A reduced discrete inf-sup condition in Lp for incompressible flows and application

ESAIM: M2AN 49 (2015) 1219–1238 ESAIM: Mathematical Modelling and Numerical Analysis DOI: 10.1051/m2an/2015008 www.esaim-m2an.org A REDUCED DISCRETE INF-SUP CONDITION IN Lp FOR INCOMPRESSIBLE FLOWS AND APPLICATION Tom´ as Chac´ on Rebollo1, Vivette Girault2, Macarena G´ omez M´ armol3and Isabel S´ anchez Mu˜ noz4 Abstract. In this work, we introduce a discrete specific inf-sup condition to estimate the Lpnorm, 1<p<+∞, of the pressure in a number of fluid flows. It applies to projection-based stabilized finite element discretizations of incompressible flows, typically when the velocity field has a low regularity. We derive two versions of this inf-sup condition: The first one holds on shape-regular meshes and the second one on quasi-uniform meshes. As an application, we derive reduced inf-sup conditions for the linearized Primitive equations of the Ocean that apply to the surface pressure in weighted Lpnorm. This allows to prove the stability and convergence of quite general stabilized discretizations of these equations: SUPG, Least Squares, Adjoint-stabilized and OSS discretizations. Mathematics Subject Classification. 35Q35, 65N12, 76D05. Received June 12, 2014. Revised October 15, 2014. Published online July 6, 2015. 1. Introduction Stabilized methods are designed to provide stable discretizations with reduced computational complexity of several sources of spurious instabilities that may arise in the discretization of incompressible flows (that may be due either to incompressibility or to large convection, Coriolis or reaction terms, among others). In this paper, we focus on the treatment of the incompressibility constraint. Concretely, we deal with projectionstabilized methods, introduced by Blasco and Codina in [3] by means of a local L2projection, that produce discretizations with high-order accuracy. This method was extended to the local projection-stabilization methods that use element-wise L2projections instead of the global L2projection, while satisfying some orthogonality properties. Among the many references on different versions of local projection-stabilization, let us quote Braack andBurman[4], Braack et al. [5], Ganesan et al. [17], Knobloch [19], Matthies et al. [21], Roos et al. [23]. Keywords and phrases. Inf-sup condition, Finite element method, Stabilized method, Incompressible flows, Primitive equations of the Ocean. 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico and Instituto de Matem´aticas de la Universidad de Sevilla (IMUS), Apdo. de correos 1160, Universidad de Sevilla, 41080 Sevilla, Spain. 2Laboratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie & C.N.R.S, UMR 7598, Paris 6, 4 Place Jussieu, 75252 Paris cedex 05, France. 3Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Apdo. de correos 1160, Universidad de Sevilla, 41080 Sevilla, Spain. 4Departamento de Matem´atica Aplicada I, Carretera de Utrera Km 1, Universidad de Sevilla, 41013 Sevilla, Spain. [email protected] Article published by EDP Sciences c EDP Sciences, SMAI 2015 1220 T. CHAC ´ ON REBOLLO ET AL. A special class of projection-stabilized methods is the interior penalty method, in which stabilization is achieved by introducing inter-element jumps of the terms to be stabilized. This method is equivalent to a projectionstabilized method, where the L2projection operator is replaced by the Oswald (cf. [22]) quasi-interpolant operator on the discrete velocity space (cf. [7–9]). A further simplification is introduced in [15] where the local projection operator is replaced