L2-PROJECTED LEAST-SQUARES FINITE ... - Semantic Scholar

L2 -PROJECTED LEAST-SQUARES FINITE ELEMENT METHODS FOR THE STOKES EQUATIONS HUO-YUAN DUAN†∗ , PING LIN

†,

SAIKRISHNAN P.† , AND ROGER C. E. TAN†

Abstract. Two new L2 least-squares (LS) finite element methods are developed for the velocitypressure -vorticity first-order system of the Stokes problem with Dirichlet velocity boundary condition. A key feature about these new methods is that a local or almost local L2 projector is applied to the residual of the momentum equation. Such L2 projection is always defined onto the linear finite element space, no matter which finite element spaces are used for velocity-pressure-vorticity variables. Consequently, the implementation of this L2 -projected LS method is almost as easy as that of the standard L2 LS method. More importantly, the former has optimal error estimates in L2 -norm, with respect to both the order of approximation and the required regularity of the exact solution for velocity using equal-order interpolations and for all three variables (velocity, pressure and vorticity) using unequal-order interpolations. Numerical experiments are given to demonstrate the theoretical results. Key words. the Stokes equation, velocity-pressure-vorticity least-squares finite element method, L2 projection, mass-lumping AMS subject classifications. 65N30

1. Introduction. The least-squares (LS) mixed finite element method is widely used in seeking numerical solution of partial differential equations arising from fluid and solid mechanics, cf. [30, 28, 31, 15, 18, 20, 24, 26, 19, 32, 14, 21, 22, 23, 27, 13]. In a broad sense, the LS method is to minimize the residual, measured in some Sobolev norms, of a mixed first-order system of partial differential equations. The mixed first-order system is obtained by introducing one or more additional physically important fields such as stress/pressure/vorticity besides displacement/velocity as unknown variables. There are many advantages about LS methods. The LS method may be viewed as a classical Ritz’s method of coercive type [29] and is not subject to the so-called inf-sup condition [9, 2]. Its resulting linear system is symmetric positive definite and can be solved by standard iterative methods such as conjugate gradient method. In addition, the standard finite element spaces can be employed for each unknown variable. Readers may refer to [13, 12] for more details on LS methods. In this paper we shall introduce and study new LS methods for the Stokes problem written as a system of equations of first-order, where velocity, pressure and vorticity appear as unknown variables. This system involves relatively few unknowns and is widely employed in engineering practice. Let us first review several LS methods developed in the last decade for the velocitypressure-vorticity Stokes system. The most widely used LS method is the standard L2 LS method ([15, 13]), where the LS functional is the squared L2 -norms of the residual of the first-order system. This method is easy to implement and performs very well in many engineering applications (cf. [13, 20, 19, 31, 30]). However, for the important case of Dirichlet velocity boundary condition, this method is not optimal in the usual sense [16, 17], for example, for equal-order continuous interpolations, the L2 -error bound for velocity is not optimal with respect to both the order of approximation and the required regularity. In the case of convex polygon, no error ∗ Corresponding

author. of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543 (e-mails: [email protected], [email protected], [email protected] and [email protected]) † Department

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Huo-yuan Duan, Ping Lin, Saikrishnan P. and Roger C. E. Tan

