Finite-element-based Faedo Galerkin weak solutions to the Navier Stokes equations in the three-dimensional torus are suitable

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1 J. Math. Pures Appl. 85 6) Finite-element-based Faedo Galerkin weak solutions to the Navier Stokes equations in the three-dimensional torus are suitable J.-L. Guermond 1 Department of Mathematics, Texas A&M University 3368 TAMU, College Station, TX , USA Received 17 November 4 Available online 16 November 5 Abstract It is shown in this paper that Faedo Galerkin weak solutions to the Navier Stokes equations in the three-dimensional torus are suitable provided they are constructed using finite-dimensional spaces having a discrete commutator property and satisfying a proper inf sup condition. Low order mixed finite element spaces appear to be acceptable for this purpose. This question was open since the notion of suitable solution was introduced. 5 Elsevier SAS. All rights reserved. Résumé Dans cet article il est montré que les solutions faibles de Faedo Galerkin des équations de Navier Stokes, en dimension trois dans le tore, sont acceptables si elles sont construites à partir d espaces de dimension finie possédant une propriété de commutateur discret et satisfaisant une certaine condition de compatibilité. Les espaces d éléments finis de bas degré satisfont ces hypothèses. Cette question était ouverte depuis l introduction de la notion de solution faible acceptable. 5 Elsevier SAS. All rights reserved. MSC: 35Q3; 65N35; 76M5 eywords: Navier Stokes equations; Suitable weak solutions; Faedo Galerkin approximation; Finite element approximation 1. Introduction 1.1. Position of the problem This paper is concerned with the regularity of weak solutions of the Navier Stokes equation in the threedimensional torus Ω: t u + u u + p ν u = f in Q T, u = inq T, 1.1) u t= = u, u is periodic, address: guermond@math.tamu.edu J.-L. Guermond). 1 On leave from LIMSI CNRS-UPR 351), BP 133, 9143, Orsay, France /$ see front matter 5 Elsevier SAS. All rights reserved. doi:1.116/j.matpur.5.1.4

2 45 J.-L. Guermond / J. Math. Pures Appl. 85 6) where Q T = Ω,T). Henceforth we assume f L,T;[H 1 Ω)] 3 ) and u H ={v L Ω) 3 ; v = ; v n is periodic}. To the present time, the best partial regularity result is the so-called Caffarelli ohn Nirenberg theorem [4,9] proving that the one-dimensional Hausdorff measure of the set of singularities of a suitable weak solution is zero. One intriguing hypothesis on which this result is based is that the weak solution must be suitable. The notion of suitable weak solution has been introduced by Scheffer [1] and boils down to the following: Definition 1.1 Scheffer). A weak solution to the Navier Stokes equation u, p) is suitable if u L,T;[H 1 Ω)] 3 ) L,T;[L Ω)] 3 ), p L 5/4 Q T ) and the local energy balance, ) ) ) ) t u + u + p u ν u + ν u) f u, 1.) is satisfied in the distributional sense. Although it has been proved recently by He Cheng [8] that the result of the CN theorem also holds for weak solutions it is not known whether indeed weak solutions are suitable. Two important questions arise a this points: 1) Are suitable weak solutions unique? ) Are the solution constructed by the Faedo Galerkin method suitable? see, e.g., [1], [, p. 77], [9, p. 45]). The purpose of the present work is to give a partial answer to the second question which seems to have been open since Scheffer introduced the notion of suitable solution. The main result of this paper is that, yes indeed, in the three-dimensional torus the Faedo Galerkin weak solutions to the Navier Stokes equations are suitable provided the finite-dimensional spaces involved in the construction have a discrete commutator property and satisfy a proper inf sup condition. It is shown that, contrary to high order Fourier-based spectral methods, low order mixed finite element spaces are acceptable for this purpose. The paper is organized as follows. In Section we introduce the discrete setting and we define the Galerkin approximation to 1.1). In Section 3 we derive a priori estimates. A key estimate is derived for the pressure in Lemma 3.. This estimate is intimately linked to the fact that we are working in the three-dimensional torus. Generalizing this estimate or a similar one with Dirichlet boundary conditions and using finite elements still seems to be challenging at the present time. The main result of this paper is reported in Section 4 where we show that the Galerkin solution converges up to sequences) to a suitable weak solution to 1.1), see Theorem 4.1. The key to this result is that, contrary to approximation spaces based on trigonometric polynomials, finite element spaces have a discrete commutator property, see Definition Notations and conventions Henceforth Ω denote the three-dimensional torus. As usual, we denote by W s,p Ω) the Sobolev spaces of functions in L p Ω) with partial derivatives of order up to s in L p Ω) when s is integer and W s,p Ω) is defined by interpolation otherwise. We do not make notational distinctions between vector- and scalar-valued functions. For s 1, W s,p # Ω) denotes the functions in W s,p Ω) that are periodic. In the following c is a generic constant which may depend on the data f, u, ν, Ω, T.Thevalueofc may vary at each occurrence.. The Galerkin approximation.1. The discrete setting For the time being we do not particularize the setting to the torus. Let X be a closed subspace of [H 1 Ω)] 3 think of [H 1 Ω)]3 if homogeneous Dirichlet boundary conditions are enforced or think of [H 1 # Ω)]3 if periodicity is enforced). Let M = L = Ω), where L = Ω) is composed of those functions in L Ω) that are of zero mean. To construct a Galerkin approximation of the Navier Stokes equations, we assume that we have at hand two families of finite-dimensional spaces, {X h } h>, {M h } h> such that X h X and M h M. The velocity is approximated in X h and the pressure in M h. To avoid irrelevant technicalities we assume M h H 1 Ω) M.

