µκβ· ∇(p−qh)) +((U−U_h)− ∇(u−u_h), (U−V_h)− ∇(u−v_h))

≤ C kU−Uhk²+ku−uhk²₁+kp−phk_βββ²

· kU−Vhk²+ku−vhk²₁+kp−qhk²_βββ ,

whereCdepends onΩ. Note that the last inequality holds becauseT is positive definite and symmetric. Canceling the common factor on both sides and squaring the remains, we have the conclusion.

Corollary 4.1. Suppose that µ satisfies the same assumption of Theorem 3.1. Assume that (U,u, p)∈

H^r(Ω)ⁿ²×H₀^s+1(Ω)ⁿ×H^k+1(Ω)

∩ Vis the solution of the minimization prob- lem forG₁in (3.3) and(U_h,u_h, p_h)is the unique minimizer ofG₁overV_h. Then

kU−Uhk² + ku−uhk²₁ + kp−phk_βββ² (4.6)

≤C h^2rkUk²_r+h^2skuk²_s+1+h^2kkpk²_k+1 , whereCdepends onΩ.

Proof. By (4.1),(4.2),(4.3) and Theorem 4.1, we can get the above result.

Note that one may get easily a similar approximation result for minimizing G₂ functional as Corollary 4.1. We notice that the inequality (4.5) can be expected to give an optimal rate of convergence. As any choice of subspacesUh,S0,handWh, (4.6) shows that the error bounds for velocity flux and velocity obtained here are the best approximations in the subspacesUhandS0,h, but one may see that theL²error bound for pressure is one order less than the best approximation in the subspaceWh. It is noted in [1] that when the mixed finite element methods are applied to (1.1), taking the so called MINI elements for velocity and pressure which are subject to the inf-sup condition, the optimal convergence rate corresponding to the error bound (4.5) may not be obtained.

In order to evaluate functionalsG1 andG2 on finite dimensional subspaces, it is required to replace the operator T by a computable and equivalent operator T_h. Let T_h on the finite dimensional subspaceS_0,hofH₀¹(Ω)ⁿbe the discrete Galerkin solution operator corresponding toT. Since a slight modification on arguments for the functionalG₁can be applied to the functional G₂, we mainly focus on evaluation of the functionalG₁from now on. Assume that there is a good preconditionerB_h:S_0,h−→ S_0,h, which is symmetric, positive definite with respect toL²inner product and equivalent toT_h, that is,

c(Thv,v)≤(Bhv,v)≤C(Thv,v), for all v∈ S0,h. (4.7) Define

Th=h²I+Bh, whereIdenotes the identity operator and

Gh(U,u, p;1 µf, g)

=

Th(−∇ ·U+ρ

µβ· ∇u+∇p−1

µf), −∇ ·U+ρ

µβ· ∇u+∇p− 1 µf

+ (∇ ·u+µκβ· ∇p−g , ∇ ·u+µκβ· ∇p−g) + (U− ∇u,U− ∇u). LetQhbe theL²(Ω)ⁿorthogonal projection operator ontoS0,h. We further assume that

kQ_huk₁≤Ckuk₁ for all u∈H₀¹(Ω)ⁿ. (4.8) The symmetric properties of operatorsB_handT_hwith respect toL²inner product yieldB_h= B_hQ_h and T_h = T_hQ_h. Thus (4.7) holds for any v ∈ L²(Ω)ⁿ. Hence, using (4.8) and the definition of discrete solution operatorT_h(refer to (4.4)), it follows that

c||Qhu||²_−1,0≤(u, Thu)≤C||u||²_−1,0 for all u∈H₀⁻¹(Ω)ⁿ. (4.9) It is easy to check that, from the approximation result (4.2) with s = 1and Cauchy-Schwarz inequality,

||v−Q_hv||_−1,0≤Ch||v||, for all v∈L²(Ω)ⁿ, (4.10)

so that, (4.7), (4.9) and (4.10) imply that

||f||_−1,0≤C(f,T_hf) for all f ∈L²(Ω)ⁿ. (4.11) Lemma 4.1. Suppose that the assumption of Theorem 3.1 holds and assume thatVhsatisfies the inverse inequalities such thathkUhk1≤CkUhk,hkuhk1 ≤Ckuhk, andhk∇phk ≤Ckphk.

For any(Uh,uh, ph)∈ Vh, there are two positive constantscandCindependent ofhsuch that c(||Uh||²+||uh||²₁+||ph||_βββ²)≤Gh(Uh,uh, ph;0,0) (4.12) and

G_h(U_h,u_h, p_h;0,0)≤C(||U_h||²+||u_h||²₁+||p_h||_βββ²). (4.13) Proof. SinceT_h=h²I+B_h, using (4.7) and definitions ofT_h(refer to (4.4)), we have

(T_hv,v)≤C(h²kvk²+kvk²_−1.0) for all v∈L²(Ω)ⁿ. (4.14) Then (4.14), triangle inequality and inverse inequality imply that

( T_h (−∇ ·U_h+_µ^ρβ· ∇u_h+∇p_h),−∇ ·U_h+_µ^ρβ· ∇u_h+∇p_h )

≤ C(kUhk²+ (_µ¹)²kρβk²_∞kuhk²₁+kphk²+k − ∇ ·Uh+_µ^ρβ· ∇uh+∇pk−1,0)

≤ C(kUhk²+kuhk²₁+kphk²+k − ∇ ·Uh+_µ^ρβ· ∇uh+∇pk−1,0),

where the bound of¹_µkρβk_∞is used. This argument completes the upper bound (4.13). The lower bound (4.12) can be proved easily by (4.11) and Theorem 3.1.

Because of this lemma, the similar proofs of Theorem 4.1 and Lemma 4.1 lead to:

Theorem 4.2. Suppose that µ satisfies the same assumption of Theorem 3.1. Assume that (U,u, p)∈ Vis the solution of the problem (3.3). Let(Uh,uh, ph)∈ Vhbe the unique minimizer ofGh. Then

kU−Uhk² + ku−uhk²₁+kp−phk_βββ² (4.15)

≤ C inf

(Vh,vh,qh)∈Vh kU−Vhk²+ku−vhk²₁+kp−qhk_βββ² , whereCdepends onΩ.

Corollary 4.2. Suppose that µ satisfies the same assumption of Theorem 3.1. Assume that (U,u, p)∈

H^r(Ω)ⁿ²×H₀^s+1(Ω)ⁿ×H^k+1(Ω)

∩ Vis the solution of the problem (3.3). Let (U_h,u_h, p_h)∈ V_hbe the unique minimizer ofG_h. Then

kU−U_hk² + ku−u_hk²₁+kp−p_hk²_βββ (4.16)

≤ C h^2rkUk²_r+h^2skuk²_s+1+h^2kkpk²_k+1 , whereCdepends onΩ.

We briefly note that the inequality (4.15) can also give an optimal rate of convergence and the inequality (4.16) shows that the error bounds for velocity flux and velocity are the best approximations on the subspacesUhandSh.

This work was sponsored by KOSEF under grant number 1999-2-103-002-3 and the Brain Korea 21. The authors thank Prof. J. Kweon and the anonymous referees for their valuable suggestions.

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