estimates are available. Also, no improved error estimates are obtained for unequalorder continuous interpolations (See [33] for some numerical results). The reason that the standard L2 LS method suffers from suboptimal error estimates in the case of velocity Dirichlet boundary condition may be the following: the coercivity for vorticity and pressure is measured in L2 norm, whereas their first-order derivatives appeared in the term resulting from the momentum equation suggest that the continuity condition cannot be obtained in the same measure, which prevents from obtaining optimal error estimates. To recover the optimal error estimates one has to do some modifications to that term of the momentum equation. There have been two important methods which can overcome the difficulty from the term of the momentum equation. One is the Bochev-Gunzburger(BG) method, where a factor h2 is put in front of the term of the momentum equation. Alternatively, the BG method may be scaled as the one with a factor h−2 put in front of terms of the incompressibility condition and of the vorticity equation [18]. The other is the H −1 LS method which may be viewed as a modified version of the BG method by introducing an additional term of the momentum equation, where a precondtioner Bh (or an operator of the finite element solution) for the Dirichlet problem of a second-order elliptic equation is applied [14, 32, 21]. In the BG method the effects from the term of the momentum equation can be eliminated because of the factor h2 and optimal error estimates can be derived with the use of unequal-order continuous interpolations [18, 16]. In the H −1 method, when Bh satisfies a spectral equivalence (See the equation (2.15) in [[21], page 941]), the coercivity and the optimal error estimates can be established. These are excellent efforts in achieving optimal error estimates of LS methods. However, there are still rooms for improvement. The BG method does not give optimal L2 -error estimates for the velocity, excludes the use of linear element ([33, 18, 16]) and has a condition number O(h−4 ). This is due to the fact that it lacks a coercivity uniform in mesh sizes or that in its scaled version the scale factor h−2 worsens its continuity condition. The H −1 method can provide optimal L2 error bounds for the velocity, but there is a restriction on Bh (see the equation (3.24) in [[21], page 947]). There are few examples of Bh known to satisfy that restriction. Also, due to the fact that Bh is defined onto Uh (the approximating space for the velocity), Bh varies with Uh accordingly. This may complicate implementation issues when Uh is of higher order elements. Our new idea presented in the paper is to add an L2 projected term of the momentum equation to the BG method, or to use an L2 projector to replace the preconditioner in the H −1 method. With the L2 projection term, the uniform coercivity holds (see Theorem 3.1), and the error estimates are optimal (see Theorem 3.3 and Theorem 4.1), for velocity using equal-order interpolations and for all three variables (velocity, pressure and vorticity) using unequal-order interpolations. Also, the condition number is of O(h−2 ) (see Corollary 3.2) and the implementation of this L2 -projected LS method is almost as easy as that of the standard L2 LS method, since the L2 projection is local or almost local and is always defined onto the linear finite element space, no matter which finite element spaces are used for velocity-pressurevorticity variables. Note that, although the L2 projection is “fixed” onto the linear element space, this does not cause any consistency problem and does not affect the order of the error estimates. We provide two methods according to the definitions of L2 projectors applied to the term of the momentum equation. One is called the local L2 projection method (I) and the other the mass-lumping L2 projection method (II). The L2 projection

L2 -Projected Least-Squares Finite Element Methods for the Stokes Equations

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in the method (I) is always element-by-element defined onto the discontinuous linear element space; in the method (II) the L2 projection is always defined by using the mass-lumping technique [1] onto the continuous linear element space. Note that the L2 projection in method (II) is almost local because the resulting matrix of this L2 projection is diagonal. Standard equal-order or unequal-order finite elements, with lower-order finite elements for pressure and vorticity enriched with element or edge(face) or both bubbles, are employed for approximating velocity, pressure and vorticity variables. Note that the role of the bubbles for lower-order elements for pressure and vorticity is to make Assumption (A2) (see equations (3.35) and (3.36)) holds(cf. Remark 3.2 and Theorem 3.4). Our L2 projection plays a critical role(see equation (3.24)) in the derivation of a uniform coercivity (see equations(3.6)). All the first-order derivatives of pressure and vorticity appear only in L2 -projected and h2 -weighted terms. Due to Assumption (A2), the errors associated with the L2 -projected term can be made zero (See equation (3.47)), while the h2 -weighted term is obviously consistent in terms of both the order of approximation and the regularity of the exact solution. Therefore, optimal error estimates can be achieved. Also, an O(h−2 ) condition number is obtained. Of course, we may project onto a higher order element space. But obviously the linear element is simpler and the L2 projection can be easily implemented. For the method (I) we could even consider to define the local L2 projection onto a piecewise constant space and almost all our techniques of analysis may still work. However, an optimal L2 error bound for velocity cannot be obtained because the interpolation result of the linear element has to be used in the derivation (See equation (4.20)). We finally remark that in deriving the L2 error estimates for the velocity we assume that the domain is a convex polygon as usual [5, 29]. For such a domain some known regularity results for Stokes and elasticity problems(cf. [7, 6, 8, 10]) is used. The outline of this paper is as follows. In section 2, we recall the first-order system of the velocity-pressure-vorticity Stokes problem, and formulate L2 projected methods. In section 3, we establish coercivity and error bounds and verify an important assumption (Assumption (A2)). In section 4, the L2 error bound for velocity is obtained. In section 5, numerical results are presented to support our theoretical analysis. 2. Problem formulation. 2.1. First-order system of the Stokes problem. Let Ω ⊂ Rd (d = 2, 3) be a bounded domain with boundary Γ and f ∈ (L2 (Ω))d . We consider the Stokes problem: Find velocity u and pressure p such that (2.1)