by a quasi-local approximation operator that does not need to satisfy any orthogonality property. This method has a more compact stencil while retaining the same optimal accuracy as all projection-stabilization methods, in the sense that its convergence order is optimal with respect to the degree of the finite element spaces. In the present work, we extend the method introduced in [15] to stabilize the discretization of the pressure in flows where the velocity has a low accuracy, typically in some space W1,s(Ω)with1<s<2. Then the pressure has only Lp(Ω) regularity, where pis the conjugate exponent of s. This is the case, for instance, of the weak solutions of the Primitive equations of the Ocean, that we consider as an application of our general setting. We introduce a specific inf-sup condition in Lpnorms, which is the main technical contribution of this paper. As in [15], the derivation of this condition faces the difficulty of the reduced number of degrees of freedom of the buffer space. This difficulty is solved by a finite-dimensional argument of equivalence of norms (cf. Lem. 2.4). We derive two versions of this inf-sup condition: the first one holds on shape-regular meshes and the second one on quasi-uniform meshes. As an application, we derive reduced inf-sup conditions for the linearized Primitive equations of the Ocean that apply to the surface pressure in weighted Lpnorms. This allows to prove the stability and convergence of quite general stabilized discretizations of these equations: SUPG, Least Squares, Adjointstabilized and OSS discretizations. This condition generalizes a similar one for L2weighted norms introduced in [14]. The paper is structured as follows: in Section 2we introduce the inf-sup condition in Lpnorms for shaperegular meshes, as well as a simplified condition for quasi-uniform meshes. These conditions are applied to stabilized discretizations of a linearized version of the Primitive equations in Section 3. In this section, reduced inf-sup conditions for the surface pressure are deduced, for both shape-regular and quasi-uniform meshes. Also, the stability and convergence of the stabilized discretizations are proved by means of these conditions. 2. Inf-sup condition for shape-regular meshes Let Ω⊂Rd(d= 2 or 3) be a bounded domain. We assume that Ωis a polygon if d= 2 or a polyhedron if d=3.Let{Th}h>0be a family of conforming triangulations of ¯ Ωformed by simplicial elements, where the parameter hdenotes the largest diameter of the elements of Th.Weassumethefollowing. Hypothesis 2.1. The family {Th}h>0is shape-regular in the sense of Ciarlet [16]and no element of Thhas 3nodes(when d=2)or 4 nodes (when d=3)on the boundary of Ω. We decompose Ωinto a finite union of macro-elements: Ω= R  i=1 Oi, such that each Oiis the support of the piecewise affine basis function associated to the node i.Notethatany element K∈T hbelongs to at most Mmacro-elements, where Mis independent of h. For all i,1≤i≤R,weset hi=max K⊂Oi{hK}and ρi=min K⊂Oi{ρK}, where hKdenotes the diameter of the element Kof Thand ρKthe diameter of the ball inscribed in K. As the mesh is regular, then it is locally uniformly regular (or locally quasi-uniform) and this implies that there exist positive constants C1and C2independent of h, such that for all K∈T hand for all ifor which K⊂O i, C1ρi≤hK≤hi,hi ρi≤C2.(2.1) A REDUCED DISCRETE INF-SUP CONDITION IN LPFOR INCOMPRESSIBLE FLOWS AND APPLICATION 1221 This implies immediately that C1 C2 hi≤hK≤hi.(2.2) For a domain O⊂Rd,wedenoteby· k,p,Oand |·| k,p,Othe norm and, respectively, the seminorm in Wk,p(O). In Lp(Ω)/R,wealsodenoteby·0,p,Ω the quotient norm in order to simplify the notation. For all p∈[1,+∞)andforallv∈Lp(Ω)d, we define vh,p =R  i=1 hp ivp 0,p,Oi 1 p .