An analysis technique for stabilized finite element solution of incompressible flows

Mathematical Modelling and Numerical Analysis ESAIM: M2AN Mod´elisation Math´ematique et Analyse Num´erique Vol. 35, No1, 2001, pp. 57–89 AN ANALYSIS TECHNIQUE FOR STABILIZED FINITE ELEMENT SOLUTION OF INCOMPRESSIBLE FLOWS ∗ Tom´ as Chac´ on Rebollo1 Abstract. This paper presents an extension to stabilized methods of the standard technique for the numerical analysis of mixed methods. We prove that the stability of stabilized methods follows from an underlying discrete inf-sup condition, plus a uniform separation property between bubble and velocity finite element spaces. We apply the technique introduced to prove the stability of stabilized spectral element methods so as stabilized solution of the primitive equations of the ocean. Mathematics Subject Classification. 65N30, 76M10. Received: November 24, 1999. Revised: October 19, 2000. 1. Introduction and motivation This paper deals with the numerical analysis of the solution of incompressible flow problems by stabilized finite elements. We shall be interested in the Oseen equations (Stokes equations plus a linear transport term), also called in some works “linearized Navier-Stokes equations”. Stabilized methods provide efficient and computationally cheap techniques to solve incompressible fluids. Historically, these methods have been the object of a specific analysis, different from that of mixed methods. Indeed, the proof of stability is not based upon the existence of a discrete velocity – pressure inf-sup condition, but rather upon specific arguments that strongly rely on the elementwise regularity of finite element functions. Based upon such kind of arguments, the papers of Hugues, Franca and Balestra [21] and Hughes and Franca [20] contained an error analysis that was improved in Brezzi and Douglas [8] and in Pierre [25]. In Franca and Stenberg [15] a general stability and error analysis technique was introduced, which was summarized in Franca, Hugues and Stenberg [16]. Also, the paper of Tobiska and Verf¨urth [27] develops an analysis of stability and convergence for the solution of Navier-Stokes equations by stabilized methods. Another way of analysis is suggested by the relationship between stabilized and mixed methods. In Franca and Frey [14] it is proved that the Streamline Upwind/Petrov-Galerkin (SUPG) method is equivalent to the standard mixed method constructed with the mini-element. This equivalence is understood in the sense that both methods yield the same formulation if the degrees of freedom associated to the bubbles are eliminated by static condensation. This equivalence yields the stability of SUPG method from that of the mixed method Keywords and phrases. Oseen equations, finite elements, mixed methods, stabilized methods, discrete inf-sup condition, spectral methods, primitive equations. ∗This research was partially supported by Spanish M. E. C. Project MAR97-1055-C02-02 and by EU HCM ERBCHB ICT 94.1823 Grant. 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla. C/ Tarfia, s/n. 41080 Sevilla, Spain. e-mail: [email protected] c EDP Sciences, SMAI 2001 58 T. CHAC ´ ON REBOLLO constructed with the mini-element. It is a direct consequence of the fact that this element satisfies the discrete inf-sup condition. Such analysis is essentially performed in Chac´on Rebollo [10]. We address in this paper the question of whether this way of analysis may be applied to stabilized methods other than SUPG. We develop a technique for the numerical analysis of stabilized methods that gives a positive answer to that question. Concretely, we prove the existence of an underlying discrete inf-sup condition from which we deduce the stability of stabilized methods. Once this point has been set up, our technique allows to analyze stabilized methods as if they where mixed methods (Th. 1). They appear as internal approximations of a weak formulation, whose stability relies on an inf-sup condition. Then, our analysis may be applied to more complex situations, where we use the tools provided by functional analysis to obtain gains with respect to the standard analysis. We include in this paper two of such applications: •To prove the stability of a spectral element approximation of the generalized Stokes equations, introduced in Gervasio and Saleri [19]. Here, we obtain L2estimates for the pressure, while the standard analysis, used in that paper, allows only to estimate a seminorm of the pressure gradient. •To solve a linear model of primitive equations of the ocean by stabilized finite elements. For such equations, there is some lack of regularity for the convection term, so that the pressure has only Lpregularity, for some p∈(1,2). In this case, we obtain Lpestimates for the discrete pressure, and prove convergence in H1×Lpnorm to the continuous solution. The standard analysis in this context would be quite difficult to be carried on. Our analysis may also be applied to nonlinear flows. For instance, in Chac´on Rebollo and Dom´ınguez Delgado [11], it is applied to the analysis of the approximation of Navier-Stokes equations by stabilized methods, in parallel to the analysis of their approximations by mixed methods. Stability and error