−∆u + 5p = f ,

5 · u = 0,

in Ω,

u = 0,

on Γ.

Let 5× denote the curl operator. Introducing the vorticity ω = 5 × u ∈ (L2 (Ω))2d−3 and noting that −∆u = 5 × 5 × u − 5 5 ·u and 5 · u = 0, we can write (2.1) as the first-order system: (2.2)

5 × ω + 5 p = f,

ω = 5 × u,

5 · u = 0,

in Ω,

along with a Dirichlet boundary condition and a pressure mean-zero condition: Z (2.3) u|Γ = 0, p = 0. Ω

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Huo-yuan Duan, Ping Lin, Saikrishnan P. and Roger C. E. Tan

Below we shall use the standard Sobolev spaces H01 (D) and H s (D), with norm k · ks,D and semi-norm | · |s,D , where D is some Lipschitz subdomain of Ω, D will be omitted from the notation when D = Ω. We shall use (·, ·)0,D for the inner 0 product of L2 (D) (= H R (D)). When D = Ω, (·, ·) := (·, ·)0,Ω . We shall also define 2 2 L0 (Ω) := {q ∈ L (Ω); Ω q = 0}. Throughout this paper we always assume that Ω is a Lipschitz polygon (polyhedron in R3 ) and that Ch is a regular triangulation of Ω (tetrahedrons in R3 ), with diameters hK ≤ h for all triangular elements K ∈ Ch . 2.2. Local L2 -projection method (I). Introduce (2.4)

Zh := {v ∈ (L2 (Ω))d ; v|K ∈ (P1 (K))d , ∀K ∈ Ch },

where P1 (K) denotes the space of linear polynomials on K. For given g ∈ (L2 (Ω))d ˘ h (g) ∈ Zh by we define a function R Z Z ˘ h (g) v = (2.5) R g v ∀v ∈ (P1 (K))d , ∀K ∈ Ch . K

K

Let (2.6)

Uh ⊂ (H01 (Ω))d ,

Ph ⊂ L20 (Ω),

Wh ⊂ (L2 (Ω))2 d−3

be continuous piecewise polynomial spaces on Ch for velocity, pressure and vorticity, respectively. We define an LS functional on Uh × Ph × Wh by X ˘ h (5 × ω + 5 p − f ) k2 + JhI (u, p, ω; f ) :=k R h2K k 5 × ω + 5 p − f k20,K 0 K∈Ch

+ k ω − 5 × u k20 + k 5 · u k20 .

(2.7)

We consider a minimization problem: Find (uh , ph , ωh ) ∈ Uh × Ph × Wh such that (2.8)

JhI (uh , ph , ωh ; f ) =

inf

(vh ,qh ,zh )∈Uh ×Ph ×Wh

JhI (vh , qh , zh ; f ).