(2.3) Given an integer l≥1, we denote by Pl(K) the space of polynomials of degree smaller than, or equal to, l defined on an element K∈T hand define the following finite element spaces: Vl h={vh∈C0(¯ Ω) such that vh|K∈Pl(K),∀K∈T h}, Xh=(Vl h∩H1 0(Ω))d, Mh=Vm h/R, Vl h(Oi)={vh∈C0(Oi) such that vh|K∈Pl(K),∀K∈T hsuch that K⊂O i}, Xh(Oi)=(Vl h(Oi)∩H1 0(Oi))d. We recall now the inverse inequalities in finite element spaces, that we use frequently in the sequel. Lemma 2.2. Let p1and p2be numbers in [1,∞],andl1and l2two non-negative integers such that l1≥l2and l1−d p1≥l2−d p2. [Local inverse inequalities] ([2], Prop. 4.2) For all K∈T h, ∀v∈Pl(K),|v|Wl1,p1(K)≤Cρ l2−l1−d p2 Kh d p1 K|v|Wl2,p2(K),(2.4) where Cis a constant independent of K. [Global inverse inequalities] ([6],Thm4.5.11) If p1≥p2and Zhis a finite element space of polynomials in each K,then ∀wh∈Zh, K∈Th |wh|p1 Wl1,p1(K)1 p1 ≤Cρ l2−l1−d p2+d p1 min  K∈Th |wh|p2 Wl2,p2(K)1 p2 ,(2.5) where Cis a constant independent of hand ρmin =inf K∈Th{ρK}. Throughout this work, Crepresents a constant that is always independent of hbut may vary from an inequality to another. Let us now consider an interpolation or projection operator Jh:L2(Ω)d→(Vl−1 h)d,(2.6) and set J∗ h=Id−Jh. Our main result is the following. 1222 T. CHAC ´ ON REBOLLO ET AL. Theorem 2.3. Assume that Hypothesis 2.1 holds. Then for any p∈(1,+∞)there exists a constant γp>0 independent of hsuch that for all qh∈Mh, γpqh0,p,Ω ≤sup vh∈Xh (∇·vh,q h) |vh|1,s,Ω +R  i=1 sup vh∈Xh(Oi) |(∇·vh,q h)Oi|p |vh|p 1,s,Oi 1 p +J∗ h(∇qh)h,p (2.7) where sis the conjugate exponent of p(1 p+1 s=1). To prove this theorem we need the following auxiliary result. Lemma 2.4. Under Hypothesis 2.1, there exists a positive constant Cindependent of h, such that for all i, 1≤i≤R, ∀gh∈Vl−1 h(Oi),gh0,p,Oi≤Csup vh∈Vl h(Oi)∩H1 0(Oi) (gh,v h)Oi vh0,s,Oi ,(2.8) where p∈(1,+∞)and sis the conjugate exponent of p. Proof. For p=s= 2 this result was proved in [15]. We extend it here to any conjugate pair of exponents p and s. We denote by {aj}the nodes of Ththat belong to Oiand we define the function wh∈V1 h(Oi)∩H1 0(Oi)such that: wh|K∈P1(K), for all K⊂O i; wh(aj)=1ifajis a interior node of Oi; wh(aj)=0ifajis a node belonging to ∂Oi. Because each macro-element Oiis the support of one piecewise affine basis function and no element of Thhas 3nodes(whend= 2) or 4 nodes (when d= 3) on the boundary of Ω, there exists at least one interior node in Oi. So, this function whis well-defined and it is positive in the interior of Oi. Let gh∈Vl−1 h(Oi)andvh=ghwh. Then, vh∈Vl h(Oi)∩H1 0(Oi) and it satisfies |(gh,v h)Oi|≥Cgh2 0,Oi,(2.9) with Ca positive constant independent of h(cf. [15], Lem. 3.2). Given K⊂O i,letˆgK=gh|K◦FK,whereFKis the affine mapping that transforms the reference element ˆ Konto K. The shape regularity of the mesh implies that gh2 0,Oi= K⊂OiK|gh|2=C K⊂Oi|K|ˆ K|ˆgK|2≥C|Oi| K⊂Oiˆ K|ˆgK|2.(2.10) Denote by Nithe number of elements of Th|Oiand consider the norm in RNi: (x1,...,x Ni)p,Ni=⎛ ⎝ Ni  j=1 |xj|p⎞ ⎠ 1 p . As the grids are regular, Ni≤N(independent of h)and (x1,...,x Ni)p,Ni=(x1,...,x Ni,0,...,0)p,N . By the equivalence of norms in RN, there exists a constant Cp(independent of h) such that (x1,x 2,...,x Ni,0,...,0)2,N ≥Cp(x1,x 2,...,x Ni,0,...,0)p,N .(2.11) A REDUCED DISCRETE INF-SUP CONDITION IN LPFOR INCOMPRESSIBLE FLOWS AND APPLICATION 1223 Then, applying (2.11)with xj=ˆ K|ˆgKj|p1 p ∀j=1,...,N i, we have  K⊂Oiˆ K|ˆgK|21 2 ≥Cp K⊂Oiˆ K|ˆgK|p1 p . Thus, from (2.10) and the shape regularity of the mesh gh2 0,Oi≥C|Oi|CpCs K⊂Oiˆ K|ˆgK|p 1 p K⊂Oiˆ K|ˆgK|s 1 s ≥C K⊂Oi|K|ˆ K|ˆgK|p1 p K⊂Oi|K|ˆ K|ˆgK|s1 s . That is, gh2 0,Oi≥Cgh0,p,Oigh0,s,Oi.