HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS. PART I: THE STOKES PROBLEM

Transcription

1 MATHEMATICS OF COMPUTATION Volume 75, Number 254, Pages S (05) Article electronically publised on December 16, 2005 HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS. PART I: THE STOKES PROBLEM JESÚS CARRERO, BERNARDO COCKBURN, AND DOMINIK SCHÖTZAU Abstract. We devise and analyze a new local discontinuous Galerkin (LDG) metod for te Stokes equations of incompressible fluid flow. Tis optimally convergent metod is obtained by using an LDG metod to discretize a vorticity-velocity formulation of te Stokes equations and by applying a new ybridization to te resulting discretization. One of te main features of te ybridized metod is tat it provides a globally divergence-free approximate velocity witout aving to construct globally divergence-free finite-dimensional spaces; only elementwise divergence-free basis functions are used. Anoter important feature is tat it as significantly less degrees of freedom tan all oter LDG metods in te current literature; in particular, te approximation to te pressure is only defined on te faces of te elements. On te oter and, we sow tat, as expected, te condition number of te Scur-complement matrix for tis approximate pressure is of order 2 in te mes size. Finally, we present numerical experiments tat confirm te sarpness of our teoretical a priori error estimates. 1. Introduction Tis is te first in a series of papers in wic we propose and analyze ybridized, globally divergence-free local discontinuous Galerkin (LDG) metods. In tis paper, we present te main ideas in a simple setting and consider te Stokes equations u +gradp = f, div u =0 inω; u = u D on Γ = Ω; wic model te flow of a viscous, incompressible fluid. Here Ω is taken to be a bounded polygonal domain in R 2, f L 2 (Ω) 2 a given source term, and u D H 1/2 (Γ) 2 is a prescribed Diriclet datum satisfying te usual compatibility condition (1.1) u D nds=0, Γ wit n denoting te outward normal unit vector on te boundary Γ of te domain Ω. We devise an LDG metod tat is locally conservative, optimally convergent and provides an approximate velocity wic is globally divergence-free, tat is, wic Received by te editor June 9, 2004 and, in revised form, February 9, Matematics Subject Classification. Primary 65N30. Key words and prases. Divergence-free elements, local discontinuous Galerkin metods, ybridized metods, Stokes equations. Te second autor was supported in part by te National Science Foundation (Grant DMS ) and by te University of Minnesota Supercomputing Institute. Te tird autor was supported in part by te Natural Sciences and Engineering Researc Council of Canada. 533 c 2005 American Matematical Society License or copyrigt restrictions may apply to redistribution; see ttp://

2 534 JESÚS CARRERO, BERNARDO COCKBURN, AND DOMINIK SCHÖTZAU belongs to te space H(div 0 ;Ω)={ v L 2 (Ω) 2 :div v =0inΩ}. One of te advantages of working wit finite-dimensional subspaces of H(div 0 ;Ω) is tat te approximate velocity as a smaller number of degrees of freedom in comparison wit standard finite element spaces wit similar approximation properties. Anoter advantage is tat te pressure can be completely eliminated from te equations, and te resulting problem for te velocity becomes a simple second-order elliptic problem. An LDG metod for te velocity can be devised in a most straigtforward way; te pressure can be easily computed once te velocity is obtained. On te oter and, finite-dimensional subspaces of H(div 0 ; Ω) are extremely difficult to deal wit because teir basis functions are complicated to construct and do not ave local support. Te main contribution of tis paper is to sow tat tis difficulty can be overcome by a simple ybridization of te LDG metod. Tis ybridization is similar to tat of mixed metods for second-order elliptic problems; see [1], [8], [9], and [12]. It is acieved as follows. First, we relax te continuity of te normal components of te approximate velocity across interelement boundaries. Tis allows us to use velocity spaces wose elements are completely discontinuous and divergence-free inside eac element. In tis procedure, owever, te pressure as to be reintroduced in te equations as te corresponding Lagrange multiplier. Furtermore, new equations ave to be added to te system to render it solvable. Tese equations enforce te continuity of te normal components of te new approximate velocity. Tus, altoug te approximate velocity is divergencefree only inside eac element, te additional equations automatically ensure tat it is also globally divergence-free. In tis way, te proposed ybridization allows one to base te approximate velocity on te space H(div 0 ; Ω) witout aving to work wit globally divergence-free finite-dimensional spaces. Anoter advantage of tis approac is tat te resulting ybridized LDG metod produces a remarkably small Scur-complement matrix for te pressure, given tat tis variable is only defined on te edges of te triangulation; te values of te pressure inside te elements can be computed in an element-by-element fasion. Te price to pay for tese advantages is te fact tat te condition number of te Scur-complement matrix for te pressure is of order 2, instead of being of order one, as is typical of classical mixed metods. Fortunately, good preconditioners can be obtained by using te tecniques developed in [5]. We also refer te reader to [24] for Scwarz preconditioners for ybridized discretizations of second-order problems. We carry out te numerical analysis of te resulting ybridized LDG metod. We sow tat it is well defined and derive optimal a priori error bounds in te classical norms. Tus, for approximations using polynomials of degree k 1, te error in te velocity, measured in an H 1 -like norm, is proven to converge wit te optimal order k wit respect to te mes size. Similarly, we sow tat te L 2 - errors in te vorticity and te pressure converge wit te optimal order k wen tese variables are taken to be piecewise polynomials of degree k 1. Te same rates are obtained if all te unknowns are approximated by polynomials of degree k. We display numerical results sowing tat tese results are sarp. Finally, let us point out tat, altoug ere we use te LDG approac to discretize te Stokes problem, our teoretical results are equally valid for any oter License or copyrigt restrictions may apply to redistribution; see ttp://