Taking variations in (2.7) with respect to (vh , qh , zh ), we obtain the weak statement of problem (2.8): Find (uh , ph , ωh ) ∈ Uh × Ph × Wh such that  I Lh ((uh , ph , ωh ); (vh , qh , zh )) :=    ˘ h (5 × ωh + 5 ph ), R ˘ h (5 × zh + 5 qh ))  (R    + P h2 (5 × ω + 5 p , 5 × z + 5 q ) h h h h 0,K K (2.9) K∈Ch   +(ωh − 5 × uh , zh − 5 × vh ) + (5 · uh , 5 · vh )     ˘ h (5 × zh + 5 qh )) + P h2 (f , 5 × zh + 5 qh )0,K  = (f , R K K∈Ch

holds for all (vh , qh , zh ) ∈ Uh × Ph × Wh . 2.3. Mass-lumping L2 projection method (II). Introduce (2.10)

V0,h := Zh ∩ (H01 (Ω))d ,

where Zh is defined in (2.4). Let (·, ·)h denote an inner product in V0,h and the induced norm in V0,h is given by (2.11)

1/2

k v kh := (v, v)h .

Remark 2.1 (·, ·)h is usually taken as an approximation of (·, ·). For example, when Ch consists of 2D triangles, we may take (·, ·)h as the quadrature scheme:

L2 -Projected Least-Squares Finite Element Methods for the Stokes Equations

(u, v)h :=

(2.12)

5

3 X area(K) X u(i) v(i), 3 i=1

K∈Ch

where i = 1, 2, 3 denote vertices of the triangle K. In the literature [1], (·, ·)h replacing (·, ·) is called mass-lumping. The matrix associated with (·, ·)h is diagonal. For given w ∈ (L2 (Ω))2d−3 and p ∈ L2 (Ω) we define two functions Rh (5 × w) ∈ V0,h and Sh (5 p) ∈ V0,h , respectively, by (2.13) (2.14)

(Rh (5 × w), vh )h = (w, 5 × vh ) ∀vh ∈ V0,h , (Sh (5 p), vh )h = −(p, 5 · vh ) ∀vh ∈ V0,h . ¯ h (g) ∈ V0,h by For given g ∈ (L2 (Ω))d we define a function R ¯ h (g), vh )h = (g, vh ) ∀vh ∈ V0,h . (2.15) (R ¯ h are all linear operators. In addition, if w ∈ Remark 2.2 Clearly, Rh , Sh and R (H 1 (Ω))2d−3 and p ∈ H 1 (Ω), we have ¯ h (5 × ω) + R ¯ h (5 p) = R ¯ h (5 × w + 5 p). (2.16) Rh (5 × w) + Sh (5 p) = R We consider the case of Ph and Wh possibly being discontinuous or being linear and quadratic continuous elements and we define an LS functional on Uh × Ph × Wh : X ¯ h (f ) k2h + JhII (u, p, ω; f ) :=k Rh (5 × ω) + Sh (5 p) − R h2K k 5 × ω + 5 p − f k20,K (2.17)

+

X E∈Eh

Z |[p]|2 +

hE E

Z

X

hE

E∈Eh

E

K∈Ch

|[w]|2 + k ω − 5 × u k20 + k 5 · u k20 ,

where Eh denotes the collection of interior edges (faces in R3 ), [p] is the jump in p across E, hE is the length or diameter of E. With JhII , in the same way as that for (2.8) and (2.9), we can consider an LS minimization problem and then obtain its weak statement: Find (uh , ph , ωh ) ∈ Uh × Ph × Wh such that  II Lh ((uh , ph , ωh ); (vh , qh , zh )) :=     (RhP (5 × ωh ) + Sh (5 ph ), Rh (5 × zh ) + Sh (5 qh ))h     + h2 (5 × ωh + 5 ph , 5 × zh + 5 qh )0,K   K∈Ch K R R P P (2.18)  + E∈E hE E [ωh ] [zh ] + E∈E hE E [ph ] [qh ]  h h     +(ωh − 5 × uh , zh − 5 × vh ) + (5  P· uh2, 5 · vh )    = (f , Rh (5 × zh ) + Sh (5 qh )) + hK (f , 5 × zh + 5 qh )0,K K∈Ch