(2.12) Moreover, vh0,s,Oi≤gh0,s,Oiwh0,∞,Oi≤gh0,s,Oi.(2.13) Then, from (2.9) and taking into account (2.12)and(2.13)weobtain Cgh0,p,Oi≤(gh,v h)Oi vh0,s,Oi , whence we deduce (2.8).  ProofofTheorem2.3.We adapt Verf¨urth’s technique (cf. [24]). Given qh∈Mh, by Amrouche and Girault (cf. [1]), there exists a constant C>0 independent of hsuch that Cqh0,p,Ω ≤sup v∈[W1,s 0(Ω)]d−{0} (∇·v,q h) |v|1,s,Ω · So there exists v∈[W1,s 0(Ω)]dsuch that 1 2Cqh0,p,Ω ≤(∇·v,q h) |v|1,s,Ω ·(2.14) Since the family of grids is regular, following the standard finite element interpolation theory (cf. [2]or[6], Sect. 4.8), there exists an interpolate of v,vh∈Xh, such that |vh|1,s,Ω ≤C|v|1,s,Ω,(2.15) v−vh0,s,K ≤Ch K|v|1,s,ωK,(2.16) where ωKdenotes the union of all elements of Ththat intersect K. Let us rewrite the r.h.s. of (2.14)as (∇·v,q h) |v|1,s,Ω =(∇·vh,q h) |v|1,s,Ω +(∇·(v−vh),q h) |v|1,s,Ω ·(2.17) 1224 T. CHAC ´ ON REBOLLO ET AL. Using (2.15), (∇·vh,q h) |v|1,s,Ω ≤C(∇·vh,q h) |vh|1,s,Ω ·(2.18) Also, because qhbelongs to H1(Ω)and(v−vh)·n=0on∂Ω, (∇·(v−vh),q h)=−(v−vh,∇qh) ≤ K∈Thv−vh0,s,K ∇qh0,p,K ≤ K∈Th Ch K|v|1,s,ωK∇qh0,p,K (using (2.16)) ≤C K∈Th|v|s 1,s,ωK1 s K∈Th hp K∇qhp 0,p,K1 p ≤C|v|1,s,Ω  K∈Th hp K∇qhp 0,p,K1 p . (2.19) Then from (2.14), combining (2.17)−(2.19), we have Cqh0,p,Ω ≤sup vh∈Xh (∇·vh,q h) |vh|1,s,Ω + K∈Th hp K∇qhp 0,p,K1 p .(2.20) As each element K∈T hbelongs to some macro-element Oi,  K∈Th hp K∇qhp 0,p,K ≤C R  i=1 hp i∇qhp 0,p,Oi. So, from (2.20) Cqh0,p,Ω ≤sup vh∈Xh (∇·vh,q h) |vh|1,s,Ω +∇qhh,p.(2.21) To estimate the last term in (2.21), we use the relation Jh+J∗ h=Id and write ∇qh0,p,Oi≤Jh(∇qh)0,p,Oi+J∗ h(∇qh)0,p,Oi.(2.22) Since Jh(∇qh)|Oi∈(Vl−1 h(Oi))dwe can apply the inf-sup condition (2.8) to each of its components, Jh(∇qh)0,p,Oi≤Csup vh∈Xh(Oi) (Jh(∇qh),vh)Oi vh0,s,Oi· Using again Jh+J∗ h=Id, |(Jh(∇qh),vh)Oi|≤|(∇qh,vh)Oi|+|(J∗ h(∇qh),vh)Oi| ≤|(∇·vh,q h)Oi|+J∗ h(∇qh)0,p,Oivh0,s,Oi, as (∇qh,vh)Oi=−(∇·vh,q h)Oibecause vh=0on ∂Oi.So, Jh(∇qh)0,p,Oi≤Csup vh∈Xh(Oi) |(∇·vh,q h)Oi| vh0,s,Oi +J∗ h(∇qh)0,p,Oi.(2.23) A REDUCED DISCRETE INF-SUP CONDITION IN LPFOR INCOMPRESSIBLE FLOWS AND APPLICATION 1225 Thus, from (2.22)and(2.23) ∇qhp h,p ≤CR  i=1 sup vh∈Xh(Oi) hp i|(∇·vh,q h)Oi|p vhp 0,s,Oi+J∗ h(∇qh)p h,p.(2.24) The local inverse inequality (2.4) between W1,s(K)andLs(K)oneachK⊂O iyields |vh|1,s,Oi≤Cρ −1 ivh0,s,Oi. This inequality and (2.1)implythat hp i vhp 0,s,Oi≤C1 |vh|p 1,s,Oi , and we obtain ∇qhp h,p ≤CR  i=1 sup vh∈Xh(Oi) |(∇·vh,q h)Oi|p |vh|p 1,s,Oi+J∗ h(∇qh)p h,p.(2.25) Finally, (2.7) follows from (2.21)and(2.25).  2.1. The case of uniformly regular meshes The inf-sup condition (2.7) may be simplified if the grids are uniformly regular. We assume the following. Hypothesis 2.5. The family {Th}h>0is uniformly regular (also called quasi-uniform): there exist positive constants αand βindependent of hsuch that βh≤hK≤αρ K,∀K∈T h.(2.26) We define the space Y=v∈H1(Ω)dsuch that v·n= 0 a.e. on ∂Ω, and consider an interpolation or projection operator Ih:L2(Ω)d→Yh,where Xh⊆Yh⊆(Vl h)d∩Y. (2.27) We shall denote I∗ h=Id−Ih. In this case the inf-sup condition (2.7) reduces to a simpler one. This is stated as follows. Theorem 2.6. Assume that Hypothesis 2.5 holds. Then for any p∈(1,+∞)there exists a constant λp>0 independent of hsuch that for all qh∈Mh, λpqh0,p,Ω ≤sup vh∈Yh (∇·vh,q h) |vh|1,s,Ω +hI∗ h(∇qh)0,p,Ω,(2.28) where sis the conjugate exponent of p. Proof. As in Theorem 2.3 we obtain (2.20) because this estimate only requires that the grids be regular. Moreover, as Xh⊆Yh, Cqh0,p,Ω ≤sup vh∈Yh (∇·vh,q h) |vh|1,s,Ω +h∇qh0,p,Ω.