3 HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS 535 stable and consistent DG discretization as described in te unifying analysis for second-order problems proposed in [2]. Tis paper is organized as follows. In Section 2, we present a brief overview of te development of divergence-free metods for te Stokes problem, in order to place our work into perspective. Te ybridized LDG metod is ten described in Section 3. In Section 4, we present our main teoretical results, namely, optimal error estimates for te velocity and te vorticity (Teorem 4.1) and optimal error estimates for te pressure (Teorem 4.2). We also outline te main steps of te proof of tese estimates wile te details of some of te auxiliary results are given in Section 6. In Section 5, we discuss a modification of te matrix equations tat allows us to use minimization algoritms to actually solve te resulting linear systems. We furter state an upper bound on te condition number of te Scur-complement matrix for te pressure (Teorem 5.2). Tis bound follows from an inf-sup condition wose proof is contained in Section 7. Ten in Section 8, we present numerical experiments tat confirm tat te teoretical orders of convergence are sarp. Finally, we end in Section 9 by discussing some extensions of te proposed metod. 2. A istorical overview of te main ideas for enforcing incompressibility To properly motivate te devising of ybridized LDG metods, we briefly discuss te evolution of some of te main ideas and tecniques used to deal wit te incompressibility condition div u = 0. To simplify te exposition, we take u D = Enforcing exact incompressibility. By considering finite-dimensional subspaces V H0 1 (Ω) 2 H(div 0 ; Ω), exactly incompressible velocity approximations u V can be readily defined by requiring tat (grad u, grad v) =( f, v), v V. Unfortunately, since te early beginnings of te development of finite element metods for incompressible flow, it was clear tat te construction of suc finitedimensional spaces V was an extremely difficult goal to acieve. Indeed, in is pioneering work of 1972, Fortin [20] was able to construct spaces of tat type, but tey turned out to be complex elements of limited applicability, as Crouzeix and Raviart said in teir seminal paper of 1973, [18]. In an effort to be able to use simpler metods, tese autors proposed an alternative approac Enforcing weak incompressibility. Crouzeix and Raviart [18] sacrificed te exact verification of te incompressibility condition and opted instead for enforcing it only weakly; te pressure must ten be considered simultaneously wit te velocity. In te case of conforming metods, for example, we take ( u,p ) in a finite-dimensional space V Q H0 1(Ω)2 L 2 (Ω)/R, and determine it by requiring tat (grad u, grad v) (p, div v) =( f, v), (q, div u )=0, ( v, q) V Q. Of course, we can still try to solve only for te velocity in te above metod. Indeed, since te approximate velocity u belongs to te finite-dimensional space Z = { v V } :(q, div v) =0, q Q, License or copyrigt restrictions may apply to redistribution; see ttp://

4 536 JESÚS CARRERO, BERNARDO COCKBURN, AND DOMINIK SCHÖTZAU it can be caracterized as te only element of Z suc tat (grad u, grad v) =( f, v), v Z. Te pressure can tus be eliminated from te equations and recovered once te velocity u is computed (see [10] and [11]) by solving (p, div v) =( f, v) (grad u, grad v), v V. Te solvability of tis equation is guaranteed by a discrete inf-sup condition for te metod; see te books [9] and [22], and te recent article [27]. Unfortunately, bases for te space Z of weak incompressibility are also very difficult to construct. Bases for spaces of tis type were constructed by Griffits in 1979 [25], by Tomasset in 1981 [38], bot for te two-dimensional case, and by Hect in 1981 [28] for te tree-dimensional case and te nonconforming metod using piecewise-linear functions. However, tey ave a support tat is not necessarily local; tis as a negative impact on te sparsity of te stiffness matrix. Yet anoter problem wit tis approac was discovered in 1990 by Dörfler [19] wo sowed tat te condition number of te stiffness matrix for te velocity is of order 4. Because of all tese difficulties, tis approac was never very popular Weak incompressibility can ensure exact incompressibility. Of course, exact incompressibility follows from weakly imposed incompressibility if div V Q. Tis condition is satisfied, for example, if we take V to be te space of continuous vector fields wic are polynomials of degree k on eac triangle and W te space of discontinuous functions wic are piecewise polynomials of degree k 1. In 1985, Scott and Vogelius [37] sowed tat te constant of te inf-sup condition for tis conforming metod is independent of te mes size, provided k 4(and a minor condition on te triangulation olds). Tis implies tat te approximate velocity u is globally divergence-free and tat te solution ( u,p ) is optimally convergent; see oter cases in [6]. Lower order polynomial spaces wit tis property do not exist, as was proven in 1975 by Fortin [21]. Extensions of tese results to te tree-dimensional case seem to be quite callenging and remain an interesting open problem. Oter examples of ow weak incompressibility implies exact incompressibility can be found in te book by Gunzburger [26] Locally incompressible approximations. In 1990, Baker, Jureidini and Karakasian considered locally incompressible velocities, tat is, tey took V { v L 2 (Ω) 2 : v K H(div 0 ; K), K T }, wit T being a triangulation of Ω. Since te use of tis space rendered impossible te satisfaction of te continuity constraints typical of conforming and nonconforming metods, te autors were led to use completely discontinuous approximations for te velocity; te approximate pressures were, owever, continuous. Tis optimally convergent metod was extended in te late 90 s to te Navier Stokes equations by Karakasian and Katsaounis [30] and by Karakasian and Jureidini [29] wit excellent results. Recently, several DG metods for incompressible flow ave been proposed in te literature, all of wic impose te incompressibility condition weakly; see [4], [17], [16], [14], [15], [23] and [36]. General DG metods face an important difficulty wen applied to te Navier Stokes equations. In tis case, te appearance of te nonlinear convection introduces a penomenon tat is not present in te Stokes or Oseen License or copyrigt restrictions may apply to redistribution; see ttp://