holds for all (vh , qh , zh ) ∈ Uh × Ph × Wh . Remark 2.3 The method (I) is simpler than the method (II), but the latter applies to lower order continuous elements and discontinuous elements for pressure and vorticity. Note that using linear or quadratic elements for pressure and vorticity without ˘ h term and the h2 factor, the method (I) is the standard L2 LS method [15] the R which does not have optimal error estimates. In addition, we remark that without the L2 projection, our method reduces to the Bochev-Gunzburger method [18], and when replacing the L2 projector by a preconditioner (or an operator of the finite element solution) for the Dirichlet problem of a second-order elliptic equation, the H −1 method [14, 32] is obtained.

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Huo-yuan Duan, Ping Lin, Saikrishnan P. and Roger C. E. Tan

3. Coercivity and error bounds in energy norm. We shall give a unified analysis for coercivity and error bounds in energy norm for methods (I) and (II). In what follows C represents a generic positive constant independent of h and may take different values at different occurrences. 3.1. Coercivity analysis. In this subsection we investigate the coercivity. Proposition 3.1 ([2][11]) Let Ω be a bounded connected domain with a Lipschitzcontinuous boundary Γ. Then (3.1)

k v k21 ≤ C {k 5 × v k20 + k 5 · v k20 }

Proposition 3.2 ([25]) (3.2)

∀v ∈ (H01 (Ω))d .

Under the assumption on Ω as in Proposition 3.1, we have inf

sup

q∈L20 (Ω) 06=v∈(H 1 (Ω))d 0

(5 · v, q) ≥ C. k v k1 k q k0

Lemma 3.1 Let X be p a given Hilbert space, with inner product (·, ·)X and corresponding norm k · kX = (·, ·)X . For any two elements A ∈ X, B ∈ X and for any α ∈ R , we have (3.3)

k A − B k2X ≥ α(1 − α/2) (k A k2X + k B k2X ) − 2α(A, B)X .

Proof. (3.3) follows from the sum of the two equations: k A − B k2X =k A − αA − B k2X +α(2 − α) k A k2X −2α (A, B)X , k A − B k2X =k A − B + αB k2X +α(2 − α) k B k2X −2α (A, B)X . ¤ For the following analysis we recall the well-known Young’s inequality |a| |b| ≤ ² |a|2 +

1 |b|2 4²

∀a, b ∈ R, ∀² > 0

and Green’s formulae of integrating by parts Z (5 × v, φ)0,D − (v, 5 × φ)0,D = φ v × n ∀v ∈ (H 1 (D))d , φ ∈ (H 1 (D))2 d−3 , ∂D Z (5 · v, q)0,D + (v, 5 q)0,D = q v · n ∀v ∈ (H 1 (D))d , q ∈ H 1 (D), ∂D

where n denotes the exterior unit normal to ∂ D, D is a Lipschitz subdomain of Ω. We also introduce a notation µZ ¶ X X (3.4) |(p, ω)|2(Ch ,Eh ) := h2K k 5 × ω + 5 p k20,K + hE |[p]|2 + |[ω]|2 . K∈Ch

E∈Eh

E

Assumption (A1) There exists C1 > 0, C2 > 0 independent of h such that (3.5)

C1 k vh k0 ≤k vh kh ≤ C2 k vh k0

∀vh ∈ V0,h .

L2 -Projected Least-Squares Finite Element Methods for the Stokes Equations

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Remark 3.1 Taking (2.12) as an example. Assumption (A1) can be easily shown by considering each triangle separately (see also [[5], page 157]). Theorem 3.1 Assuming (A1) for the method (II). Let Jh stands for JhI or JhII . Then, for all (u, p, ω) ∈ Uh × Ph × Wh , Jh (u, p, ω; 0) ≥ C {k u k21 + k p k20 + k ω k20 }.