estimates are derived. This analysis also applies to nonlinear stabilized methods, such as the optimal one introduced in Russo [26]. Up to our knowledge, the standard analysis is unable to handle nonlinear stabilization, which turns out to be rather simple to manage with our technique. Another possible application is the analysis of the solution of Oseen equations by the reduced Q1/Q1stabilized methods introduced in Knobloch and Tobiska [22]. This is a new family of computationally cheap methods that may be directly analyzed with our analysis. In fact, all hypothesis of Theorem 1 are readily proved to be satisfied, using the analysis developped in that paper. We would like to point out that the analysis technique that we introduce is rather complex from a technical point of view. However, we think that it is worth to be used, as it essentially reduces the difficulties of the analysis of stabilized methods to that of mixed method. Moreover, we have tried to present the technique in a systematic way, so that it may be applied to situations other than the considered here, with relative ease. The paper is organized as follows. In Section 2 we introduce an abstract discretization of Oseen equations, whose stability is analyzed in Section 3. In Section 4, we apply the abstract theory to stabilized methods. Section 5 is devoted to the analysis of spectral element stabilized methods. Finally, in Section 6 we solve a linear model of primitive equations of the ocean by stabilized finite elements. 2. Abstract discretization In this Section we introduce an abstract discretization of Oseen equations which is the base of our analysis. Let us consider a connected bounded domain Ω ⊂Rd(d= 2 or 3), with Lipschitz-continuous boundary Γ. We are given a “driving” velocity field u:Ω−→ Rd, that we assume to be divergence-free. Our purpose is to solve numerically the following boundary value problem:    Find y:Ω−→ Rd,p:Ω−→ Rsuch that u·∇y−ν∆y+∇p=f,∇·y=0 inΩ, y=0 onΓ. (1) UNIFIED MIXED AND STABILIZED SOLUTIONS 59 Here, ν>0 is the viscosity coefficient, and f∈H−1(Ω)dis a given source term. Only homogeneous Dirichlet boundary conditions are considered, in order to not introduce nonessential difficulties in our derivation. Let us define the bilinear form on H1 0(Ω)d×H1 0(Ω)d, a(w,v)=(u·∇w,v)+ν(∇w,∇v),∀w,v∈H1 0(Ω)d,(2) where we denote by (·,·)theL2scalar product, either for scalar, vector or tensor functions. If we assume that u∈[Lp(Ω)]dfor some p>d,and∇·u=0,thena(·,·) is well defined and is continuous and H1 0(Ω)d-elliptic; i.e., it verifies a(w,v)≤M(u)|w|1|v|1,a(v,v)≥ν|v|2 1∀v,w∈H1 0(Ω)d.(3) Here, we have denoted by |·| 1the H1(Ω)dseminorm. Also, M(u)=C(kuk0,p +ν) for some constant C appearing from Sobolev injections, where k·k0,p denotes the Lpnorm. The form a(·,·) defines a linear bounded operator Afrom H1 0(Ω)dinto H−1(Ω)d,givenby hAw,vi=a(w,v),∀w,v∈H1 0(Ω)d. Thus, Aw=u·∇w−ν∆w. The standard mixed formulation of problem (1) reads as follows: (Obtain (y,p)∈H1 0(Ω)d×L2 0(Ω) such that B(y,p;v,q)=hf,vi,∀(v,q)∈H1 0(Ω)d×L2 0(Ω); (4) where B(y,p;v,q)=a(y,v)−(p, ∇·v)−(∇·y,q). Also, h·,·i stands for the H−1(Ω)d−H1 0(Ω)dduality, and L2 0(Ω) is the subspace of L2(Ω) given by L2 0(Ω) = {q∈L2(Ω) such that ZΩ qdx=0}· The pair of spaces (H1 0(Ω)d,L 2 0(Ω)) verifies the continuous inf-sup condition (cf. Girault and Raviart [17]). Then, due to properties (3), problem (4) has a unique solution that depends continuously on the data f. In order to describe our abstract discretization of problem (4) we shall consider two families of subspaces {Yh}h>0and {Zh}h>0of H1 0(Ω)dand another family of subspaces {Mh}h>0of L2 0(Ω), all of them of finite dimension. These spaces may be, for instance, standard finite element spaces. We shall also consider a family of bilinear continuous forms on H1 0(Ω)d×H1 0(Ω)d,{Sh(·,·)}h>0. These forms are assumed to be coercive in H1norm on Zh. We shall denote by Rhthe “static condensation” operator Rh:H−1(Ω)d→Zh, defined as follows. Given ϕ∈H−1(Ω)d,Rh(ϕ) is the only element of Zhthat satisfies Sh(Rh(ϕ),zh)=hϕ, zhi,∀zh∈Zh.(5) 60 T. CHAC ´ ON REBOLLO We discretize problem (4) by Obtain (yh,p h)∈Yh×Mhsuch that Bh(yh,p h;vh,q h)=Fh(vh,q h),∀(vh,q h)∈Yh×Mh;(6) where Bh(w,r;v,q)=B(w,r;v,q)−Sh(Rh(Bv+∇q),Rh(Aw+∇r)) ; Fh(v,q)=hf,vi−Sh(Rh(Bv+∇q),Rh(f)) ; where Bdenotes the operator Bw=−u·∇w+εν∆w,∀w∈H1 0(Ω)d, foragivenε∈R. We shall use method (6) as an abstract framework to analyze various standard stabilized methods. To describe these methods, we shall consider affine-equivalent finite element spaces, as described in Hughes, Franca and Balestra [21]. Assume that the domain Ω is polyhedric. Let us consider a triangulation Thof Ω formed by either simplicial or parallelepipedic elements. We assume that the elements of Thare affine-transformed of a reference element K∗(either the unit simplex or parallelepiped), in the sense of Ciarlet [13]. Given an integer number k≥0, and an element K∈T h, denote by Pk(K) the space of polynomials of degree smaller than, or equal to, k, defined on K. Also, denote by Qk(K) the space of polynomials of degree smaller than, or equal to, k, in each variable, defined on K. Denote by Rk(K)eitherPk(K), if Kis a triangle or tetrahedron, or Qk(K) if Kis a quadrilateral or hexaedron. Given two integer numbers m≥1, l≥0, consider the following finite element spaces. Y(m) h=nv∈H1 0(Ω)d|v|K∈[Rm(K)]d,∀K∈T ho;(7) M(l) h=q∈L2 0(Ω) |q|K∈Rl(K),∀K∈T h,(8) or M(l) h=q∈L2 0(Ω) ∩C0(Ω) |q|K∈Rl(K),∀K∈T h.