3 J.-L. Guermond / J. Math. Pures Appl. 85 6) Let π h : [L Ω)] 3 X h be the L -projection onto X h. We assume that X h and M h are compatible in the sense that there is c> independent of h such that q h M h, π h q h L c q h L..1) A first consequence of this hypothesis is that X h and M h satisfy the so-called LBB condition, see, e.g., [7]. That is to say: Lemma.1. Assume that.1) holds, X h and M h are such that q h, v h ) = q h,v h ) for all q h M h and all v h X h, and there exists C h : [H 1 Ω)] 3 X h an H 1 -stable interpolation operator such that C h v v L ch v H 1 for all v [H 1 Ω)] 3, then there is a constant c independent of h such that q h, v h ) inf sup c..) q h M h v h X h q h L v h H 1 Proof. See Appendix A.1. The operator C h can be, e.g., the Clément interpolation operator [6] or the Scott Zhang operator [13]. Lemma.. Hypothesis.1) holds in either one of the following situations: i) X h is composed of P 1 -Bubble H 1 -conforming finite elements and M h is composed of P 1 H 1 -conforming finite elements i.e., the so-called MINI element). ii) X h is composed of P H 1 -conforming finite elements and M h is composed of P 1 H 1 -conforming finite elements i.e., the so-called Hood Taylor element), and no tetrahedron has more than 3 edges on Ω. Proof. See Appendix A.. We now particularize the functional setting to the torus. We assume that X =[H# 1 Ω)]3, i.e., X h [H# 1 Ω)]3, and to minimize technicalities we assume M h H# 1 Ω) L = Ω). Moreover, we assume that there is an interpolation operator J h : H# Ω) M h such that for all q H# Ω), q Jh q) L ch q H..3) We also make the following key hypotheses: There is c independent of h such that for all v [H 1 # Ω)]3, v π h v L = inf v w h L ch v H 1, w h X h.4) π h v H 1 c v H 1..5) In addition to the above interpolation properties, we assume that the following inverse inequality holds in X h : There is c> independent of h such that v h H 1 ch 1 v h L, v h X h..6) Remark.1. i) The above interpolation and stability results.4),.5) hold only with periodic boundary conditions. In the case of Dirichlet boundary conditions, i.e., X h [H 1 Ω)]3, the above results are not true; in this case we only have v π h v L ch 1/ v H 1 and π h v H 1 ch 1/ v H 1 for all v [H 1 Ω)] 3. This limitation is the main obstacle to the extension of the results stated in the remainder of the paper to more general boundary conditions. ii) The inequality.6) holds whenever the family of spaces {X h } h> is composed of finite element spaces based on mesh families that are quasi-uniform, see, e.g., [5]. We define the map ψ h : H # Ω) M h such that for all q in H # Ω), ψ hq) solves: πh ψ h q), r h ) = q, rh ), r h M h..7) Observe that the above problem has a unique solution since the bilinear form π h q h, r h ) is coercive owing to hypothesis.1).