(2.29) In order to bound the last term in (2.29), we argue as in the proof of Theorem 2.3 but now we do not need to follow a local argument. Using the relation Ih+I∗ h=Id, ∇qh0,p,Ω ≤Ih(∇qh)0,p,Ω +I∗ h(∇qh)0,p,Ω.(2.30) 1226 T. CHAC ´ ON REBOLLO ET AL. As Ih(∇qh)∈Lp(Ω)d,thereexistsv∈Ls(Ω)dsuch that (Ih(∇qh),v)=Ih(∇qh)0,p,Ω v0,s,Ω. Now, let vhbe the L2orthogonal projection of von Yh. As the mesh is uniformly regular, the L2projection is stable in the Lsnorm (cf. [25]): vh0,s,Ω ≤Cv0,s,Ω. Then, as Ih(∇qh)∈Yh, (Ih(∇qh),vh)=(Ih(∇qh),v)≥CIh(∇qh)0,p,Ω vh0,s,Ω. Thus, Ih(∇qh)0,p,Ω ≤Csup vh∈Yh (Ih(∇qh),vh) vh0,s,Ω · Using again Ih+I∗ h=Id and (∇qh,vh)=−(∇·vh,q h) because vh∈Y,wehave Ih(∇qh)0,p,Ω ≤Csup vh∈Yh |(∇·vh,q h)| vh0,s,Ω +I∗ h(∇qh)0,p,Ω.(2.31) Therefore, from (2.30)and(2.31), h∇qh0,p,Ω ≤Csup vh∈Yh h|(∇·vh,q h)| vh0,s,Ω +hI∗ h(∇qh)0,p,Ω.(2.32) The global inverse inequality (2.5) between W1,s(Ω)andLs(Ω) and the quasi-uniformity of the mesh implies |vh|1,s,Ω ≤Ch −1vh0,s,Ω. Using this estimate in (2.32), we obtain h∇qh0,p,Ω ≤Csup vh∈Yh |(∇·vh,q h)| |vh|1,s,Ω +hI∗ h(∇qh)0,p,Ω.(2.33) Finally, (2.28) follows from (2.29)and(2.33).  Remark 2.7. At the angular corner points of the boundary, the condition v·n= 0 implies that v=0. So, when the domain Ωis approximating a curved domain the space Yhin (2.27)coincideswithXh.Inthe application to the Primitive equations of the Ocean studied in the next section, the boundary of the domain has a flat component (the surface) and the space Xhis strictly contained into Yh. 3. Application to the Primitive equations of the Ocean We study the fluid in a domain Ω={(x,z)∈Rdsuch that x∈ω, −D(x)≤z≤0}, where ωis a bounded domain in Rd−1and D:ω→Ris a Lipschitz-continuous non-negative function that represents the depth. The boundary is split as ∂Ω =Γs∪Γb,whereΓs={(x,0) ∈Rdsuch that x∈ω} represents the ocean surface, and Γbrepresents the ocean bottom and, eventually, sidewalls. A REDUCED DISCRETE INF-SUP CONDITION IN LPFOR INCOMPRESSIBLE FLOWS AND APPLICATION 1227 We consider a linearized version of the steady reduced Primitive equations model (cf. [20]). The problem consists in finding a horizontal velocity field u:Ω→ Rd−1and a surface pressure p:ω→ Rsuch that ⎧ ⎪ ⎨ ⎪ ⎩ W·∇u−μΔu+∇xp+ϕu⊥=fin Ω, ∇x·u=0 inω, u|Γb=0,μ z∂zu|Γs=g. (3.1) Here W:Ω→ Rdis a given convection velocity and u:ω→ Rd−1is defined by u(x)=0 −D(x) u(x,s)ds. (3.2) Also μis the viscosity coefficient, that we assume to be isotropic for simplicity. The term ϕu⊥is due the Coriolis acceleration and it appears only when d=3.Inthiscase,u=(u1,u 2)andu⊥=(−u2,u 1). The function ϕ is defined by ϕ=2θsin φ,whereθis the angular rotation rate of the earth and φis the latitude. The source term ftakes into account variable density effects, due to variations of temperature and salinity, and grepresents the wind tension at the surface. This model includes the rigid-lid assumption, stating that the free surface is flat (z= 0 in our case), and that the vertical velocity vanishes at the surface (cf. [13]). This last condition and the incompressibility of the velocity U=(u,u z) are used to define the vertical velocity uz:Ω→ Rfrom the horizontal velocity: uz(x,z)=0 z∇x·u(x,s)ds. (3.3) To define weak solutions of problem (3.1), we consider the spaces W1,s b(Ω)={v∈W1,s(Ω) such that v|Γb=0},for an integer s≥1. In particular H1 b(Ω)=W1,2 b(Ω). We denote H−1 b(Ω) the dual space of H1 b(Ω)d−1and H−1 2(Γs) the dual space of H1 2(Γs)d−1. We also consider the spaces Lp D(ω)=q:ω→ Rsuch