(9) We consider the following stabilized methods. (Find (yh,p h)∈Y(m) h×M(l) hsuch that BS(yh,p h;vh,q h)=FS(vh,q h),∀(vh,q h)∈Y(m) h×M(l) h;(10) where BS(w,r;v,q)=B(w,r;v,q)−X K∈Th τK(Bv+∇q;Aw+∇r)K; FS(v,q)=hf,vi− X K∈Th τK(Bv+∇q,f)K, where the τKare given stabilizing coefficients, and (·,·)Kdenotes the inner product in L2(K)d.When l=m= 1, method (10) is independent of the actual value of the coefficient ε, and it is known as Streamline UNIFIED MIXED AND STABILIZED SOLUTIONS 61 Upwind/Petrov-Galerkin (SUPG) method. For other values of m≥1andl≥0, when ε=−1,0and1, method (10) is respectively known as Adjoint stabilized (AdS), generalized SUPG and Galerkin-Least Squares (GaLS) method. Typically, the coefficients τKare continuous functions of the local P´eclet number on element K, PeK=UKhK νwith UK=ZK|u|p1/p ; τK(PeK)=AhK UK min(PeK,P)=       Ah2 K νif PeK≤P, AP hK UK if PeK>P; (11) where Ais a numerical constant and Pis a preset threshold for the P´eclet number. This allows on one hand to introduce some suitable stabilization of high frequence components of the transport operator (of order hK), due to convection dominance (Large PeK). Also, this introduces low levels of numerical diffusion (of order h2 K) in regions where diffusion is dominant (Low PeK). On the other hand, this stabilizes the spurious modes of the pressure gradient. Also, for reasons of computability, in practice the convection velocity uis replaced in the stabilizing terms by some stable interpolate uh∈Y(m) h. We shall assume it so in our analysis. The standard analysis of stabilized methods, summarized in Franca, Hughes and Stenberg [16], states that SUPG and GaLS methods are stable for any positive coefficients τK, and that AdS and generalized SUPG methods are stable if the τKare small enough. The obtention of optimal bounds for these coefficients to ensure stability requires the computation of the best constant CIin the inverse inequality CIX K∈Th h2 Kk∆vhk2 K≤k∇vhk2 0,∀vh∈Y(m) h.(12) That analysis applies to either continuous pressures combined with velocities of arbitrary interpolation degree, or to discontinuous pressures combined with high-degree interpolation velocities. Concretely, it holds under the following condition: Either M(l) h⊂C0(¯ Ω),or m≥n, (13) where n=dif This formed by triangles or tetrahedra, and 2ifThis formed by quadrilaterals or hexaedra. In Tobiska and Verf¨urth [27] this restriction is removed by introducing in the structure of the method some additional terms that take into account interelement pressure jump terms. However, it seems that method (10), without these jump terms, is not able to stabilize the discretization of discontinuous pressures combined with low-degree velocities. In this paper we shall analyze methods satisfying condition (13). Our analysis also applies to general discretizations that do not necessarily satisfy this condition. However, its proof requires a rather lengthy derivation that shall appear in a forthcoming paper. Notice that method (6) applies to general internal approximations of H1 0(Ω)dand L2 0(Ω), while stabilized methods only apply to approximations by piecewise smooth functions. We are, thus, considering a genuine generalization of stabilized methods. In the next two Sections we first develop a stability and convergence analysis for the abstract method (6) which extends the standard analysis of mixed methods, and next apply it to analyze the stabilized methods (10). 62 T. CHAC ´ ON REBOLLO 3. Analysis of abstract method In this section we prove that the stability of the abstract method (6) follows from a discrete inf-sup BrezziBabuˇska condition, similarly to mixed methods. The stability of abstract method (6), in addition to the inf-sup condition, requires the following hypotheses on the new elements appearing in method (6): Hypothesis 1. There exists a constant C0>0 independent of hsuch that |yh|1+|zh|1≤C0|yh+zh|1,∀yh∈Yh,zh∈Zh,∀h>0.(14) Hypothesis 2. There exist two constants νs>0,M s>0 such that |Sh(wh,vh)|≤Ms|wh|1|vh|1,Sh(vh,vh)≥νs|vh|2 1,∀wh,vh∈Zh. Both hypotheses play a crucial role in the obtention of estimates for both velocity and pressure, and thus in the proof of stability of method (6). Hypothesis 1 is a generalization of the well known H1 0-orthogonality between piecewise affine and bubble finite elements. Hypothesis 2 is a generalization of the fact that the stabilizing coefficients in (11) are of order h2 K. Let us recall the definition of stability for method (6) (cf. Babuˇska [3], Brezzi [7]): Definition 1. Method (6) is said to be stable on Yh×Mhif there is a constant γ>0 independent of hsuch that for any (w,r)∈Yh×Mh, sup (v,q)∈Yh×Mh (v,q)6=(0,0) Bh(w,r;v,q) |v|1+kqk0≥γ(|w|1+krk0); sup (v,q)∈Yh×Mh (v,q)6=(0,0) Bh(v,q;w,r) |v|1+kqk0≥γ(|w|1+krk0).  