(3.6)

Proof. We consider the method (II) here. The argument remains unchanged for the method (I). One only needs to note that, in the method (I), Ph × Wh are continuous and (·, ·) is in place of (·, ·)h . In the proof we only need to deal with k ω−5×u k20 and k Rh (5×ω)+Sh (5 p) k2h . The proof is divided into three steps. In the first two steps we find lower bounds for k ω−5×u k20 + k Rh (5×ω)+Sh (5 p) k2h . In the last step we use the mesh-dependent terms to obtain (3.6). Step 1. Let α > 0 be a constant to be determined. From Lemma 3.1 we have (3.7)

k ω − 5 × u k20 ≥ α (1 − α/2) {k 5 × u k20 + k ω k20 } − 2 α (ω, 5 × u).

˜ ∈ V0,h as the Cl´ement-interpolant [2, 4] of u ∈ Uh and we have We take u à !1/2 Z X X −2 −1 2 2 ˜ − u k0,K + ˜ k1 ≤ C k u k1 . hK k u (3.8) hE |˜ u − u| +ku K∈Ch

E∈Eh

E

We also have (3.9)

˜ ) − 2 α (ω, 5 × (u − u ˜ )) −2 α (ω, 5 × u) = −2 α (ω, 5 × u ˜ )h − 2 α (ω, 5 × (u − u ˜ )), = −2 α (Rh (5 × ω), u

and (3.10)

˜ )h + k Rh (5 × ω) + Sh (5 p) k2h −2 α (Rh (5 × ω), u ˜ k2h −α2 k u ˜ k2h +2 α (Sh (5 p), u ˜ )h , =k Rh (5 × ω) + Sh (5 p) − α u

where ˜ k2h −α2 k u (3.11)

(3.12)

˜ k20 ≥ −α2 C k u (by Assumption (A1)) 2 ≥ −α C k u k21 (by (3.8)) ≥ −α2 C k 5 × u k20 −α2 C k 5 · u k20 (by Proposition 3.1),

˜ )h 2 α (Sh (5 p), u

˜) = −2 α (p, 5 · u ˜ )) = −2 α (p, 5 · u) + 2 α (p, 5 · (u − u

and (3.13)

−2α(p, 5 · u) ≥ −²1 k p k20 −

α2 C k 5 · u k20 ²1

(by the Young’s inequality). Here ²1 > 0 is a constant to be determined later. Therefore, summarizing (3.7) and (3.9)-(3.13), we get  k ω − 5 × u k20 + k Rh (5 × ω) + Sh (5 p) k2h ≥      α(1 −nα/2) k ω k20 +α [1 − α (1/2 + C)] k 5o× u k20 (3.14) ˜ )) + (p, 5 · (u − u ˜ )) +2 α − (ω, 5 × (u − u   ´ ³ 2    −²1 k p k20 − α C + α2 C k 5 · u k20 , ²1

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Huo-yuan Duan, Ping Lin, Saikrishnan P. and Roger C. E. Tan

where

(3.15)

n o   ˜ ˜ 2 α − (ω, 5 × (u − u )) + (p, 5 · (u − u )) =   P    ˜ )0,K −2 α (5 × ω + 5 p, u − u    K∈C h R  P P R   ˜ ) × nE + 2 α ˜ ) · nE , −2 α [ω] (u − u [p] (u − u   E E  E∈Eh E∈Eh  ≥ −2 α C |(p, ω)|(Ch ,Eh ) Ã !1/2    P −2 P −1 R   2 2  ˜ k0,K + ˜| × hK k u − u hE E |u − u    K∈Ch E∈Eh     ≥ −2 α C k u k1 |(p, ω)|(Ch ,Eh ) ≥ −α2 k u k21 −C |(p, ω)|2(Ch ,Eh )    ≥ −α2 C k 5 × u k20 −α2 C k 5 · u k20 −C |(p, ω)|2(Ch ,Eh ) .