5 HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS 537 equations, namely, tat energy-stable DG metods, like te metods proposed in [29] and in [23], cease to be locally conservative because te incompressibility condition is enforced only weakly Exactly incompressible, locally conservative LDG metods. To recover te igly valued property of local conservation for DG metods for te Navier Stokes equations, a new way to deal wit te incompressibility condition was introduced by Cockburn, Kanscat and Scötzau in [15]. It is based on te observation tat, for mixed DG discretizations of te incompressibility condition, a globally divergence-free velocity can be easily computed by using an element-byelement post-processing of te approximate velocity u. Since te velocity u is only weakly incompressible, tis gives anoter way of enforcing te incompressibility condition strongly by only enforcing it weakly. For tis locally conservative, optimally convergent DG metod, te velocity is taken to be a piecewise polynomial of degree k 1 and te pressure a piecewise polynomial of degree k 1. Tis is te first metod to produce globally divergencefree velocities by using polynomials of degree equal or bigger tan one, in bot two and tree space dimensions Hybridized, exactly incompressible LDG metods. Te LDG metod we propose in tis paper is devised in an effort to reduce te number of unknowns of te LDG metod in [15] wile maintaining te exact incompressibility of its approximate velocity. Tis is acieved in two steps. First, we devise an LDG metod wic uses approximate velocities in spaces V suc tat V H(div 0 ;Ω). Ten, we ybridize te metod. Tat is, we base its velocity approximation on spaces of te form V { v L 2 (Ω) 2 : v K H(div 0 ; K), K T }, and still get te approximate velocity in V. Te spaces V are remarkably simpler to deal wit from te implementational point of view. It must be empasized tat it is not known ow to carry out a similar ybridization for classical conforming and nonconforming metods for te Stokes problem. Let us briefly discuss te nature of tis difficulty. Wen dealing wit te LDG metod proposed in tis paper, te original velocity spaces are suc tat V H(div 0 ; Ω), and ence to perform an ybridization, we only ave to deal wit te continuity of te normal components across interelement boundaries. On te oter and, for conforming metods, te velocity spaces satisfy te inclusion V H0 1 (Ω) 2, and ence teir ybridization must involve te more difficult issue of ow to andle te continuity of te wole function and not only tat of its normal component. For classical nonconforming metods, a similar situation arises. 3. Te ybridized LDG metod In tis section, we introduce te ybridized LDG metod. After introducing te notation we are going to use, we first describe te LDG metod wit globally divergence-free velocities and ten present its ybridization. Finally, we sow ow to compute te pressure in an element-by-element fasion in terms of te approximate solution of te ybridized LDG metod. License or copyrigt restrictions may apply to redistribution; see ttp://

6 538 JESÚS CARRERO, BERNARDO COCKBURN, AND DOMINIK SCHÖTZAU 3.1. Preliminaries. To define te metod, we need to introduce some notation. Let us begin wit te notation related to te triangulation of te domain. We denote by T a regular and sape-regular triangulation of Ω into triangles and set = max =max K T { K },were K is te diameter of te triangle K. Wewrite (p, ψ) T = K T (p, ψ) K, were (p, ψ) K = K pψdx. Furter, we denote by E I te set of interior edges of T and by E B te set of boundary edges. We define E = E B EI.IfF is a subset of E,weusetenotation f,φ F = e F f,φ e, were f,φ e = e fφds.for an element K T, te boundary K will be understood as a subset of E. Next, let us deal wit te notation associated wit weak formulations. Tus, if ϕ satisfies te equation O ϕ =Φ,wereO is a first-order differential operator and ϕ is a scalar- or vector-valued function, we can write, for any element K T, tat (Oϕ, ψ) K =(ϕ, O ψ) K + ϕ, ψ n K K =(Φ,ψ) K, were ψ is any smoot function, O is te formal adjoint operator to O and te symbol stands for te corresponding multiplication operator on te boundary (wit respect to te outward unit normal vector n K on K). Adding over te triangles of te triangulation, we obtain (ϕ, O ψ) T + ϕ, ψ n K K, =(Φ,ψ) T. K T Next, we define te jump operator [ ] by { ϕ (ψ n) on boundary edges in E B [ϕ (ψ n)] =, ϕ + (ψ + n K +)+ϕ (ψ n K ) on interior edges in E I. Here ϕ ± and ψ ± denote te traces of ϕ and ψ on te edge e = K + K taken from witin te interior of te triangles K ±.Wecantusrewriteteabove equation as (ϕ, O ψ) T + 1, [ϕ (ψ n)] E =(Φ,ψ) T. Finally, to be able to define te discrete traces of te LDG metod, it only remains to introduce te average { } of te traces, namely, { ϕ on boundary edges in E B {ϕ } =, 1 2 (ϕ+ + ϕ ) on interior edges in E I. We are now ready to define te LDG metod An LDG metod for te vorticity-velocity formulation. We first introduce te LDG metod tat is based on velocities in H(div 0 ; Ω). To introduce its weak formulation, we rewrite te Stokes system as (3.1) ω curl u =0 inω, (3.2) curl ω +gradp = f in Ω, (3.3) div u =0 inω, (3.4) u = u D on Γ, were ω is te scalar vorticity, ω =curl u = 1 u 2 2 u 1, and curl ω is te vectorvalued curl given by curl ω =( 2 ω, 1 ω). License or copyrigt restrictions may apply to redistribution; see ttp://

7 HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS 539 Next, we multiply equation (3.1) by a test function σ L 2 (Ω), equation (3.2) by v H(div 0 ; Ω), suc tat v n = 0 on Γ, and equation (3.4) by nq wit q L 2 (Γ). We assume tat te test functions v and σ are smoot inside te triangles K but migt be discontinuous in Ω. Ten, after integrating by parts and making use of te identities a b = a 1 b 2 a 2 b 1,c a = c ( a 2,a 1 ), and a c = c a, weobtain (ω, σ) T ( u, curl σ) T u, [σ n] E =0, (ω, curl v) T + ω, [ v n] E =( f, v) T, u n, q E B = u D n, q E B. Here we ave also used te continuity of te tangential components of u and ω. Note tat te pressure p does not appear in te equations; tis is due to te fact tat v H(div 0 ; Ω). Moreover, only te information about te tangential component of te Diriclet boundary condition u D appears in te first equation, wereas te information about its normal component is contained in te tird equation. Finally, if q is a constant, we ave u D n, q E B = q u D n, 1 E B =0, by te compatibility condition (1.1) on u D. Tus, in order to ensure tat te formulation is well defined, we must take q modulo constants. Motivated by te fact tat te exact solution satisfies te above equations, we define te LDG approximation (ω, u ) in te finite-dimensional space Σ k V k by requiring tat (3.5) (ω,σ) T ( u, curl σ) T u, [σ n] =0, E (3.6) (ω, curl v) T + ω, [ v n] E =( f, v) T, (3.7) u n, q E B = u D n, q E B for all (σ, v, q) Σ k V k Qk /R wit v n =0onΓ,were Σ k = { σ L 2 (Ω) : σ K P k 1 (K), K T }, V k = { v H(div 0 ;Ω): v K P k (K), K T }, Q k = { q L 2 (Γ) : q e P k (e), e E B }. Here P l (D) denotes te space of polynomials of degree l on D. Te quantities u and ω are approximations to te traces of u and ω ; tey must be suitably defined to render te metod stable and optimally convergent. On interior edges in E I, tese discrete traces are cosen as (3.8) ω = {ω } + E [ω n] + D [ u n], u = { u } + E [ u n]. Similarly, on boundary edges in E B,wetake (3.9) ω = ω + D ( u u D ) n, u = u D. Here D and E are functions defined on E and E I, respectively. Te function D as a stabilization role, wereas te proper definition of te function E can reduce te sparsity of te resulting matrices and migt even ave a positive impact on te accuracy of te approximation in some cases; see [13]. Tis completes te definition of te LDG metod wit globally divergence-free velocity spaces. License or copyrigt restrictions may apply to redistribution; see ttp://