We now state our basic stability result. Theorem 1. Assume that the pairs of spaces {(Yh+Zh,M h)}h>0satisfy a uniform discrete Brezzi-Babuˇska condition, and that Hypotheses 1 and 2 hold. Assume that at least one of the two following sentences hold: i) Zhand Yhare orthogonal with respect to the H1 0(Ω)dinner product and νs>0,or ii) νs≥1−ε 22 ν,whenε6=1,orνs>0when ε=1. Then, the abstract method (6) is stable. From this theorem we deduce the main result of this paper: Theorem 2. Assume that the family of triangulations {Th}h>0is regular. Assume that condition (13) holds. Then, the stabilized method (10) coincides with an abstract method (6) constructed with a finite element space Zhof bubble functions and a bilinear form Sh, verifying 1. The pairs of spaces {Yh+Zh,M h}h>0satisfy a uniform discrete inf-sup condition. 2. The pairs of spaces {Yh,Z h}h>0satisfy Hypothesis 1. 3. The forms {Sh}h>0satisfy Hypothesis 2. UNIFIED MIXED AND STABILIZED SOLUTIONS 63 As a consequence, •GaLS and SUPG methods are stable for any A>0in (11) •The general stabilized method (10) is stable if A≤A02 ε−12 ,whereA0is a computable positive constant. In particular, AdS method is stable if A≤A0, and generalized SUPG method is stable if A≤4A0. Thus, under our analysis, the stability of stabilized methods follows from a discrete inf-sup condition, similarly to mixed methods. We shall prove this result in Section 4. In addition, we shall prove that the constant A0 depends on the aspect ratio of the grid and on the reference elements of spaces Yhand Mh,andshallgive computable fine estimates for this constant. Proof of Theorem 1. Velocity estimate. We shall treat separately cases i)andii). i) Assume that spaces Zhand Yhare orthogonal with respect to the H1 0(Ω)dinner product. In this case, all methods (10) coincide, independently of the actual value of ε, as such orthogonality implies Rh(∆w)=0,∀w∈ Yh. Consider a pair (wh,r h)∈Yh×Mh. Define ch=Rh(Awh+∇rh).As Rh(∆vh)=0,then ch=Rh(−Bwh+∇rh). Consequently, Bh(wh,r h;wh,−rh)=a(wh,wh)+Sh(ch,ch)≥ν|wh|2 1+νs|ch|2 1. ii) Consider a pair (wh,r h)∈Yh×Mh. Define ch=Rh(Awh+∇rh). Then, B(wh,r h;wh,−rh)=a(wh,wh)+Sh(ch,ch)+(1−ε)νSh(Rh(∆wh),ch) (15) =a(wh,wh)+Sh(ch,ch)−(1 −ε)ν(∇wh,∇ch) Due to Hypothesis 1, |(∇yh,∇zh)|≤(1 −δ0)|yh|1|zh|1,∀yh∈Yh,zh∈Zh,where δ0=2 C2 0·(16) Then, using Young’s inequality, (15) implies B(wh,r h;wh,−rh)≥˜ν|wh|2 1+˜νs|ch|2 1,(17) where ˜ν=ν[1 −(1 −δ0)|1−ε| 2µ],˜νs=νs−ν(1 −δ0)|1−ε| 2µ−1 for any µ>0. When ε=1,˜ν=νand ˜νs=νs>0. When ε6=1,wemaychoose ν νs |1−ε| 2(1 −δ0)<µ< 2 |1−ε|(1 −δ0)−1, and then ˜ν>0, ˜νs>0. Denote S=sup (v,q)∈Yh×Mh (v,q)6=(0,0) Bh(wh,r h;v,q) |v|1+kqk0· Then, in all cases ˜ν|wh|2 1+˜νs|ch|2 1≤(|wh|1+krhk0)S, (18) 64 T. CHAC ´ ON REBOLLO where for case i) we define ˜ν=νand ˜νs=ν. Pressure estimate. Consider a nonzero element vh∈Yh.Wehave (rh,∇·vh)=−Bh(wh,r h;vh,0) + a(wh,vh)−Sh(Rh(Bvh),ch).(19) Remark that Bh(wh,r h;−vh,0) ≤S|vh|1. Observe also that Sh(Rh(Bvh),ch)=hBvh,chi=−(u·∇vh,ch)−εν(∇vh,∇ch) ≤[M(u)+|ε−1|ν]|vh|1|ch|1. Consequently, (rh,∇·vh)≤{S+M(u)|wh|1+[M(u)+|ε−1|ν]|ch|1}|vh|1≤ ≤C1(S+|wh|1+|ch|1)|vh|1, where C1=max{1,M(u)+|ε−1|ν}. Also, given a nonzero element zh∈Zh, (rh,∇·zh)=−h∇rh,zhi=−Sh(Rh(∇rh),zh) =Sh(Rh(Awh),zh)−Sh(ch,zh) (20) ≤M s[|Rh(Awh)|1+|ch|1]|zh|1 ≤M sν−1 s|Awh|−1+|ch|1|zh|1 ≤C2(|wh|1+|ch|1)|zh|1, where C2=Msmax{ν−1 sM(u),1}. Then, using Hypothesis 1, (rh,∇·(zh+vh)) ≤C3(S+|wh|1+|ch|1)(|vh|1+|zh|1) ≤C0C3(S+|wh|1+|ch|1)|vh+zh|1,(21) where C3=max{C1,C 2}. Now, we use the discrete inf-sup condition: There exists a constant α>0 such that αkqhk0≤sup xh∈Yh+Zh (qh,∇·xh) |xh|1 ,∀qh∈Mh. Therefore, krhk0≤C4(S+|wh|1+|ch|1),(22) where C4=α−1C0C3. Conclusion. Combining (18) and (22) and applying Young’s inequality yields ˜ν|wh|2 1+˜νs|ch|2 1≤C4S2+[(1+C4)|wh|1+C4|ch|1]S ≤1 2[(1 + C4)ε1|wh|2 1+C4ε2|ch|2 1] +[C4+1 2(1 + C4)ε−1 1+1 2C4ε−1 2]S2, for any ε1>0, ε2>0. Let us take ε1=˜ν 1+C4 ,ε2=˜νs C4 . Then, ˜ν|wh|2 1+˜νs|ch|2 1≤C2 5S2,(23) UNIFIED MIXED AND STABILIZED SOLUTIONS 65 where C5=2C4+(1 + C4)2 ˜ν+C2 4 ˜νs1/2 .Thus, |wh|1≤C5 √˜νS, |ch|1≤C5 √˜νs S. (24) Combining now (23) with (22), we obtain krhk0≤C6S, where C6=C4+C4C51 √˜ν+1 √˜νs.(25) From (24) and (25) we finally deduce S≥γ(|wh|1+krhk0+|ch|1),where γ=C6+C5 √˜ν+C5 √˜νs−1 ·(26) The proof of the second inequality in Definition 1 follows from similar arguments.  The following result closes the equivalence between discrete inf-sup condition and stability of method (6). Thus, the stability analysis of mixed method and method (6) are fully parallel. Theorem 3. Assume that abstract method (6) is stable for some νs>0. Assume that Hypothesis 1 and 2 hold. Then, the pairs of spaces {Yh+Zh,M h}h>0satisfy the discrete Brezzi-Babuˇska condition. We omit the proof of this result as it again follows from arguments similar to those used in the proof of Theorem 1. The stability of form Bhyields the well-possedness of our method, and allows to derive error estimates, similarly to the standard analysis of mixed methods: Corollary 1. Under the hypotheses of Theorem 1, problem (6) admits a unique solution (yh,p h)∈Yh×Mh, that verifies, for some constant C>0, |yh|1+kphk0+|zh|1≤Ckfk−1,(27) and |y−yh|1+kp−phk0+|zh|1≤Cinf vh∈Yh|y−vh|1+inf qh∈Mhkp−qhk0,(28) where zh=Rh(Ayh+∇ph−f). Remark 1. From this result, the “bubble” space Zhappears as a control space for high-frequency components of the residual Ayh+∇ph−f. In fact, (28) shows that the high frequency components of the residual which are representable on Zh,via the condensation operator Rh, are bounded. 