that ω D(x)|q(x)|pdx<∞ and Lp D,0(ω)=Lp D(ω)/R, for all p∈(1,+∞). Problem (3.1) is a linearized version of the non-linear Primitive equations that in fact may be used as an intermediate step to prove its well posedness. We thus assume that the velocity field is W=(w,w z)with w∈H1 b(Ω)d−1and wz(x,z)=0 z∇x·w(x,s)ds. (3.4) The weak formulation of problem (3.1)thatweconsideris:Givenf∈H−1 b(Ω)andg∈H−1 2(Γs), ⎧ ⎨ ⎩ Find (u,p)∈H1 b(Ω)d−1×L 3 2 D,0(ω) such that B((u,p),(v,q)) = L(v),∀(v,q)∈W1,3 b(Ω)d−1×L2 D,0(ω), (3.5) 1234 T. CHAC ´ ON REBOLLO ET AL. and similarly sup Vh∈Xh(Oi) (∇·Vh,˜qh)Oi |Vh|1,s,Oi≤sup vh∈Uh(Oi) (∇x·vh,q h)Oi |vh|1,s,Oi·(3.42) Also, considering that ∇˜qh=(∇xqh,0), we have J∗ h(∇˜qh)0,p,Oi=Π∗ h(∇xqh)0,p,Oi.(3.43) Then, (3.39) follows from (3.40)bytakingintoaccount(3.41)–(3.43).  We assume the following local stability properties of the operator Πh: Hypothesis 3.6. The operator Πhsatisfies for all v∈L2(Ω)d−1, Πh(v)0,K ≤Cv0,wK,(3.44) Πh(v)0,3 2,K ≤Cv0,3 2,wK,(3.45) where wKis the union of all elements of Ththat intersect K. This hypothesis is stronger than Hypothesis 3.3, but is also verified by local L2projection or Lagrange interpolation-based operators. However it is not verified by the global L2(Ω) projection operator. Observe that as a consequence of (3.44), the operator Πhis also stable with respect to the norm ·τ: Πh(v)τ≤Cvτ,∀v∈L2(Ω)d−1.(3.46) Theorem 3.7. Assume that Hypotheses 2.1,3.1 and 3.6 hold. Then, the discrete problem (3.13)and (3.14) with l=1in (3.7)admits a unique solution (uh,p h)∈U h×P hwhich is bounded in H1 b(Ω)d−1×L 3 2 D,0(ω), satisfying the estimates |uh|1,Ω ≤C μfH−1 b(Ω)+gH−1 2(Γs),(3.47) phL 3 2 D,0(ω)≤C(1 + 1 μ+1 √μ)(1+|w|1,Ω)fH−1 b(Ω)+gH−1 2(Γs),(3.48) Π∗ h(Wh·∇uh+∇xph+ϕu⊥ h)τ≤C √μfH−1 b(Ω)+gH−1 2(Γs),(3.49) where Cis a constant independent of h. Moreover, the sequence {(uh,p h)}h>0contains a subsequence which is weakly convergent in H1 b(Ω)d−1× L 3 2 D,0(ω)to a solution of the continuous problem (3.5). If this solution belongs to W1,3 b(Ω)d−1, then the convergence is strong. Proof. The estimations (3.47)and(3.49) are obtained in the same way as in Theorem 3.4 but the estimate of the pressure is now based on the inf-sup condition (3.39)withp=s=2,whend=2,orp=3 2and s=3,when d= 3. We indicate the estimates in this last case. To estimate the first summand in (3.39), we start from (3.33). We treat all terms in the same way as before but to bound the term Π∗ h(Wh·∇vh+ϕv⊥ h)τwe have to argue locally because in this case we only can use local inverse inequalities: From (3.46), Π∗ h(Wh·∇vh+ϕv⊥ h)τ≤CWh·∇vh+ϕv⊥ hτ. Moreover, Wh·∇vh2 τ= K∈Th τKWh·∇vh2 0,K ≤α2 K∈Th h2 KWh2 0,K ∇vh2 0,∞,K. A REDUCED DISCRETE INF-SUP CONDITION IN LPFOR INCOMPRESSIBLE FLOWS AND APPLICATION 1235 Then, using (3.11), the local inverse inequality (2.4) between W1,∞(K)andW1,3(K), and (2.1), we derive Wh·∇vhτ≤C|w|1,Ω |vh|1,3,Ω. Likewise, ϕv⊥ h2 τ= K∈Th τKϕv⊥ h2 0,K ≤α2 K∈Th h2 Kϕ2 0,∞,Ω vh2 0,K ≤α2ϕ2 0,∞,Ω hvh2 0,Ω. So, ϕv⊥ hτ≤Cϕ0,∞,Ω |vh|1,3,Ω. Also from (3.46), Π∗ Uh(fh)τ≤Cfhτ. In this way, we obtain (3.35). To estimate the second summand in (3.39), we consider the variational formulation (3.13)and(3.14)with vh∈U h(Oi)andqh= 0. Then, (∇x·vh,p h)Oi=(Wh·∇uh,vh)Oi+μ(∇uh,∇vh)Oi+(ϕu⊥ h,vh)Oi +(Π∗ h(Wh·∇vh+ϕv⊥ h),Π∗ Uh(ch))τ,Oi−(Π∗ h(Wh·∇vh+ϕv⊥ h),Π∗ Uh(fh))τ,Oi−L(vh).