4 454 J.-L. Guermond / J. Math. Pures Appl. 85 6) Lemma.3. There exists c> independent of h such that for all q in H # Ω), ψ h q) q ) L ch q H,.8) πh ψ h q) H 1 c q H..9) Proof. See Appendix A.3... The discrete problem Denote by V the closed subspace of [H# 1 Ω)]3 that is composed of the vector fields in [H# 1 Ω)]3 that are solenoidal. Define the space: V h = { v h X h ; v h,q h ) =, q h L Ω) }..1) Since V h is not a subspace of V, i.e., V h is not composed of solenoidal vector-fields, we modify the nonlinear term as follows. We introduce a bilinear operator nl h L[H# 1 Ω)]3 ) ;[H# 1 Ω)] 3 ). We assume that nl h satisfies the following continuity property: nlh v, v) H 1 c v H 1 v L 3..11) We define the trilinear form b h L[H 1Ω)]3 ) 3 ; R) such that b h u,v,w)= nl h u, v), w H 1,H 1. We assume that b h satisfies the following property: b h u,v,v)=, v V +V h..1) For instance, an admissible form of the nonlinear term is as follows see, e.g., [14]), nl h u, v) = u v + 1 v u..13) Let h : L Ω) M h be a linear L -stable interpolation operator i.e., h z z for all z L Ω)), then another admissible form of the nonlinear term is: nl h u, v) = u) v + 1 h u v) )..14) The discrete problem we henceforth consider is as follows: Seek u h C [,T]; X h ) with t u h L,T; X h ) and p h L [,T]; M h ) such that for all v h X h,allq h M h,a.e.t [,T]: { t u h,v h ) + b h u h,u h,v h ) p h, v h ) + ν u h, v h ) = f,v h, u h,q)=,.15) u h t= = I h u, where I h : L Ω) V h is a L -stable interpolation operator; that is to say, I h z z for all z [L Ω)] 3 actually, weak convergence is enough). Note that for all v h in X h the approximate momentum equation holds in L,T). 3. A priori estimates 3.1. Energy estimates Owing to.1), we have the usual a priori energy estimates on u h, namely max u h t) L + u h L H 1 ) c, 3.1) from which we deduce the following: t T Lemma 3.1. Under the above assumptions on f and u, there is c, independent of h, such that u h L r H /r ) + u h L r L q ) c, with 3 q + r = 3, r, q 6. 3.)

5 J.-L. Guermond / J. Math. Pures Appl. 85 6) Proof. This result is standard and is a consequence of the interpolation inequality see, e.g., Lions and Peetre [11]), v H /r c v 1 /r v /r, when r, and the embedding H /r Ω) L q Ω) for 1/q = 1/ /3r). L H Pressure estimate Now we want to deduce aprioriestimates on the pressure p h. The main tool we are going to use is a duality argument. We define q = ) 1 p h and we test the momentum equation with π h ψ h q)). Lemma 3.. Under the above assumptions, there is c, independent of h, such that p h L 4/3,T ;L ) c. 3.3) q, φ) = p h,φ), φ H# 1 Owing to standard regularity results, q H c p h L. 3.4) ) Let us test the momentum equation with π h ψ h q)); note that π h ψ h q)) is an admissible test function since π h ψ h q)) X h. 3) We first take care of the pressure term. The definition of q together with that of ψ h q) yield: p h, π h ψ h q) ))) = p h,π h ψ h q) )) = p h, q) = p h L. 3.5) 4) The contribution of the time derivative is zero since t u h,π h ψ h q) )) = t u h, ψ h q) )) = t u h ), ψ h q) ) =, 3.6) owing to the fact that t u h V h and ψ h q) M h. 5) We take care of the viscous term as follows. Using the stability estimate.9) we infer: u h, π h ψ h q) ))) uh L πh ψ h q) ) H 1 c u h L q H. Then the stability estimate 3.4) implies: uh, π h ψ h q) ))) c u h L p h L. 3.7) 6) For the nonlinear term we proceed as follows: b h uh,u h,ψ h q) ) = nl h u h,u h ), π h ψ h q) nl h u h,u h ) H 1 π h ψ h q) H 1. Using the bound.11) together with the estimates.9), 3.4), we obtain: bh uh,u h,ψ h q) ) c uh L 3 u h H 1 p h L. 3.8) 7) We proceed similarly as above for the source term, f,π h ψ h q) ) f H 1 πh ψ h q) ) H 1 c f H 1 q H. That is to say: f,π h ψ h q) ) c f H 1 p h L. 3.9) 8) Combining 3.5) 3.9), we deduce: p h L c ν u h H 1 + u h L 3 u h H 1 + f H 1) ph L. Then, as a consequence of the bound 3.), we infer: This completes the proof. p h 4/3 L c T u h 4 L 3 + u h H 1 + f H 1 ) c.