4. Application to stabilized methods In this section we prove that stabilized methods (10) may be formulated as particular cases of abstract method (6), and then apply the general stability analysis of Section 3. Our derivation starts from the construction of virtual bubbles developped in Baiocchi et al. [4]. Let us recall the main result of that paper, that we adapt to our context. Consider a Hilbert space ( H,(·,·)H). Given a subset Bof Hof finite dimension, we define the abstract static condensation operator R:H0→Bby: Given ϕ∈H0,R(ϕ) is the only element of Bthat satisfies (R(ϕ),ζ)H=hϕ, ζi,∀ζ∈B. 72 T. CHAC ´ ON REBOLLO where BH(wH,r H;vH,q H)=B(wH,r H;vH,q H)−X K∈Th τK(BvH+∇qH;AwH+∇rH)N,K; FH(vH,q H)=hf,vHi− X K∈Th τK(BvH+∇qH,f)N,K. The essential difference between SSE method and stabilized method (10) is that the L2inner products (·,·)K that appear in (10) in the stabilizing terms are here replaced by the discrete inner products (·,·)N,K. In Gervasio and Saleri [19], the discrete inner product (·,·)His also used to approximate the integral terms appearing in form B. Here, for simplicity we prefer to consider the above discretization. However, we may extend our analysis to the actual discretization considered in that paper if the pressures are approximated by piecewise polynomials of degree at most N−1(seeRem.5). In Gervasio and Saleri [19], the stabilizing coefficients τKare still given by (11), with P=2N2 m,A=m 4N4,for some m>0. The parameter mis determined in that paper in order to obtain uniform-in-time stability of the linear problems that arise after time discretization. We shall simply assume that the stabilizing coefficients τKare given by (11). Our analysis allows to state the following result: Theorem 5. Assume the triangulations {Th}h>0are regular. Then, the SSE method (43) is stable for any A>0if ε=1, and for 0<A<2 1−ε2 ˆ A0if ε6=1,where ˆ A0is a computable positive constant. As a consequence, if f∈C0(Ω)d, problem (43) admits a unique solution that satisfies |yH|1+kpHk0≤CkfkC0,(44) for some constant C>0independent of H. Proof. We proceed as in the proof of Theorem 2. Step 1: Embedding of SSE method in abstract method. Let us define the local interpolation operator IK N:C0(K)→QN(K)by (IK Nw)(P(K) ijk )=w(P(K) ijk ),i,j,k=1,···,N +1. Consider the space of piecewise continuous functions on Th, Cp,h(Ω) = {v∈L2(Ω) |v|K∈C0(K),∀K∈T h}; and define the global interpolation operator IH:Cp,h(Ω) →WHby (IHw)|K=IK N(w|K),∀K∈T h. Observe that C0(¯ Ω) ⊂Cp,h(Ω) and that IHw∈VHif w∈C0(¯ Ω). UNIFIED MIXED AND STABILIZED SOLUTIONS 73 The following representation formula holds: Lemma 4. There exists a finite-dimensional bubble finite element space ZH⊂H1 0(Ω)dsuch that , Sh(RH(IHv1),RH(IHv2)) = X K∈Th τK(v1,v2)N,K,∀v1,v2∈[Cp,h(Ω)]d; (45) where Shis the bilinear form defined by (32). This lemma is proved in the Appendix. As a consequence, for all wH,vH∈YH;rH,qH∈MH, BH(wH,r H;vH,q H)=B(wH,r H;vH,q H) (46) −Sh(RH(IH(BvH+∇qH)),RH(IH(AwH+∇rH))); FH(vH,q H)=hf,vHi−Sh(RH(IH(BvH+∇qH)),RH(IHf)) . This occurs because f∈C0(Ω)dand BvH+∇qH,AwH+∇rH∈[Cp,h(Ω)]d. Steps 2 and 3: Proof of Hypotheses 1 and 2. Hypotheses 1 and 2 have respectively been proved in the Steps 3 and 4 of the proof of Theorem 2. Step 4: Discrete inf-sup condition. Observe that if rH∈MH,thenIH(∇rH)=∇rH, because IK N(qN)=qN,∀qN∈QN(K). Then, by (45), Sh(RH(∇rH),RH(∇rH)) = X K∈Th τK(∇rH,∇rH)N,K,∀rH∈MH. By Bernardi and Maday [5], kqNk0,K ≤kqNkN,K ≤3kqNk0,K,∀qN∈QN(K).(47) These estimates are obtained by affine transportation of similar estimates obtained in the reference element. As the coefficients τKare of order h2 K, then there exists a constant C>0 such that X K∈Th h2 Kk∇rHk2 0,K ≤CSh(RH(∇rH),RH(∇rH)) . Then, by Lemma 3, the pairs of spaces {YH+ZH,M H}H>0satisfy the discrete inf-sup condition. Step 5: Conclusion. Following the proof of Theorem 1, we prove that if νs≥(1 −ε)2 4νwhen ε6=1,orνs>0whenε=1,then the form BHis stable. Then, problem (43) admits a unique solution that satisfies, for some constant C>0, |yH|1+kpHk0+|zH|1≤C(kfk−1+|RH(IHf)|1), where zH=RH(IH(AyH)+∇pH). As f∈C0(Ω)d, using (47), |RH(IHf)|1≤ν−1 skIHfk0≤ν−1 skIHfkH=ν−1 skfkH≤CkfkC0. Thus, estimate (44) follows. The remaining of the proof is similar to the conclusion of the proof of Theorem 2.  74 T. CHAC ´ ON REBOLLO Remark 5. A slight modification of the above argument allows to prove an underlying inf-sup condition and thus the stability for a stabilized full spectral element discretization of Oseen equations. Indeed, let us replace the pressure space MHby MH0, with H0=(h, (N−1)−1), for N≥2; i.e.,weconsider pressures of degree at most N−1 elementwise. We consider the following discrete problem: Obtain (yH,p H0)∈YH×MH0such that B0 H(yH,p H0;vH,q H0)=F0 H(vH,q H0),∀(vH,q H0)∈YH×MH0;(48) where B0 H(wH,r H0;vH,q H0)=1 2[(u·∇wH,vH)H−(u·∇vH,wH)H]+ν(∇wH,∇vH)H −(∇·wH,q H0)H−(rH0,∇·vH)H −X K∈Th τK(BvH+∇qH0;AwH+∇rH0)N,K; F0 H(vH,q H0)=(f,vH)H−X K∈Th τK(BvH+∇qH0,f)N,K. Then, our analysis allows to prove that the form B0 His stable. This holds because the quadrature formula ZK∗ gdx∗≃ N X i,j,k=0 ωiωjωkg(ξi,ξ j,ξ k) is exact for g∈Q2N−1(K∗). 