8 540 JESÚS CARRERO, BERNARDO COCKBURN, AND DOMINIK SCHÖTZAU Wit arguments similar to tose in [17, Proposition 2.1], it can be readily seen tat LDG metod in (3.5) (3.9) as a unique solution (ω, u ) Σ k V k,provided tat D > Te ybridized LDG metod. Next, we ybridize te LDG metod described in (3.5) (3.9). Te purpose of te ybridization is to base te approximation of te velocities on te space of locally divergence-free functions given by V k = { v L 2 (Ω) 2 : v K J k (K), K T }, were Jk (K) ={ v P k (K) : div v =0onK }. Tis space is noticeably bigger tan te space V k but as te advantage of being a set of functions wic are totally discontinuous across interelement boundaries. As a consequence, only local basis functions are needed for its implementation. On te oter and, te price we pay for tis is tat we need to reintroduce te pressure in te equations. To tis end, we define te space Q k = { q L 2 (E ): q e P k (e), e E }. Tus, we define te LDG approximation (ω, u,p ) Σ k V k Qk /R by requiring tat (3.10) (ω,σ) T ( u, curl σ) T u, [σ n] E =0, (3.11) (ω, curl v) T + ω, [ v n] E + [ v n],p E =( f, v) T, (3.12) [ u n],q E = u D n, q E B for all (σ, v, q) Σ k V k Qk /R. Note tat te pressure p now appears in (3.11); tis is because [ v n] is not necessarily equal to zero. Note also tat if q is a constant on E,weavetat,for v V k, [ v n],q E = v n K,q K = q (div v, 1) K =0, K T K T and tat, in view of (1.1), u D n, q E B = q u D n, 1 E B =0. Tus, to ensure tat te above weak formulation is well defined, we must take q in te space Q k /R. Next, we establis tat te LDG metod in (3.10) (3.12), completed wit te definition of te discrete traces (3.8) (3.9), is well defined. Moreover, we sow tat its approximate vorticity and velocity are also te solution of te original LDG metod in (3.5) (3.9). In oter words, even toug te ybridized LDG metod works wit locally divergence-free, but totally discontinuous, approximate velocities, it provides a globally divergence-free velocity field u H(div 0 ;Ω). Proposition 3.1. If D > 0 and k 1, tere is a unique solution (ω, u,p ) Σ k V k Qk /R of (3.10) (3.12) wit te numerical fluxes in (3.8) (3.9). Moreover, (ω, u ) is also te solution of (3.5) (3.9). Inparticular, u V k H(div0 ;Ω). To prove tis result, we use te following stability result wose detailed proof is presented in Section 7 below; cf. Proposition 7.1. License or copyrigt restrictions may apply to redistribution; see ttp://

9 HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS 541 Proposition 3.2. Let q Q k,werek 1. Ten for any K T tere is a velocity field v J k (K) suc tat v n K,q K q m K (q) 2 0, K, wit m K (q) denoting te mean value m K (q) = 1 K 1,q K. Here K is te one-dimensional measure of K. Proof of Proposition 3.1. To prove te existence and uniqueness of te approximate solution in (3.10) (3.12), it is sufficient to sow tat te only LDG approximation to te omogeneous Stokes problem wit u D = 0 andf = 0 is te trivial one. In tis case, by using arguments similar to tose in [17, Proposition 2.1], it can be readily seen tat (ω, u )=(0, 0), provided tat D > 0. Equation (3.11) ten becomes 0= [ v n],p E = v n K,p K, v V k. K T From te stability result in Proposition 3.2, we can find a field v V k suc tat 0= v n K,p K p m K (p ) 2 0, K. K T K T Tis immediately implies tat p must be constant on K for all K T. Hence, te multiplier p is constant on E, wic sows te first claim. To sow te second claim, note tat (3.12) implies tat te normal component of u is continuous across interelement boundaries and tat u n, q E B = u D n, q E B, q Q k. We conclude tat u is in H(div 0 ; Ω) and satisfies (3.7). Furtermore, in view of te inclusion V k V k, we obtain tat (ω, u ) is also te solution of (3.5) (3.7) Recovering te pressure. Next, we sow tat once te solution of te ybridized LDG metod is computed, it is possible to recover te pressure on te wole domain Ω in an element-by-element fasion. To see tis, note tat if we multiply te equation (3.2) by a test function v and integrate over te triangle K, weget (p, div v) K =( f, v) K (ω, curl v) K v n K,p K ω, v n K K. We ten set Q k = {q L 2 (Ω) : q K P k 1 (K), K T }, and take te pressure p Q k on eac triangle K as te element of P k 1(K) suc tat (3.13) (p, div v) K =( f, v) K (ω, curl v) K v n K,p K ω, v n K K for all v in P k (K). Proposition 3.3. Te pressure p Q k given by (3.13) is well defined for k 1. Proof. Since P k 1 (K) =div P k (K), we only ave to prove tat if div v K =0, ten te rigt-and side of (3.13) is also equal to zero. But in tat case, v J k (K), and te rigt-and side of (3.13) is zero by (3.11) defining te ybridized LDG metod. License or copyrigt restrictions may apply to redistribution; see ttp://