(3.50) All terms in the right-hand side are estimated as before but here again we argue locally. For the stabilizing term, we have: (Π∗ h(Wh·∇vh+ϕv⊥ h),Π∗ Uh(ch))τ,Oi≤Π∗ h(Wh·∇vh+ϕv⊥ h)τ,OiΠ∗ Uh(ch)τ,Oi. To estimate the first factor, we first observe that we have the local version of (3.46): Π∗ h(Wh·∇vh+ϕv⊥ h)τ,Oi≤CWh·∇vh+ϕv⊥ hτ,Oi, because vh∈U h(Oi). Now, we expand the terms Wh·∇vhτ,Oiand ϕv⊥ hτ,Oi: Wh·∇vh2 τ,Oi= K∈Oi τKWh·∇vh2 0,K ≤α2 K∈Oi h2 KWh2 0,K ∇vh2 0,∞,K ≤C|w|2 1,Ω  K∈Oi|vh|2 1,3,K , applying (3.15), (3.11), the local inverse inequality (2.4) between W1,∞(K)andW1,3(K)and(2.1). Next, ϕv⊥ h2 τ,Oi= K∈Oi τKϕv⊥ h2 0,K ≤α2 K∈Oi h2 Kϕ2 0,∞,Ω vh2 0,K ≤Cϕ2 0,∞,Ω h2 K∈Oi|vh|2 1,3,K . But  K∈Oi|vh|2 1,3,K 1 2 ≤C|vh|1,3,Oi, because the number of elements in Oiis bounded by a constant independent of hand i. 1236 T. CHAC ´ ON REBOLLO ET AL. Then, (Π∗ hWh·∇vh+ϕv⊥ h,Π∗ Uh(ch))τ,Oi≤C(1 + |w|1,Ω)|vh|1,3,OiΠ∗ Uh(ch)τ,Oi. From here, applying Jensen’s inequality, ⎡ ⎣ R  i=1 sup vh∈Uh(Oi)|(Π∗ h(Wh·∇vh+ϕv⊥ h),Π∗ Uh(ch))τ,Oi| |vh|1,3,Oi3 2⎤ ⎦ 2 3 ≤C(1 + |w|1,Ω)R  i=1 Π∗ Uh(ch)2 τ,Oi 1 2 ≤C(1 + |w|1,Ω)Π∗ Uh(ch)τ, because the number of repetitions of a given element Kin all macro-elements is bounded by a fixed constant independent of h. Hence, by (3.49), ⎛ ⎝ R  i=1 sup vh∈Uh(Oi)|(Π∗ h(Wh·∇vh+ϕv⊥ h),Π∗ Uh(ch))τ,Oi| |vh|1,3,Oi3 2⎞ ⎠ 2 3 ≤C1 √μ(1 + |w|1,Ω)fH−1 b(Ω)+gH−1 2(Γs). We proceed in a similar way for the remaining terms and obtain R  i=1 sup vh∈Uh(Oi)|(∇x·vh,p h)Oi| |vh|1,3,Oi3 2 2 3 ≤C(1 + 1 μ+1 √μ)(1+|w|1,Ω)fH−1 b(Ω)+gH−1 2(Γs). (3.51) It remains to estimate the third summand in (3.39). First, we write: Π∗ h(∇xph)h, 3 2≤Π∗ h(ch)h, 3 2+Π∗ h(Wh·∇uh)h, 3 2+Π∗ h(ϕu⊥ h)h, 3 2·(3.52) Again we argue locally to bound the first term in the r.h.s. of (3.52). By definition, Π∗ h(ch)h, 3 2=R  i=1 h 3 2 iΠ∗ Uh(ch)) 3 2 0,3 2,Oi 2 3 . To bound this term, it is convenient to use the function H(x)=  K∈Th hKχK(x), where χKis the characteritic function of K.Then(2.2) implies that Π∗ h(ch)h, 3 2≤CR  i=1 HΠ∗ h(ch) 3 2 0,3 2,Oi 2 3 . Since any element K∈T hbelongs to at most Mmacro-elements, this yields Π∗ h(ch)h, 3 2≤C K∈ThHΠ∗ h(ch) 3 2 0,3 2,K 2 3 . A REDUCED DISCRETE INF-SUP CONDITION IN LPFOR INCOMPRESSIBLE FLOWS AND APPLICATION 1237 Finally, we associate with τkthe analogue of H: T(x)=  K∈Th τKχK(x). Then we infer from (3.15) and Cauchy−Schwarz’s inequality: Π∗ h(ch)h, 3 2≤C K∈Th √TΠ∗ h(ch) 3 2 0,3 2,K 2 3 =C√TΠ∗ h(ch)0,3 2,Ω ≤C√TΠ∗ h(ch)0,2,Ω =CΠ∗ h(ch)τ,(3.53) which is bounded by (3.49). To estimate the second term in the r.h.s. of (3.52), we use first the local stability of Πh: Π∗ h(Wh·∇uh) 3 2 h, 3 2≤C R  i=1 h 3 2 i K∈OiWh·∇uh 3 2 0,3 2,wK. Then we use the local quasi-uniformity of the mesh and the local inverse inequality (2.4) between W1,6(K)and W1,2(K), ∇uh0,6,K ≤Cρ −1 K∇uh0,K. Therefore Π∗ h(Wh·∇uh) 3 2 h, 3 2≤C K∈Th h 3 2 KWh·∇uh 3 2 0,3 2,wK≤C K∈Th h 3 2 K1 ρKWh0,wk∇uh0,wK3 2 ≤C K∈ThWh 3 2 0,wk∇uh 3 2 0,wK. Hence, Π∗ h(Wh·∇uh)h, 3 2≤C K∈Th Wh 3 2 0,K∇uh 3 2 0,K2 3 ≤C K∈Th Wh6 0,K1 6 |uh|1,Ω ≤CWh0,Ω|uh|1,Ω ≤C|w|1,Ω |uh|1,Ω,(3.54) by another application of Jensen’s inequality and (3.11). Similarly, we bound the last term in the r.h.s. of (3.52) using again (3.45)andobtain Π∗ h(ϕu⊥ h)h, 3 2≤Chϕ0,∞,Ω |uh|1,Ω.(3.55) Then, from (3.52) and taking into account (3.53)–(3.55): Π∗ h(∇xph)h, 3 2≤1 μ(1 + |w|1,Ω)+ 1 √μfH−1 b(Ω)+gH−1 2(Γs).(3.56) Finally, we derive (3.48)from(3.39) by combining (3.35), (3.51)and(3.56). The rest of the proof follows as in Theorem 3.4. Remark 3.8. When l>1in(3.7), Theorems 3.4 and 3.7 also hold if the constant α2in (3.15) is small enough. Inthiscase,wehavealsotoboundthetermΔuhτ.(See[14] for details). Acknowledgements. This research was partially supported by the Spanish Ministerio de Educaci´on y Ciencia Research and EU Feder Fund Project MTM2012 36124-C02-1. 1238 T. CHAC ´ ON REBOLLO ET AL. 