6. Solution of linear primitive equations In this section we apply our analysis to the solution of a linear model for the primitive equations of the ocean by a penalty stabilized technique. This model includes the main difficulty of these equations: The vertical convection is degenerated. This makes the pressure to be only in some space Lpfor 1 <p<2. We prove a discrete inf-sup condition in this norm, and prove the convergence of the approximated solutions to a weak solution of the continuous problem. To describe our model equations, let us consider a connected 2D bounded domain ω⊂R2, and a piecewise continuous function D:¯ω→Rsuch that D(x)>0, ∀x=(x1,x 2)∈ω. This function represents the sea depth. We consider the domain Ω={(x,z)∈R3|x∈ω, −D(x)<z<0}, which is intended to represent a piece of the ocean with flat surface. To avoid some technical complexities, we shall assume that ωis polygonal and Dis piecewise affine on some triangulation of ¯ω, so that Ω is polyhedric. Our analysis can be extended to piecewise C1depth functions, similarly to the analysis of the approximation of primitive equations by mixed methods (cf. Chac´on Rebollo and Guill´en Gonz´alez [12]). We assume the domain Ω to be Lipschitz continuous. This occurs, for instance, if the normal derivative of Dsatisfies ∂D ∂n ≤αfor some α<0 a. e. on the part of ∂ω where D=0. NoticethatDmay be zero partially or totally on ∂ω. Also, that we allow the sea bottom to have vertical walls when Dhas a jump, and sidewalls if D>0 on a part of ∂ω. We also consider the following subsets of ∂Ω: Γs={(x,0) ∈R3|x∈¯ω},(sea surface), Γb=∂Ω−Γs(sea bottom and, eventually, sidewalls). UNIFIED MIXED AND STABILIZED SOLUTIONS 75 We assume known a convection velocity W=(w1,w 2,w 3)onΩ,suchthat (w=(w1,w 2)∈H1(Ω)2,w 3∈L2(Ω), ∇·W=0 inΩ,w 3|Γs=0,w 3·n3|Γb=0,w|Γb=0,(49) where n3denotes the third component of the outward normal to ∂Ω, n=(n1,n 2,n 3). We are thus forcing the incompressibility of the sea water (Boussinesq’s hypothesis). The first boundary condition means that we assume the sea surface to not move in the vertical direction (rigid lid hypothesis), while the second and third ones are rather technical boundary conditions, meaning that we treat the whole Γbas a solid wall. We also assume known a distributed source term f, representing the effects of temperature, salinity and Coriolis force (assumed to be constant on the whole domain for simplicity), and a “surface wind tension” g.We set the following problem:                Obtain y:¯ Ω→R2,y=(y1,y 2),(horizontal velocity) and p:ω→Rsuch that (surface pressure) W·∇y−ν∆y+∇Hp=fin Ω, ∇H·hyi=0 inω, y|Γb=0,ν ∂y ∂n|Γs =g. (50) Here, ∇H=(∂1,∂ 2) stands for the horizontal gradient, and the symbols h·i denote vertical mean, hyi(x)=Z0 −D(x) y(x,z)dz, for x∈ω. In this problem the surface pressure pacts as a Lagrange multiplier associated to the condition ∇H·hyi=0. Problem (50) is a reduced version of a linear model of the primitive equations of the ocean (introduced in Lions, Temam and Wang [24]), that reads as follows:                        Obtain a velocity field (y,y 3):¯ Ω→R3, and a pressure P:Ω→Rsuch that W·∇y−ν∆y+∇HP=fin Ω, ∇·(y,y 3)=0 in Ω, ∂3P=−ρg in Ω, y|Γb=0,ν ∂y ∂n|Γs =g, y3·n3|Γb=0,y 3|Γs=0. (51) Here, ρrepresents the sea water density, assumed to be constant, and gthe acceleration of the gravity. This model is formally obtained from the Navier-Stokes equations by neglecting in the vertical momentum equation all forces (convection, diffusion and Coriolis) but the gravity. This leads to the hydrostatic pressure approximation. A rigorous derivation of this approximation is found in Besson and Laydi [2], as an asymptotic limit as the ratio between vertical and horizontal dimensions tends to zero. The physically meaningful – nonlinear – problem would be to find a “fixed point” of equations (50), in the sense that y=w. This justifies the choice of regularity and boundary conditions satisfied by w(see (49)). Equations (50) may be viewed as a model problem for the nonlinear primitive equations, much as the Oseen equations are a linear model for the Navier-Stokes equations. 76 T. CHAC ´ ON REBOLLO Remark 6. Problems (50) and (51) are equivalent. The key point for this equivalence is the following: If a horizontal velocity y=(y1,y 2)∈H1(Ω)2satisfies y|Γb=0,then h∇H·yi=∇H·hyi. As a consequence, there exists a vertical velocity y3∈L2(Ω) such that ∇·(y,y 3)=0in Ω,y 3|Γs=0and y3·n3|Γb=0 if and only if y3(x,x 3)=Z0 x3∇H·y(x,s)dsin Ω,(52) and ∇H·hyi=0 in ω. This allows to eliminate the vertical velocity y3from problem (51). Also, the condition ∂3P=−ρg allows to recover the pressure Pfrom the surface pressure p,by P(x,z)=ρgz+p(x).