10 542 JESÚS CARRERO, BERNARDO COCKBURN, AND DOMINIK SCHÖTZAU Tis concludes te presentation of te metod. Let us point out tat tis metod as a considerable smaller number of degrees of freedom tan te oter LDG metods for incompressible fluid flow, a typical example of wic is te one considered in [15]. To see tis, let us recall tat te metod in [15] provides approximations for ( σ 1, σ 2, u, p), since to define it, we need to rewrite te Stokes system as follows: (3.14) σ i grad u i =0, div σ i + i p = f i i=1,2, div u =0 inω. Instead, te LDG metod proposed in tis paper only provides approximations for (ω, u, p). Tus, we see tat te LDG metod under consideration approximates a considerably smaller number of variables. Of course, an extension of te LDG metod in [15] to te system of equations (3.1) (3.4) we are using to define our LDG metod is not difficult to devise and analyze. Conversely, te ybridized LDG metod proposed in tis paper can be straigtforwardly extended to te system in (3.14). 4. Error analysis of te metod In tis section, we present two a priori error estimates for te ybridized LDG metod. Te first, Teorem 4.1, states tat te vorticity ω and velocity u converge at an optimal rate. Te second, Teorem 4.2, states tat te pressure at te interior of te elements, p, is also optimally convergent. Tese are our main teoretical results Te error estimates. Te error estimates we obtain are for te ybridized LDGmetodforwicwetake (4.1) D e = d 1 e e E, wit e denoting te lengt of te edge e and d being a positive parameter tat is independent of te mes size, and (4.2) E e C e E I, wit a constant C independent of te mes size. We use te norm v 2 1, = grad v 2 0,K + 1 e [ v n] 2 0,e. K T e E Here, for a =(a 1,a 2 )and b =(b 1,b 2 ), we denote by a b te matrix given by [ a b] ij = a i b j. Teorem 4.1. Let (ω, u, p) be te exact solution of te Stokes system, and let (ω, u,p ) Σ k V k Qk /R be te approximation given by te ybridized LDG metod wit k 1. If u H s+1 (Ω) 2 for s 1, ten ω ω 0 + u u 1, C min{k,s} u s+1, were te constant C is independent of te mes size and te exact solution. Teorem 4.2. Let p be te exact pressure of te Stokes system and let p Q k be its approximation given by te post-processing metod (3.13) wit k 1. If u H s+1 (Ω) 2 and p H s (Ω) for s 1, ten p p L 2 (Ω)/R C min{k,s} [ ] u s+1 + p s, were te constant C is independent of te mes size and te exact solution. License or copyrigt restrictions may apply to redistribution; see ttp://

11 HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS 543 Remark 4.3. Te convergence rates in te above results are optimal wit respect to te approximation properties of te underlying finite element spaces. Te same convergence rates are obtained if te approximate vorticity ω is taken to be a piecewise polynomial of degree k, tat is, if it belongs to te space Σ k+1. Remark 4.4. Note tat, in te first result in Teorem 4.1, noting is said about te error of te approximation provided by te pressure on te mes edges, p. Tis reflects te fact tat te original LDG metod in (3.5) (3.7) does not involve any pressure variable in its definition. Wile it can be readily proven, using te inf-sup condition in Teorem 5.1 below, tat, on quasi-uniform meses, p converges to p wit order k 1 inanl 2 -like norm, te numerical experiments of Section 8 actually indicate tat p converges to p wit order k. However, since p is a piecewise polynomial of degree k, tis observed rate of convergence is suboptimal wit respect to te approximation properties of Q k. A complete teoretical understanding of tis result, certainly linked wit te fact tat te vorticity and te velocity converge wit order k, still remains to be acieved Te compact form of te metod. To facilitate te analysis we rewrite te ybridized LDG metod in a compact from. We do tis by fully exploiting te fact tat te vorticity can be easily eliminated from te equations; we follow [2]. For v V (), were V () = V k + [ H 1 (Ω) 2 H(div 0 ;Ω) ], we define te lifted element L E ( u) Σ k by (4.3) (L E ( u),σ) T = E[ u n], [σ n] + {σ }, [ u n] E σ Σ k. Similarly, te lifting U D Σ k of te boundary datum u D is given by (U D,σ) T = u D,σ n E B σ Σ k. We note tat, for te exact velocity u, weave (4.4) L E ( u) = U D. Tis can easily be seen using te fact tat te jump of u vanises over interior edges and tat u = u D on boundary edges. By integration by parts, it is easy to see tat te first equation (3.10) in te definition of te ybridized LDG metod can be rewritten as (4.5) ω =curl u + L E ( u )+U D on eac K T. Next, we use tis expression to eliminate te vorticity from te equations. Tus, from te definition of te numerical fluxes and te lifting L E, te second equation (3.11) can be expressed as (ω, curl v + L E ( v)) T + D[ u n], [ v n] E + [ v n],p E =( f, v) T + D( u D n), v n E B, and, using (4.5), becomes (4.6) (curl u + L E ( u ), curl v + L E ( v)) T + D[ u n], [ v n] E + [ v n],p E =( f, v) T (U D, curl v + L E ( v)) T + D( u D n), v n E B. E I License or copyrigt restrictions may apply to redistribution; see ttp://