Thus, (3.14) becomes  2 2   k ω − 5 × u k0 +2 k Rh (5 × ω) + Sh (5 p) kh ≥ 2 α(1 − α/2) k ω k0 +α [1 − α (1/2 (3.16) ³ 2+ 2 C)] k 5´× u k0  α C 2  −²1 k p k20 −C |(p, ω)|2 k 5 · u k20 . (Ch ,Eh ) − ²1 + α 2 C Step 2. Let β > 0 be a constant to be determined. From Proposition 3.2 we can find v∗ ∈ (H01 (Ω))d such that (5 · v∗ , p) =k p k20 ,

(3.17)

k v∗ k1 ≤ C k p k0 .

˜ ∗ ∈ V0,h as the Cl´ement-interpolant [2,4] of v∗ and we have We take v Ã

X

h−2 K

k v˜∗ − v∗ k20,K +

K∈Ch

X E∈Eh

!1/2

Z h−1 E

|v˜∗ − v∗ |2

+ k v˜∗ k1 ≤ C k v∗ k1 .

E

(3.18) We can write

(3.19)

k Rh (5 × ω) + Sh (5 p) k2h =k Rh (5 × ω) + Sh (5 p) + β v˜∗ k2h n o −β 2 k v˜∗ k2h −2 β (Rh (5 × ω), v˜∗ )h + (Sh (5 p), v˜∗ )h ,

where (3.20)

−β 2 k v˜∗ k2h ≥ −β 2 C k v˜∗ k20 ≥ −β 2 C k v∗ k21 ≥ −β 2 C k p k20 ,

n o   −2 β (Rh (5 × ω), v˜∗ )h + (Sh (5 p), v˜∗ )h =      −2 β {(ω, 5 × v˜∗ ) − (p, 5 · v˜∗ )}     = 2 β (p, 5 · v∗ ) − 2 β (ω, 5 × v∗ ) (3.21) +2 β {(ω, 5 × (v∗ − v˜∗ )) − (p, 5 · (v∗P − v˜∗ ))}   2 ∗  = 2 β k p k0 −2 β (ω, 5 × v ) + 2 β (5 × ω + 5 p, v∗ − v˜∗ )0,K    K∈C h  P R P R   [ω] (v∗ − v˜∗ ) × nE − 2 β [p] (v∗ − v˜∗ ) · nE  +2 β E E E∈Eh

E∈Eh

L2 -Projected Least-Squares Finite Element Methods for the Stokes Equations

with  P 2β (5 × ω + 5 p, v∗ − v˜∗ )0,K    K∈C h  P R P R    +2 β [ω] (v∗ − v˜∗ ) × nE − 2 β [p] (v∗ − v˜∗ ) · nE ≥  E E   E∈E E∈E h h  !1/2 Ã R P −2 P −1  −2 β C |(p, ω)|(Ch ,Eh ) hK k v˜∗ − v∗ k20,K + hE E |v˜∗ − v∗ |2    K∈Ch E∈Eh     ≥ −2 β C |(p, ω)|(C ,E ) k v∗ k1    ≥ −2 β C |(p, ω)| h h k p k ≥ −β 2 k p k2 −C |(p, ω)|2 0 (Ch ,Eh ) 0 (Ch ,Eh ) (3.22) and (3.23) −2 β (ω, 5 × v∗ ) ≥ −2 β C k ω k0 k v∗ k1 ≥ −²2 k ω k20 −

C β2 k p k20 . ²2

Here ²2 > 0 is also a constant to be determined later. Summarizing (3.19)-(3.23), we get

(3.24)

k Rh (5 × ω) + Sh (5 p) k2h ≥ β [2 − β (C + 1 + C/²2 )] k p k20 −²2 k ω k20 −C |(p, ω)|2(Ch ,Eh ) .

Step 3. From (3.16) and (3.24) taking 0