(53) A rigourous proof of this equivalence may be found in Lewandowski [23]. To give a variational formulation to problem (51), let us define the spaces Vk={v=(v1,v 2)∈W1,k(Ω)2|v|Γb=0}for k≥1,integer; Lα D(ω)={q:ω→Rmeasurable such that Zω D(x)|q(x)|αdx<+∞}for α≥1; Lα D,0(ω)=Lα D(ω)/R(quotient space). Spaces Lα D(ω)andLα D,0(ω) are Banach spaces – reflexive if 1 <α<+∞–, respectively endowed with the norms kqkLα D(ω)=Zω D(x)|q(x)|αdx1/α , kqkLα D,0(ω)=inf c∈Rkq+ckLα D(ω). Space Lα D(ω) is isomorphic, and, more specifically, isometric, to the space Lα(∂3,Ω) = {q∈Lα(Ω) such that ∂3q=0}. Indeed, we identify each q∈Lα D(ω) with its extension to Ω as a constant function with respect to the x3variable. Then, we have kqkLα D(ω)=kqkLα(Ω). Moreover, if we consider the space Lα 0(∂3,Ω) = Lα(∂3,Ω)/R,thenLα D,0(ω)andLα 0(∂3,Ω) also are isomorphic, and kqkLα D,0(ω)=kqkLα 0(Ω),∀q∈Lα D,0(ω). We further assume f∈V0 2and g∈H−1/2(Γs)d, the dual space of [H1/2(Γs)]d. This space is well defined as Γsis C∞. UNIFIED MIXED AND STABILIZED SOLUTIONS 77 We consider the following weak formulation of problem (51): (Obtain (y,p)∈V2×L3/2 D,0(ω) such that B(PE)(y,p;v,q)=F(v); ∀(v,q)∈V4×L2 D,0(ω); (54) where B(PE)(y,p;v,q)=hW·∇y,viV0 4−V4+ν(∇y,∇v)Ω−(p, ∇H·hvi)ω −(∇H·hyi,q)ω, F(v)=hf,viV0 2−V2+hg,viH−1/2(Γs)−H1/2(Γs). This form is well defined, due to the following: Lemma 5. The following statements hold. i) Consider a function W=(w,w 3)∈V2×L2(Ω) such that ∇·W=0,w3|Γs=0.Then,∀u∈V2, W·∇u∈V0 kfor k≥3,and kW·∇ukV0 k≤ˆ Ck|w|1,Ω|u|1,Ω,(55) for some constant ˆ Ck>0. ii) If w∈Vkfor some k≥1,thenhwi∈[W1,k(ω)]2and ∂ihwi=h∂iwi,i=1,2. Proof. i) Observe that, given w∈V2,andw3∈L2(Ω) such that ∂3w3=−∇H·w,andw3|Γs=0wehave w3(x,x 3)=Z0 x3∇H·w(x,s)ds.Thus, kw3k0,Ω+k∂3w3k0,Ω≤C1|w|1,Ω,(56) for some constant C1>0. Now, if Wis smooth, we see by integrations by parts that for u∈V2and v∈Vk, ZΩ (W·∇u)·vdxdx3=ZΩ [(w·∇Hu)·v−∂3w3u·v−w3u·∂3v]dxdx3. Then, we may define the duality hW·∇u,viby hW·∇u,vi=ZΩ [(w·∇Hu)·v−∂3w3u·v−w3u·∂3v]dxdx3.(57) Using (56), |hW·∇u,vi| ≤ C2(kwk0,4;Ω|u|1,Ωkvk0,4;Ω +|w|1,Ωkuk0,4;Ωkvk0,4;Ω (58) +kwk0,Ωkuk0,6;Ωk∂3vk0,3;Ω )≤ˆ Ck|w|1,Ω|u|1,Ω|v|1,k;Ω. This proves that W·∇u∈V0 k. Next, consider a field W=(w,w 3)∈V2×L2(Ω) with ∇·W=0,w3|Γs=0. Then, there exists a sequence {wn}n≥1⊂[D(¯ Ω)]2such that wn=0onΓ b, which converges to win V2. This is proved by a standard argument (for instance, by symmetrization with respect to Γs)usingthat∂Ωis Lipschitz-continuous. Let Wn=(wn,w 3n), with w3n(x,x 3)=Z0 x3∇H·wn(x,s)ds. 78 T. CHAC ´ ON REBOLLO Following Dautray and Lions [18], Chapter XXI, we may ensure that if a function z∈L2(Ω) is such that ∂3z∈L2(Ω), then the trace of zon Γsbelongs to H1/2(Γs). Moreover, a Poincar´e inequality holds if z|Γs=0: kzk0,Ω≤C3k∂3zk0,Ω, for some constant C3>0. Therefore, kw3−w3nk0,Ω≤C3k∇H·(w−wn)k0,Ω, and w3nconverges to w3in L2(Ω). Thus, we may pass to the limit in the r.h.s. of (57), and define W·∇uas a linear form on Vk. Now, passing to the limit in (58) we deduce W·∇u∈V0 kand estimate (55). ii) Consider w∈Vk.Asw∈[Lk(Ω)]2, one readily proves hwi∈[Lk(ω)]2. Also, let ϕ∈D(ω). Then, if wis smooth, for i=1,2, Zωh∂iwi(x)ϕ(x)dx=ZΩ ∂iw(x,x 3)ϕ(x)dxdx3= (59) =Z∂Ω niwϕd(∂Ω) −ZΩ w(x,x 3)∂iϕ(x)dxdx3= (60) =−Zωhwi(x)∂iϕ(x)dx,=−Zω ∂ihwi(x)ϕ(x)dx,(61) as ni=0onΓ sand w=0onΓ b.Thus,∂ihwi=h∂iwiand w∈[W1,k(ω)]2. If wis any element of Vk, the same results follows from a density argument similar to that of the proof of statement i) above.  Remark 7. Any solution (y,p) of problem (54) is a weak solution of problem (50) in the distribution sense. Furthermore, if we recover the vertical velocity y3by (52), and the physical pressure Pby (53), then the couple ((y,y 3),P) is a solution of problem (51) in the distribution sense. We shall discretize problem (54) by a penalty stabilized method, of Brezzi and Pitk¨aranta’s kind (cf. [9]). Consider a triangulation Chof ¯ωsuch that Dis affine on each triangle T∈C h. Consider also a partition Phof ¯ Ω by sets of the form PT={(x,x 3)∈R3,such that x∈T, −D(x)≤x3≤0}for some triangle T∈C h. Notice that if a triangle T∈C his not adjacent to ∂ω, or if it is adjacent to ∂ω and D>0on ¯ T,thenits associated set PTis a triangular prism with upper base T×{0}and possibly non-horizontal lower base. However, if Tis adjacent to ∂ω and D= 0 on a part of ∂T,thenPTis a non-prismatic polyhedron. We shall consider a triangulation Thof Ω constructed by subdividing each element Phinto tetrahedra. Let us define the finite element spaces, Vh={vh∈C0(¯ Ω) |vh|K∈P1(K),∀K∈T h}; (62) Yh={vh∈V2 h|vh|Γb=0}; ˜ Nh={qh∈C0(¯ω)|qh|T∈P1(T),∀T∈C h};Nh=˜ Nh/R. We introduce the following discretization of (54): Obtain (yh,p h)∈Yh×Nhsuch that B(PE) h(yh,p h;vh,q h)=F(vh); ∀(vh,q h)∈Yh×Nh;(63) UNIFIED MIXED AND STABILIZED SOLUTIONS 79 where B(PE) h(uh,r h;vh,q h)=B(PE)(uh,r h;vh,q h)+ X K∈Th τ(c) K(Wh·∇uh,Wh·∇vh)K −X T∈Ch τ(p) T(∇Hrh,∇Hqh)T. The stabilizing coefficients for convection τ(c) Kare assumed to be still given by (11). This will provide some stabilization of the convective derivative. Also, to ensure the stability of the pressure discretization we shall assume that the stabilizing coefficients for pressure τ(p) Tsatisfy the following condition: There exist two constants β1>0,β 2>0 such that β1h2 TZT Ddx |T|≤τ(p) T≤β2h2 TZT Ddx |T|,∀T∈C h.(64) Observe that these inequalities make sense as we assume D>0onω. In the stabilizing terms of (63), we replace the convection velocity W=(w,w 3)bysomeinterpolateWh=(wh,w 3h)∈Yh×Vh, satisfying for some constant C>0, |Wh|1≤C|w|1.