12 544 JESÚS CARRERO, BERNARDO COCKBURN, AND DOMINIK SCHÖTZAU Finally, by introducing te bilinear forms (4.7) A ( u, v) =(curl u + L E ( u), curl v + L E ( v)) T + D[ u n], [ v n] E, (4.8) B ( v, q) = [ v n],q E, as well as te functionals (4.9) F ( v) =( f, v) T (U D, curl v + L E ( v)) T + D( u D n), v n E B, (4.10) G (q) = u D n, q E B, we are led to consider te following mixed formulation: Find ( u,p ) V k Qk /R suc tat { A ( u, v) + B ( v, p ) = F ( v) v V (4.11) k, B ( u,q) = G (q) q Q k /R. Te formulation in (4.11) is te compact form of te LDG metod tat we are going to use in our error analysis. Furtermore, in order to write te post-processing procedure (3.13) in a compact form, we introduce te space V k = { v L 2 (Ω) 2 : v K P k (K), K T }. Now, if we set B ( v, p) = (p, div v) T, ten te recovered pressure p is te element of te space Q k given by B ( v, p )=F ( v) A ( u, v) B ( v,p ) for all v V k. Here we note tat te definition of te form A can be straigtforwardly extended to te te space V() = V k+[ H 1 (Ω) 2 H(div 0 ;Ω) ] by extending te lifting operator L E to an operator V() Σ k, using te same definition. Tis is te framework we are going to use to carry out te analysis of te metod Stability properties. Next, we state te main stability properties of te forms A, B and B. Te proofs of te most difficult properties are presented in full detail in Section Continuity. We start by noting te following continuity properties: (4.12) A ( u, v) C A,cont u 1, v 1,, u, v V(), (4.13) B ( v,q) C B,cont v 1, q L2 (E ;)/R, v V(), q Q k, (4.14) B ( v, q) C B,cont v 1, q 0, v V(), q Q k, wit continuity constants C A,cont, C B,cont and C B,cont tat are independent of te mes size. Here we use te norms (4.15) q 2 L 2 (E ;) = e q 2 0,e, e E q L 2 (E ;)/R =inf L c R 2 (E ;). To see te continuity of A, we first note tat tere olds (4.16) L E ( v) 2 0 Clift 2 1 e [ v n] 2 0,e, v V(), e E License or copyrigt restrictions may apply to redistribution; see ttp://

13 HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS 545 wit a constant C lift independent of te mes size; see [33, Section 3] or [34, Proposition 4.2] for details. Property (4.12) ten follows from te above estimate and an application of te Caucy Scwarz inequality. Te continuity properties in (4.13) and (4.14) are readily obtained from te weigted Caucy Scwarz inequality Coercivity of A. Next, we discuss te coercivity properties of te form A. Te following trivial stability result olds: (4.17) A ( v, v) C [ K T ( curl v 2 0,K + div v 2 0,K) + 1 e [ v n] 2 ] 0,e, v V k ; e E see [2] or [34, Lemma 4.5]. Here we ave used te fact tat functions in V k are locally incompressible. Ten, if we introduce te norm (see [22, Lemma I.2.5 and Remark I.2.7]) (4.18) v 2 = ( curl v 2 0,K + div v 2 0,K) + K T we immediately get tat (4.19) A ( v, v) C v 2, v V k wereweaveset 1 e e E ( [ v n] 2 0,e+ [ v n] 2 ) 0,e, H 0 (div; Ω), H 0 (div; Ω) = { v L 2 (Ω) : div v L 2 (Ω), v n =0onΓ}. Te following result sows tat te norm is actually equivalent to te norm 1,. Proposition 4.5. On V k,tenorms 1, and are equivalent uniformly in te mes size. Tat is, tere are constants C 1 and C 2 independent of te mes size suc tat C 1 v 1, v C 2 v 1, for all v V k. Te proof of Proposition 4.5 is carried out in Section 6.1. Combining Proposition 4.5 and (4.19), we obtain te following coercivity property: (4.20) A ( v, v) C A,coer v 2 1,, v V k H 0 (div; Ω), wit a coercivity constant C A,coer tat is independent of te mes size Te inf-sup condition for B. To derive te error estimates in te pressure p, we will make use of te following inf-sup condition. Tere is a constant C B,is independent of te mes size suc tat B ( v,q) (4.21) sup C B,is P v V k H v Q k q L2 (Ω)/R, q L 2 (Ω), 0(div;Ω) 1, were P Q k is te L 2 -projection onto Q k. Te proof of tis result is carried out in Section Sketc of te proofs of te error estimates. Next, we outline te main steps of te proofs of our error estimates. We begin by addressing te fact tat, after elimination of te vorticity, te metod under consideration does not ave te so-called Galerkin ortogonality property. License or copyrigt restrictions may apply to redistribution; see ttp://

14 546 JESÚS CARRERO, BERNARDO COCKBURN, AND DOMINIK SCHÖTZAU Approximate Galerkin ortogonality. It is well known tat te property of Galerkin ortogonality is crucial in te analysis of classical finite element metods. However, due to te use of te lifting operators in te definition of te form A, suc a property does not old; see [33] and [34]. Instead, we ave wat we refer to as approximate Galerkin ortogonality. Tis is stated in te following result. Lemma 4.6. Te exact solution (ω, u, p) of te Stokes problem satisfies { A ( u, v) + B ( v, p) = F ( v)+r (ω, v) v V (4.22) k, B ( u, q) = G (q) q Q k /R, as well as (4.23) B ( v,p)=f ( v) B ( v,p) A ( u, v)+r (ω, v) v V k. Here R (ω, v) is te expression R (ω, v) = E[ v n], [P Σ k ω n] E I + {P Σ k ω ω }, [ v n] E, wit P Σ k denoting te L 2 -projection onto Σ k. Proof. We first note tat te second equation in (4.22) is trivially satisfied since u is continuous across interelement boundaries and satisfies u = u D on Γ. To prove te first equation in (4.22), consider te expression Θ = A ( u, v)+b ( v, p) F ( v), for v V k. As a direct consequence of te definitions of te forms A, (4.7), B, (4.8), and F, (4.9), of te fact tat L E ( u) = U D, (4.4), of te continuity of u across interelement boundaries, and of te fact tat on te boundary u = u D,we immediately get Θ =(ω, curl v + L E ( v)) T + [ v n],p E ( f, v) T =(ω, curl v + L E ( v)) T + (grad p f, v) T since v V k, =(ω, curl v + L E ( v)) T +( curl ω, v) T by (3.2), =(ω, L E ( v)) T ω, [ v n] E by integration by parts, =(P Σ k ω, L E ( v)) T ω, [ v n] E = E[ v n], [P Σ k ω n] E I + ( {P Σ k ω } ω), [ v n] E, by te definition of te lifting L E, (4.3). Tis means tat Θ = R (ω, v) andso te first equation in (4.22) olds true. Equation (4.23) follows in a completely analogous fasion. Tis completes te proof Error analysis for te velocity and vorticity. To obtain te error estimates for te velocity and te vorticity, we introduce te set (4.24) Z ( w) ={ v V k : B ( v, q) = w n, q E B,q Q k }, were w is any given function wit a well-defined normal component in te boundary of Ω. From te equations of te mixed formulation (4.11), we can see tat te approximate velocity u can be caracterized as te element of te set Z ( u D )suc tat (4.25) A ( u, v) =F ( v), License or copyrigt restrictions may apply to redistribution; see ttp://