6 456 J.-L. Guermond / J. Math. Pures Appl. 85 6) Estimate on t u h As a consequence of Lemma 3. we infer: Corollary 3.1. Under the above assumptions, there is c independent of h such that Proof. Using the H 1 -stability of π h, we infer: t u h,v) t u h H 1 = sup v [H# 1 v Ω)]3 H 1 t u h L 4/3,T ;H 1 ) c. 3.1) t u h,π h v) = sup v [H# 1 v Ω)]3 H 1 c ν u h H 1 + nl h u h,u h ) H 1 + p h L + f H 1). Using the bound.11), we deduce: t u h,π h v) c sup v [H# 1 π h v Ω)]3 H 1 t u h H 1 c ν u h H 1 + u h L 3 u h H 1 + p h L + f H 1). t u h,v h ) c sup v h X h v h H 1 Then, the conclusion follows readily as a consequence of the bound 3.) together with the pressure estimate 3.3) Convergence to a weak solution Before proving that subsequences of u h ) converge to a weak solution, we make sure that we are solving the right problem, i.e., we now formulate consistency hypotheses on the nonlinear term. In this section s denote a real number such that 4 <s<. We denote by s and s the two real numbers such that 1/s + 1/s = 1 and 1/s + 1/s = 1/, respectively. We assume that the nonlinear term has the following consistency property: For all functions w in L,T; V) L 4,T;[L 3 Ω)] 3 ) and all sequences of functions w h ) h> in C [,T]; X h ) converging weakly to w in L,T;[H 1 # Ω)]3 ) and strongly in L s,t;[l 3 Ω)] 3 ), the following holds: nl h w h,w h ) w w, in L s,t; [ H 1 # Ω) ] 3). 3.11) Lemma 3.3. The consistency property 3.11) holds for definition.13) and for definition.14). Proof. Let v be a function in L s,t;[h 1 # Ω)]3 ). 1) Assume that nl h is defined as in.13). Observing that v L s,t;[l 6 Ω)] 3 ), we deduce that v w h v w and v w h v w in L,T;[L Ω)] 3 3 ) and L,T; L Ω)), respectively. As a result v w h, w h ) v w, w) and v w h, w h ) v w, w). Moreover, since w =, a.e. in Q T, we infer v w h, w h ). The conclusion follows readily. ) Assume that nl h is defined as in.14). The only term that poses a difficulty is h w h ), v). Integrating by parts, we rewrite this term as follows h w h ), v). Banach Steinhaus theorem implies that h is uniformly bounded, then using linearity: h wh ) h w ) L s L ) c w h w L s L ) c w h w) w h + w) L s L ) c w h w L s L 3 ) wh L L 6 ) + w L L 6 )). In the last inequality we used the fact 1/s + 1/ = 1/s. Note that w h L L 6 ) is bounded since w h converges weakly to w in L,T; L 6 Ω)). The above inequality implies h w h ) h w ) in L s,t; L Ω)). Moreover, h w ) w in L Ω) a.e. on,t), h w ) s is uniformly bounded by c w s L L

7 J.-L. Guermond / J. Math. Pures Appl. 85 6) L 1,T); hence, Lebesgues Dominated Convergence theorem implies h w ) w in L s,t; L Ω)). As a result we obtain h w h ), v) w, v). Finally, nlh w h,w h ), v Hence 3.11) holds since v is arbitrary. We have the following classical result: w) w + 1 w ) ),v = w w,v). Corollary 3.. Under the above hypotheses, u h convergences, up to subsequences, to a weak solution to 1.1) in L,T;[H 1 # Ω)]3 ) weak and in any L r,t; L q Ω) 3 ) strong 1 q<6r/3r 4), r< ), and, up to subsequences, p h converges to p in L 4/3,T; L Ω)) weak. Proof. We briefly outline the main steps of the proof for the arguments are quite standard. 1) Since u h is uniformly bounded in L,T;[H# 1 Ω)]3 ) and t u h is bounded uniformly in L 4/3,T;[H# 1 Ω)] 3 ), the Aubin Lions Compactness lemma see Lions [1, p. 57] or [15]) implies that there exists a subsequence u hl ) converging weakly in L,T;[H# 1 Ω)]3 ) and strongly in any L r,t; L q Ω)), such that 1 q<6r/3r 4), r<, and that t u hl ) converges weakly in L 4/3,T;[H# 1 Ω)] 3 ). Moreover, since p h ) is bounded uniformly in L 4/3,T; L Ω)), there exists a subsequence p hl ) converging weakly in L 4/3,T; L Ω)). Letu and p denote these limits. ) Let q L,T; L Ω)) and let q hl ) hl > be a sequence of functions in L,T; M h ) strongly converging to q in L,T; L Ω)). Then = u h l,q hl ) u, q) since u h l u in L,T; L Ω)). Asa result, u =, a.e. in Q T ; that is to say u is in L,T; V). 