(65) We now state the main result of this section. Theorem 6. Assume the convection velocity W=(w,w 3)lies in the space V2×L2(Ω) and verifies ∇·W=0, w3|Γs=0. Assume the triangulations {Th}h>0are regular. Then, the following statements hold. i) Problem (63) admits a unique solution (yh,p h)∈Yh×Nhwhich is bounded in V2×L3/2 D,0(ω). ii) The sequence {(yh,p h)}h>0contains a subsequence which is weakly convergent in V2×L3/2 D,0(ω)to a solution of (54) satisfying the estimate |y|1+kpkL3/2 D,0(ω)≤CkfkV0 2+kgk−1/2,Γs(1 + |w|1,Ω),(66) for some constant C>0independent of h. Proof. We proceed by steps. Step 1: Embedding of method (63) in abstract method. Given an element T∈C h, let us define τ(p) K=|T| ZT Ddx τ(p) T, for any element K∈T hthat be in the prism PT that lies on T. We assume that the pressures of Nhare defined on the whole Ω, as constant functions in the x3 80 T. CHAC ´ ON REBOLLO variable. Then, X K∈Th τ(p) K(∇rh,∇qh)K=X T∈Ch |T| ZT Ddx τ(p) TZPT∇Hrh·∇Hqhdxdx3 =X T∈Ch |T| ZT Ddx τ(p) T(∇Hrh)|T·(∇Hqh)|TZPT dxdx3 =X T∈Ch τ(p) T(∇Hrh,∇Hqh)T,∀rh,q h∈Nh.(67) Let us define Mh=Vh/R,whereVhis given by (62). We now apply Lemma 1: There exists a bubble finite element space B1h, generated on Thby a reference element B∗ 1⊂H1 0(K∗)3, and a bilinear coercive form S1h on H1 0(Ω)3, such that X K∈Th τ(p) K(∇rh,∇qh)K=S1h(R1h(∇rh),R1h(∇qh)),∀rh,q h∈Mh; (68) where R1his the static condensation operator on B1hwith respect to form S1h.WemayidentifyNhwith the subspace of Mhdefined by {qh∈Vh|∂3qh=0}.Then, from (67) and (68) we deduce X T∈Ch τ(p) T(∇Hrh,∇Hqh)T=S1h(R1h(∇rh),R1h(∇qh)),∀rh,q h∈Nh.(69) Also, again by Lemma 1, there exists a bubble finite element space B2h, generated on Thby a reference element B∗ 2⊂H1 0(K∗)2, and a bilinear coercive form S2hon H1 0(Ω)2, such that ∀uh,vh∈Yh, X K∈Th τ(c) K(Wh·∇uh,Wh·∇vh)K=S2h(R2h(Wh·∇uh),R2h(Wh·∇vh) ); (70) where R2his the static condensation operator on B2hwith respect to form S2h. Then, B(PE) h(uh,r h;vh,q h)=B(PE)(uh,r h;vh,q h) +S2h(R2h(Wh·∇uh),R2h(Wh·∇vh)) −S1h(R1h(∇rh),R1h(∇qh)),∀uh,vh∈Yh,∀rh,q h∈Nh. We recall that by Theorem (2) (Step 3), the forms {S2h}h>0are uniformly continuous and coercive in H1norm. Also, due to (64) and the regularity of triangulations Th, the coefficients τ(p) Kare of order h2 K. Then, the forms {S1h}h>0also are uniformly continuous and coercive. Step 2: Discrete inf-sup condition. We state the following: Lemma 6. Given α∈(1,2], there exists a constant Cα>0such that ∀qh∈Nh, CαkqhkLα D,0(ω)≤sup vh∈Yh−{0} (∇H·hvhi,q h)ω |vh|1,α0,Ω +[S1h(R1h(∇qh),R1h(∇qh))] 1/2,(71) where α0is the conjugate exponent of α. UNIFIED MIXED AND STABILIZED SOLUTIONS 81 Proof. Define the space Wh={(vh,v 3h)∈V3 h|(vh,v 3h)|∂Ω=0}. It is enough to prove that CαkqhkLα 0(Ω) ≤sup (vh,v3h)∈Wh−{0} (∇·(vh,v 3h),q h)Ω |(vh,v 3h)|1,α0,Ω (72) +[S1h(R1h(∇qh),R1h(∇qh))] 1/2,∀qh∈Vh. Indeed, if qh∈Nh,(vh,v 3h)∈Wh, (∇·(vh,v 3h),q h)Ω=(∇H·vh,q h)Ω−(v3h,∂ 3qh)Ω=(∇H·hvhi,q h)ω. Then, sup (vh,v3h)∈Wh−{0} (∇·(vh,v 3h),q h)Ω |(vh,v 3h)|1,α0,Ω =sup (vh,v3h)∈Wh−{0} (∇H·hvhi,q h)ω |(vh,v 3h)|1,α0,Ω ≤sup vh∈Yh−{0} (∇H·hvhi,q h)ω |vh|1,α0,Ω· Also, kqhkLα 0(Ω) =kqhkLα D,0(ω)if qh∈Nh. Thus, (71) follows from (72). To prove (72), consider qh∈Vh. As Ω is polyhedric, then ∂Ω is Lipschitz, and the continuous inf-sup condition in Lα(Ω) norm is satisfied (cf. Amrouche and Girault [1]): There exists a constant Dα>0 such that DαkqhkLα 0(Ω) ≤sup v∈ h W1,α0 0(Ω) i 3−{0} (∇·v,q h)Ω |v|1,α0,Ω ,∀q∈Lα 0(Ω). As [D(Ω)]3is dense in hW1,α0 0(Ω)i3, there exists a v0∈[D(Ω)]3such that 1 2DαkqhkLα 0(Ω) ≤(∇·v0,q h)Ω,|v0|1,α0,Ω=1. Following the standard finite elements interpolation theory (cf. Ciarlet [13]), there exists an interpolate v0h∈ Whsuch that |v0h|1,α0,Ω≤C1|v0|1,α0,Ω; (73) kv0h−v0k0,K ≤C1hK|v0|1,K,∀K∈T h; (74) for some constant C1>0 independent of h. Then, as qhis continuous, 1 2DαkqhkLα 0(Ω) ≤(∇·v0h,q h)Ω+(v0h−v0,∇qh)Ω ≤C1|(∇·v0h,q h)Ω| |v0h|1,α0,Ω +"X K∈Th h−2 Kkv0h−v0k2 0,K#1/2"X K∈Th h2 Kk∇qhk2 0,K#1/2 . As α0≥2, then (74) yields "X K∈Th h−2 Kkv0h−v0k2 0,K#1/2 ≤C2|v0|1,α0,Ω. 88 T. CHAC ´ ON REBOLLO where kAKkdenotes the spectral matrix norm. Then, kzhk2 0≤C∗γ2X K∈ThkAKk2|zh|2 1,K ≤C1h2|zh|2 1, for some constant C1>0. Let us now consider the interpolation estimate (cf. Br´ezis [6]), kwk0,q ≤C2kwkβ 0kwk1−β 0,6,∀w∈L6(Ω)d, if 2 ≤q≤6, with βgiven in (78). As H1 0(Ω)dis continuously embedded in L6(Ω)dif d=2ord=3, then (78) follows. ii) Consider a sequence {zh}h>0, with zh∈Zh. This sequence contains a subsequence, that we still denote in the same way, weakly convergent to some element zin H1 0(Ω)d.As H1 0(Ω)dis compactly embedded in L2(Ω)d, we may assume that this sequence converges strongly in L2(Ω)d. Recall that space Y(0) his defined by Y(0) h=nv∈L2(Ω)d|v|Kis constant,∀K∈T ho.(84) Due to standard finite element interpolation analysis, there exists a sequence {yh}h>0, with yh∈Y(0) h, strongly convergent to zin L2(Ω)d(even if the family of triangulations is not regular). Denote by Y∗the reference space that generates space Y(0) h. By hypothesis, Y∗∩Z∗={0}. Then, there exists ρ>0 such that |(z∗,y∗)K∗|≤(1 −ρ)kz∗k0,K∗ky∗k0,K∗,∀z∗∈Z∗,y∗∈Y∗. This is proved similarly to estimate (83) in Lemma 2. Thus, |(zh,yh)|=X K∈Th|det AK||(zK h,yK h)K∗|≤(1 −ρ)X K∈Th|det AK|kzK hk0,K∗kyK hk0,K∗ ≤(1 −ρ)kzhk0kyhk0. Consequently, z=0askzk2 0= lim h→0|(zh,yh)|≤(1 −ρ)kzk2 0. As the limit of any weakly convergent subsequence is necessarily zero, the the whole sequence {zh}h>0 converges weakly to zero.  References [1] C. Amrouche and V. Girault, Decomposition of Vector spaces and application to the Stokes problem in arbitrary dimensions. Czeschoslovak Math. J. 44 (1994) 109–140. [2] O. Besson and M. R. 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