15 HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS 547 for all v Z ( 0). Note tat tis is noting but a compact form of te formulation of te original unybridized LDG metod after te elimination of te vorticity. Tis follows from te definition of te metod and from te fact tat Z ( w) ={ v V k : v n, q E B = w n, q E B,q Q k }. Since, by Lemma 4.6, we ave tat A ( u, v) =F ( v)+r (ω, v), for all v Z ( 0), we proceed as in [33] and [34] and apply te tecnique proposed by Strang to analyze nonconforming metods to obtain tat ( u u 1, 1+ C ) A,cont 1 R (ω, v) inf u v 1, + sup. C A,coer v Z ( u D ) C A,coer v Z v ( 0) 1, Here we ave used te continuity of A in (4.12) and te coercivity property (4.20) on Z ( ) ( 0) V k H 0 (div; Ω). Te error estimate for te velocity ten follows immediately if we sow tat (4.26) (4.27) inf u v 1, C min{k,s} u s+1, v Z ( u D ) sup v V k R (ω, v) v 1, C min{k,s} u s+1. Te properties in (4.26) and (4.27) are proven in Section 6.3. Now, it only remains to obtain te estimate for te vorticity from tat of te velocity. Indeed, in view of te definition of te approximate vorticity (4.5) and (4.4), we ave tat, on eac element K T, ω ω =curl( u u )+L E ( u u ), and ence ( ) ω ω 0 2+Clift u u 1,, were we used te stability property (4.16) for te lifting operator L E Error analysis for te pressure. To obtain te estimate of te error p p in te approximation of te pressure in te interior of te elements, we proceed as follows. Note tat, by te definition of te approximate pressure p and Lemma 4.6, we ave B ( v,p p )= A ( u u, v)+r (ω, v), for all v V k H 0(div; Ω). Here we ave used tat B ( v,p p )=0for v V k H 0(div; Ω). Ten, from te inf-sup condition in (4.21) we obtain B ( v,p p ) sup C B,is P v V k H v Q k p p L2 (Ω)/R, 0(div;Ω) 1, were we recall tat P Q k is te L 2 -projection onto Q k. Wit te continuity property (4.12), we immediately conclude tat P Q k p p L 2 (Ω)/R C A,cont C B,is u u 1, + 1 C B,is R (ω, v) sup. v V k H v 0(div;Ω) 1, License or copyrigt restrictions may apply to redistribution; see ttp://

16 548 JESÚS CARRERO, BERNARDO COCKBURN, AND DOMINIK SCHÖTZAU Teerrorestimateforp p now follows from te estimate of te error of te velocity, te estimate (4.27) of R, and te well-known approximation result p P Q k p L2 (Ω)/R C min{k,s} p s. Tis concludes te description of te main steps in te error analysis. In Section 6 we prove in full detail te auxiliary results tat remain to be proven. 5. Te Scur-complement matrix for te pressure In tis section, we discuss a subtle but important issue related to te actual solution of te matrix equation associated to te ybridized LDG metod. Tis issue does not appear in te classical mixed metods for te Stokes system. From te weak formulation (4.11), it is easy to see tat te matrix equation of te ybridized LDG metod is of te form ( )( ) ( A B t U F =, B 0 P G) were U and P are te vectors of coefficients of te velocity u and te pressure p wit respect to teir corresponding finite element basis, respectively. A popular metod used to solve tis system of equations is te Uzawa metod, wic is noting but a simplified version of te steepest descent metod applied to te Scur-complement matrix for te pressure. Unfortunately, we cannot use suc a metod since te Scur-complement matrix for te pressure is not defined, given tat te matrix A is not invertible. Indeed, we can see in te proof of Teorem 4.1 tat its inverse is only defined in te kernel of te matrix B. Tis suggests te following remedy. Let us define te mapping M,ε : V k Qk by M,ε( v) e = ε 1 e [ v n] e for e E, for some strictly positive function piecewise-constant function ε L 2 (E ). It is ten clear tat te approximate solution ( u,p ) V k Qk /R also satisfies te weak formulation { A ( u, v)+b ( u,m,ε ( v)) + B ( v, p ) = F ( v)+g (M,ε ( v)), B ( u,q) = G (q), for all ( v, q) V k form (5.1) Qk /R. Te matrix equation of tis new formulation is of te ( Aε B t B 0 )( ) U = P ( ) Fε, G were A ε can now be proven to be symmetric and positive definite (and tus invertible). Indeed, since (5.2) B ( u, M,ε ( v)) = ε 1 e [ u n], [ v n ] e, e E te matrix A ε is readily seen to be symmetric. Furtermore, we easily see tat, for any v V k, A ( v, v)+b ( v, M,ε ( v)) 0, and tat we ave equality if and only if A ( v, v) =0andB ( v, M,ε ( v)) = 0. Te second equation implies tat v is in te kernel of te matrix B. SinceA is positive License or copyrigt restrictions may apply to redistribution; see ttp://