3) Let s be a real number such that 4 <s<. Letv be any function in L s,t;[h# 1 Ω)]3 ) and let v hl ) hl > be a sequence of functions in L s,t; X h ) strongly converging to v in L s,t;[h# 1 Ω)]3 ). Then 4) Q T t u hl v hl Q T t u v, since t u hl t u in L 4/3,T;[H# 1 Ω)] 3 ). 5) Q T u hl : v hl Q T u: v, since u hl u in L,T;[L Ω)] 3 ) L 4/3,T;[L Ω)] 3 ). 6) Q T p hl v hl Q T p v, since p hl pin L 4/3,T; L Ω)). 7) Since u hl converges weakly to u in L,T;[H# 1 Ω)]3 ) and strongly in L s,t;[l 3 Ω)] 3 ), the hypotheses of 3.11) hold; hence, u u, v). b hu hl,u hl,v hl ) 8) Finally, since u hl converges in C [,T]; L w Ω)) functions that are continuous over [,T] with value in L Ω) equipped with the weak topology) u I hl u = u hl ) u) in L Ω); hence, u) = u. 9) That u satisfies Leray s energy inequality is standard. It is a consequence of the inequality u hl u) u + u u hl. The theorem is proved. 4. Convergence to a suitable solution The main issue we address in the present work is to determine whether weak solutions are suitable in the sense of Definition 1.1. To answer this question we assume that the discrete framework satisfies the following property that we henceforth refer to as the discrete commutator property see Bertoluzza [3]). Definition 4.1. We say that X h resp. M h ) has the discrete commutator property if there is an operator P h L[H# 1 Ω)]3 ; X h ) resp. Q h LL Ω); M h )) such that for all φ in W, # Ω) resp. all φ in W 1, # Ω)) and all v h X h resp. all q h M h ), φv h P h φv h ) H l ch 1+m l v h H m φ W m+1,, l m 1, φqh Q h φq h ) L ch q h L φ W 1,.

8 458 J.-L. Guermond / J. Math. Pures Appl. 85 6) Remark 4.1. Fourier-based approximation spaces do not have the discrete commutator property since Fourier series do not have local interpolation properties. Fourier series are very accurate but they only have global interpolation properties. We also assume that the following consistency property holds for the nonlinear term: For all functions w in L,T; V) L,T;[L Ω)] 3 ) and all sequences of functions w h ) h> in C [,T]; X h ) uniformly bounded in L,T;[H 1 # Ω)]3 ) L,T;[L Ω)] 3 ) and strongly converging to w in L s,t;[l 3 Ω)] 3 ), where 3 s < 4 i.e., 4 <s 6), the following holds: b h w h,w h,φw h ) ) 1 w w, φ, φ D,T; C# Ω )). 4.1) Lemma 4.1. The consistency property 4.1) holds for definition.13) and also for definition.14) provided M h has the discrete commutator property. Proof. 1) The situation for Definition.13) is quite simple since ) b h w h,w h,φw h ) = w h w h,φw h + 1 ) 1 w h w h,φw h ) = w h w h + 1 ) w h w h,φ ) ) ) 1 = w h w h 1,φ = w h w h, φ. Then, clearly b hw h,w h,φw h ) 1 w w 1, φ) since w h w h w 1 w in L s /3,T; L 1 Ω)) L 1 Q T ). ) For definition.14) we proceed as follows: b h w h,w h,φw h ) = ) 1 w h ) w h,φw h + h wh ) )) 1,φw h = h wh ), φw h ) ) = 1 wh h wh ), φ ) 1 φh wh ) ) 1, w h = wh w h φ ) + R 1 + R, where R 1 = 1 w h h w h ) w h ), φ) and R = 1 φ h w h ), w h ). By using the same arguments as in the second part of the proof of Lemma 3.3 we infer h w h ) w in L s,t; L Ω)); that is to say, h w h ) w h inl s,t; L Ω)). Since w h φ w φ in L s,t; L Ω)), we infer R 1 as h. For R we use the fact that M h has the discrete commutator property as follows: R = 1 φ h wh ) Q h φh wh )) ), w h 1 φ h wh ) Q h φh wh )) L w h H 1 ch h wh ) L w h H 1 ch w h L w h H 1 ch w h L 4 w h H 1 ch w h 1/ w L h 3/ w L 6 h H 1 Hence ch w h 1/ L w h 1/ H 1 w h H 1 ch 1/ w h L w h H 1. R ch 1/ w h L H 1 ) w h L L ). Then clearly R ash. In conclusion b hw h,w h,φw h ) 1 w w 1, φ) since w h w h w 1 w in L s /3,T; L 1 Ω)) L 1 Q T ) and R 1 + R. That concludes the proof. The main result of the paper is stated in the following theorem: Theorem 4.1. Under the aboves hypotheses, if X h and M h have the discrete commutator property, the couple u h,p h ) convergences, up to subsequences, to a suitable solution to 1.1), sayu, p).