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] C. Bernardi, Y. Maday and F. Rapetti, Discretisations variationnelles de probl`emes aux limites elliptiques. Springer-Verlag, Berlin (2004). [3] J. Blasco and R. Codina, Stabilized finite element method for the transient Navier-Stokes equations based on a pressure gradient projection. Comput. Methods Appl. Mech. Engrg. 182 (2000) 277–300. [4] M. Braack and E. Burman, Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 43 (2000) 2544–2566. [5] M. Braack, E. Burman, V. John and G. Lube, Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853–866. [6] S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edition. Springer-Verlag, Berlin (2008). [7] E. Burman, M.A. Fern´andez and P. Hansbo, Continuous interior penalty finite element method for Oseen equations. SIAM J. Numer. Anal. 44 (2006) 1248–1274. [8] E. Burman and M.A. Fern´andez, Continuous interior penalty finite element method for the time-dependent Navier−Stokes equations: space discretization and convergence. Numer. Math. 107 (2007) 39–77. [9] E. Burman, Interior penalty variational multiscale method for the incompressible Navier-Stokes equations: Monitoring artificial dissipation. Comput. Methods Appl. Mech. Engrg. 196 (2007) 4045–4058. [10] R. Codina, Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods. Comput. Methods Appl. Mech. Engrg. 190 (2000) 1579–1599. [11] T. Chac´on Rebollo, An analysis technique for stabilized finite element solution of incompressible flows. ESAIM: M2AN 35 (2001) 57–89. [12] T. Chac´on Rebollo and F. Guill´en Gonz´alez, An intrinsic analysis of existence of solutions for the hydrostatic approximation of Navier-Stokes equations. C. R. Acad. Sci. Paris, S´erie I 330 (2000) 841–846. [13] T. Chac´on Rebollo, R. Lewandowski and E. Chac´on Vera, Analysis of the hydrostatic approximation in oceanography with compression term. ESAIM: M2AN 34 (2000) 525–537. [14] T. Chac´onRebollo,M.G´omez M´armol and I. S´anchez Mu˜noz, Numerical solution of the Primitive equations of the ocean by the Orthogonal Sub-Scales VMS method. Appl. Numer. Math. 62 (2012) 342–356. [15] T. Chac´on Rebollo, M. G´omez M´armol, V. Girault and I. S´anchez Mu˜noz, A high order term-by-term stabilization solver for incompressible flow problems.IMA J. Numer. Anal. 33-3 (2013) 974–1007. [16] Ph. Ciarlet, The Finite Element Method for Elliptic Problems. Siamm (2002). [17] S. Ganesan, G. Matthies and L. Tobiska, Local projection stabilization with equal order interpolation applied to the Stokes problem. Math. Comput. 77 (2008) 2039–2060. [18] V. Girault and J.L. Lions, Two-grid finite-element schemes for the transient Navier-Stokes equations. ESAIM: M2AN 35 (2001) 945–980. [19] P. Knobloch, A generalization of the local projection stabilization for convection-diffusion-reaction equations. SIAM J. Numer. Anal. 48 (2010) 659–680. [20] J.L. Lions, R. Temman and S. Wang, New formulation of the primitive equations of the atmosphere and applications. Nonlinearity 5(1992) 237–288. [21] G. Matthies, P. Skrypacz and L. Tobiska, A unified convergence analysis for local projection stabilisations applied to the Oseen problem. ESAIM: M2AN 41 (2007) 713–742. [22] P. Oswald, On a BPX preconditioner for P1elements. Computing 51 (1993) 125–133. [23] H.G. Roos, M. Stynes and L. Tobiska, Robust numerical methods for singularly perturbed differential equations. 2nd edition. Springer Series Comput. Math. 24 (2008). [24] R. Verf¨urth, Analysis of some finite element solutions for the Stokes problem. RAIRO Anal. Numer. 18 (1984) 175–182. [25] L.B. Wahlbin, Local behavior in finite element methods. Elsevier Science, North Holland (1991).