9 J.-L. Guermond / J. Math. Pures Appl. 85 6) Proof. To alleviate notations we still denote by u h ) and p h ) the subsequences that converge to u and p, respectively. 1) Let φ be a non-negative function in D,T; C# Ω)). Testing the momentum equation in.15) by P hu h φ), we obtain: t u h,p h u h φ) ) + b h uh,u h,p h u h φ) ) p h, P h u h φ) ) + ν u h, P h u h φ) ) f,p h u h φ) ) =. Each of the terms on the left-hand side of the equation are now treated separately in the following steps: ) For the time derivative we have: t u h,p h u h φ) ) = t u h,u h φ) + R = 1 uh, t φ ) + R, u, t φ) since where we have set R = u h,t,p h u h φ) u h φ). It is clear that 1 u h, t φ) 1 u h u in L r L 1 ) for any 1 r<. To control the residual we use the discrete commutator property and the inverse inequality.6) as follows: R = uh,t,p h u h φ) u h φ ) u h,t H 1 Ph u h φ) u h φ H 1 ch u h,t L 4/3 H 1 ) u h L 4 H 1 ) ch1/ u h,t L 4/3 H 1 ) u h 1/ L L ) u h 1/ L H 1 ). Now, it is clear that R ash. 3) Using the fact that u h is periodic and the first derivatives of φ are also periodic, the viscous term yields: uh, P h u h φ) ) = u h, u h φ) ) + R = u h,φ ) ) 1 u h, φ + R where R = u h,p h u h φ) u h φ). For the first term we proceed as follows: uh,φ ) = u h u + u),φ ) = u h u) + u h u) : u + u,φ ). Since u h uin L,T; H 1 ) and φ is non-negative, we infer lim inf u h,φ) u,φ).forthe second term we have 1 u h, φ) 1 u, φ) since u h u in L r L 1 ) for any 1 r<. Now we control the residual as follows: R = u h,p h u h φ) u h φ ) ch uh H 1. Then it is clear that R ash. In conclusion, lim inf h uh, P h u h φ) ) u,φ ) ) 1 u, φ. 4) For the pressure term we have: ph, P h u h φ) )) = p h, u h φ) ) + R 1 = p h u h, φ) + R 1 + R, where R 1 = p h, P h u h φ) u h φ)) and R = φp h u h ).ForR 1, using the discrete commutator property together with an inverse inequality.6), we have: R 1 c p h L Ph u h φ) u h φ H 1 ch p h L u h H 1 ch p h L 4/3 L ) u h L 4 H 1 ) ch1/ p h L 4/3 L ) u h 1/ L H 1 ) u h 1/ L L ).

10 46 J.-L. Guermond / J. Math. Pures Appl. 85 6) Then clearly R 1 ash. We proceed similarly for R using the fact that u h take its values in V h, R = ) φp h Q h φp h ), u h c φp h Q h φp h ) L u h H 1 ch p h L 4/3 L ) u h L 4 H 1 ) ch1/ p h L 4/3 L ) u h 1/ L H 1 ) u h 1/ L L ). Then again R ash. 5) The source term does not pose any particular difficulty, f,ph φu h ) = f,φu h +R, where R = f,p h φu h ) φu h. Clearly f,φu h f,φu since u h uin L,T;[H# 1 Ω)]3 ) and f L,T;[H# 1 Ω)] 3 ). Moreover, R f L H 1 ) P h φu h ) φu L h H 1 ) ch f L H 1 ) u h L H 1 ). Then R ash. 6) Now we pass to the limit in the nonlinear term, b h uh,u h,p h φu h ) ) = b h u h,u h,φu h ) + R, where R = b h u h,u h,p h φu h ) φu h ). Then R nl h u h,u h ) H 1 P h φu h ) φu H h 1 ch u h L 3 u h H 1 u h H 1 ch u h 1/ u L h 1/ u H 1 h H 1. That is to say, This in turn implies Appendix A. Proofs from Section A.1. Proof of Lemma.1 We start with a standard lemma: R ch 1/ u h L L ) u h L H 1 ). R ash. Then conclude using hypothesis 4.1). Lemma A.1. There are c 1 >, c independent of h such that q h, v h ) q h M h, c 1 q h L c h q h L + sup. A.1) v h X h v h H 1 Proof. Let q h be a nonzero function in M h. Since the linear mapping : [H 1Ω)]3 L = Ω) is continuous and surjective, there is β> such that for all r L = Ω) there is w [H 1Ω)]3 verifying w = r and β w H 1 r L.Letv [H 1Ω)]3 be such that v = q h and β v H 1 q h L. Then, using q h, C h v) = q h, C h v), Ω sup q h v h Ω q h C h v) Ω c q h C h v) Ω = c C hv) q h v h X h v h H 1 C h v) H 1 v H 1 v H 1 = c Ω v q h c v H 1 Ω C hv) v) q h. v H 1

11 J.-L. Guermond / J. Math. Pures Appl. 85 6) Since v [H 1Ω)]3 we integrate by parts the first term in the right-hand side: Ω sup q h v h Ω = c q h v c v h X h v h H 1 v H 1 Then using C h v) v) L ch v H 1 the results follows easily. Ω C hv) v) q h c 1 β q h v L c q h L H 1 C h v) v) L. v H 1 To prove Lemma.1, we use q h,v h ) = q h, v h ) and we proceed as follows: q h, v h ) q h,v h ) sup = sup q h,π h q h ) = π h q h L. v h X h v h H 1 v h X h v h H 1 π h q h H 1 π h q h H 1 Using the inverse inequality π h q h H 1 ch 1 π h q h L together with the hypothesis.1), we infer: q h, v h ) sup ch π h q h v h X h v h L c h q h L. H 1 Conclude using A.1). A.. Proof of Lemma. The technique of proof is adapted from that which is used to prove the standard LBB condition, see, e.g., [16,7]. Let us first prove statement i). Let q h be a member of M h.let be an element in the mesh. Let b be the bubble function associated with, i.e., b H 1), b 1, and meas) b b c where c does not depend on and h. Set v h = q h b. T h b Observe that v h = q h = meas) q h. Owing to this definition: v h, q h ) = v h q h = q h v h = q h L ). T h T h T h That is v h, q h ) = q h L. Moreover, v h L = T h q h meas) b Since bubbles functions are such that meas) b b c where c does not depend on and h, we infer: v h L c q h L. Then, using the fact that π h q h is in X h and π h is a projection: π h q h,w h ) π h q h L = sup w h X h w h L b. q h,w h ) = sup q h,v h ) c q h w h X h w h L v h L. L Hence, statement i) is proved. ) Let A ={a n } be the collection of all the vertices in the mesh. Let E i ={e l } be the collection of all the internal edges in the mesh, E ={e l } be the collection of all the edges in the mesh that are on Ω. Likewise we denote by M i ={m l } and M ={m l } the set of midedges that are internal and the set of those that are at the boundary, respectively. For an edge e l we denote by τ l one of the two unit vectors that are aligned with e l.letq h be a member of M h. Define v h X h be such that v h a n ) =, a n A, v h m l ) =, m l M, v h m l ) = τ l τl q h, m l M i.

12 46 J.-L. Guermond / J. Math. Pures Appl. 85 6) Note that this definition implies that v h [H 1Ω)]3. Using the quadrature formula: 1 φ P, φ = 5 φm l) ) 1 φa n) meas), m l M a n A where M = M i M ) and A = A, we infer: v h, q h ) = v h q h = 1 5 T h = 1 5 T h m l M i T h m l M i τl q h m l ) meas), and since each element has at least 3 internal edges, we infer: τl q h m l )τ l q h m l ) meas) v h, q h ) c T h q h meas) c q h L. Moreover it is clear that v h L c q h L. Then the conclusion follows readily as in part 1) above. This concludes the proof. A.3. Proof of Lemma.3 1) Let us first prove the estimate.8). Denote a h s, r) = π h s, r) and as,r) = s, r). It is clear that owing to the L -stability of π h, a h is continuous over H 1 Ω) + M h ) H 1 Ω) + M h ), i.e., ah s, r) s L r L. A.) It is clear that the hypothesis.1) implies the following stability estimate: There is c> independent of h such that a h q h,r h ) inf sup c. A.3) q h M h r h M h q h H 1 r h H 1 Now let us prove a consistency property. Let q be a member of H# Ω). Observe that aj h q,r h ) a h J h q,r h ) = J h q, r h π h r h ) = inf J h q w h, r h π h r h ) w h X h ) = inf Jh q q)+ q w h, r h π h r h. w h X h Since q H# Ω), q is a member of [H # 1 Ω)]3. Then using the interpolation properties.3),.4) we infer the following consistency estimate. aj h q,r h ) a h J h q,r h ) sup ch q r h M h r h H. A.4) H 1 To conclude we use the First Strang Lemma. In other words, using A.3), we write c ψh q) J h q a h ψ h q) J h q,r h ) aq,r h ) a h J h q,r h ) H 1 sup sup r h M h r h H 1 r h M h r h H 1 aq J h q,r h ) + aj h q,r h ) a h J h q,r h ) sup. r h M h r h H 1 The result follows by using A.4) together with the interpolation property.3). ) We now prove the estimate.9). Using the inverse inequality.6) together with.8) and the H 1 -stability of π h,.5), we infer: π h ψ h q) ) H 1 π h ψ h q) q ) H 1 + π h q H 1 c 1 h 1 ψ h q) q ) L + c q H c q H. This completes the proof.

13 J.-L. Guermond / J. Math. Pures Appl. 85 6) Appendix B. The discrete commutator property The goal of this section is to show that the discrete commutator property see Definition 4.1) holds for standard H 1 -conforming finite element spaces. Let T h be a regular mesh of simplices and let Z h H# 1 Ω) be the P k-lagrange finite element space based on this mesh. Let 1 p<, and let m be such that m 1ifp = 1 and m>1/p otherwise. Let P h : W m,p # Ω) Z h be the Scott Zhang interpolation operator [13]. Recall that P h is linear, is a projection onto Z h, and satisfies the following interpolation property: Lemma B.1 Scott Zhang). In addition to the above hypotheses, assume m k + 1 then for all l [,m]: v W m,p # Ω), T h, v P h v W l,p ) chm l v W m,p Δ ), where h = diam) and Δ = interior { }). As a corollary we infer the following so-called discrete commutator property see, e.g., Bertoluzza [3]). Lemma B. Bertoluzza). Let m and p be such that the assumptions of Lemma B.1 hold, then the following holds for all v h in Z h and for all φ in W m+1, Ω): φv h P h φv h ) W l,p ch 1+m l v h W m,p φ W m+1,, l m 1. Proof. We prove the result locally. Let be a cell in the mesh T h. Denote by x some point in, say the barycenter of. Letφ be a function in W 1, Ω). Define R = φ φx ). It is clear that R W 1, Ω), and R L Δ ) ch φ W 1, Ω), R W 1, Δ ) c φ W 1, Ω). Let v h be the mean value of v h over Δ, then it is clear that We have: v h v h W l,p Δ ) v h L p Δ ) c v h L p Δ ), chm l v h W m,p Δ ), l m. φv h P h φv h ) W l,p ) 1 P h )φv h ) W l,p ) + 1 P h ) φv h v h ) ) W l,p ). Let us denote by R 1 and R the two residuals in the right-hand side. To control R 1 we proceed as follows: R 1 ch 1+m l φv h W m+1,p ) ch1+m l v h L p ) φ W m+1, Ω) ch1+m l v h L p Δ ) φ W m+1, Ω). For the other residual we use the fact that P h is linear and is a projection as follows: 1 Ph ) φv h v h ) ) W l,p ) = 1 Ph ) φ φx ) ) v h v h ) ) W l,p ). As a result R = 1 P h ) R v h v h ) ) W l,p ch1 l ) R v h v h ) W 1,p Δ ) ch 1 l R L Δ ) v h v h W 1,p Δ ) + R W 1, Δ ) v h v h L p Δ )) ch 1 l h v h v h W 1,p Δ ) + v h v h L p Δ )) φ W 1, Ω) ch1+m l v h W m,p Δ ) φ W 1, Ω). Then, the desired result follows easily owing to the regularity hypothesis on the mesh which implies that sup T h {card{ T h Δ }} can be bounded from above by a constant that does not depend on h.

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