17 HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS 549 definite on tis subspace according to (4.20), te first equation implies tat v = 0. Tis sows tat te matrix A ε is positive definite. Consequently, te Scur-complement matrix for te pressure, S ε = B A 1 ε B t,can be formed and used to efficiently solve for te unknowns. To bound te condition number of S ε, we will use te following inf-sup stability result. Teorem 5.1. Tere is a constant C B,is independent of te mes size suc tat B ( v, q) sup C B,is min q L2 (E v ;)/R, q Q k, v V k 1, were min =min K T K. Te detailed proof of Teorem 5.1 can be found in Section 7. Here we sow ow tese results allow us to bound te condition numbers of A ε and S ε. Teorem 5.2. Te augmented stiffness matrix for te velocity, A ε,aswellaste Scur-complement matrix for te pressure, S ε = B A 1 ε B t, are symmetric, positive definite matrices. Moreover, teir condition number is of order 2 provided ε e = f e e for e E, and te mes is quasi-uniform. Here te positive piecewise-constant functions f and f 1 are uniformly bounded. Tus, we can apply a minimization algoritm to te Scur-complement matrix for te pressure of te new system. Note also tat new system of equations is associated wit te application of te classical augmented Lagrangian metod to te original equations. For classical mixed metods, te condition number of te Scur-complement matrix for te pressure is of order one. Te fact tat in our case it is of order 2 is directly associated wit te fact tat our approximate velocity is divergence-free. A similar result was obtained by Dörfler [19] for a nonconforming metod wose approximate velocity was weakly divergence-free. He found tat te condition number of te stiffness matrix for te velocity was of order 4 instead of order 2, as is te case for classical weakly divergence-free metods. Proof of Teorem 5.2. In wat follows, we drop te subindex ε, since in tis result, tis piecewise-constant function as been cosen. As already discussed, te matrix A is symmetric and positive definite. By construction, te Scur-complement matrix S is symmetric. Furtermore, in view of te inf-sup condition in Teorem 5.1, it is positive definite as well; see [9] for details. Next, let us prove te claimed upper bounds on te condition numbers of A and S, denoted by κ A and κ S, respectively. We start by estimating κ A,wiccan be expressed as follows: κ A = max V 0 V t AV V t V. V min t AV V 0 V t V Since V t AV = A ( v, v)+b ( v, M ( v)), by te continuity properties of A and B in (4.12) and (4.13) and te definition of M,weimmediatelyavetat V t AV C A,cont u 1, v 1,, u, v V k, License or copyrigt restrictions may apply to redistribution; see ttp://

18 550 JESÚS CARRERO, BERNARDO COCKBURN, AND DOMINIK SCHÖTZAU for a continuity constant C A,cont tat is independent of te mes size. Furtermore, te coercivity result (4.17), te definition of te operator M, and te equivalence result in Proposition 4.5 yield (5.3) V t AV C A,coer v 2 1,, v V k, for a coercivity constant C A,coer tat is independent of te mes size. We tus can conclude tat C A,coer v 2 1, V t AV C A,cont v 2 1,. Furtermore, from te Poincaré inequality in [7] and standard inverse estimates, we obtain v 2 [ ] 1, 1 v 2 0 CP 2,Cinv 2 2 min, wit constants C P and C inv tat are independent of te mes size. Similarly, by te regularity of te mes, v 2 [ ] 0 2 V t V min,c reg 2 max, C reg wit a constant C reg tat is independent of te mes size. Combining tese estimates yields ( ) ( CA,cont C 2 ) P κ A C2 inv C2 reg 2 max = O( 2 ), C A,coer provided te mes is quasi-uniform. Next, we let us bound κ S = max Q 0 Q t ( SQ Q t Q = Qt BV Q t Q = were AV = B t Q,tatis, Q t SQ Q t Q min Q 0 Q t SQ Q t Q 4 min. Let us begin by noting tat we ave )( ) ( ) B ( v, q) v 1, q 2 L 2 (E ;)/R v 1, q L 2 (E ;)/R q L 2 (E ;)/R Q t, Q (5.4) A ( v, w)+b ( v, M ( w)) = B ( w, q) w V k. Since, by te regularity of te mes, we ave q 2 L 2 (E ;)/R Q t Q [ ] 2 min, C reg 2 max, C reg wit a constant C reg independent of te mes size. Furtermore, from te continuity and inf-sup stability for B in (4.13) and Teorem 5.1, respectively, we conclude tat B ( v, q) [C B,is min,c B,cont ]. v 1, q L 2 (E ;)/R Hence, we obtain ( ) C 2 ( B,cont C A,coer C 2 ) 2 reg κ S max CB,is 2 C = O( 2 ), A,cont 4 min provided te mes is quasi-uniform and provided we sow tat [ v 1, CB,is (5.5) min, C ] B,cont. q L 2 (E ;)/R C A,cont C A,coer License or copyrigt restrictions may apply to redistribution; see ttp://

19 HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS 551 To prove property (5.5), we first take w = v in (5.4), use (5.3) and (4.13), and get v 2 1, C 1 A,coer B ( v, q) C 1 A,coer C B,cont v 1, q L 2 (E ;)/R. Furtermore, from te inf-sup condition in Teorem 5.1 and te weak form of te problem in (5.4), we conclude tat B ( w, q) C B,is min q L2 (E ;)/R sup w w V k 1, A ( v, w)+b ( v, M ( w)) = sup. w w V k 1, Hence, we obtain tat C B,is min q L2 (E ;)/R C A,cont v 1,. Tese estimates sow property (5.5) and complete te proof of Teorem Auxiliary results in te proofs of Teorems 4.1 and 4.2 In tis section, we complete te proofs of te auxiliary results tat we used in Section 4.4 to derive te error estimates in Teorems 4.1 and Te norm equivalence result in Proposition 4.5. To prove te result in Proposition 4.5, we are going to use te following result sown in [31, Teorem 2.2]. Lemma 6.1. For eac v V k tere is a function A v V k H1 0 (Ω)2 suc tat v A v 2 1, C 1 e [ v n] 2 0,e, e E wit a constant C independent of te mes size. Proof of Proposition 4.5. We first note tat tat te inequality on te rigt-and side is trivial and tat we only need to establis te one on te left-and side. To do so, we set V k,c = V k [ H 0 (div; Ω) H 0 (curl; Ω) ], were H 0 (curl; Ω) = { v L 2 (Ω) 2 :curl v L 2 (Ω), v n =0onΓ}. Clearly, [ V k H0 1 (Ω) 2] V k,c. Moreover, te result in [22, Lemma I.2.5 and Remark I.2.7] ensures te algebraic and topological equality of H 0 (div; Ω) H 0 (curl; Ω) and H0 1 (Ω) 2. Tis implies tat tere exist constants c 1 and c 2 suc tat (6.1) c 1 v 1 v c 2 v 1 v V k,c. Let V k, be te ortogonal complement of V k,c in V k wit respect to te norm. Hence, (6.2) V k = V k,c V k,. Now, fix v in V k arbitrary. We decompose v into v = vc + v, according to (6.2). We ave v 2 1, C [ v A v 2 1, + A v v c vc 2 ] 1 C [ e [ v n] 2 0,e + A v v c 2 + v c 2 ]. e E 1 License or copyrigt restrictions may apply to redistribution; see ttp://

20 552 JESÚS CARRERO, BERNARDO COCKBURN, AND DOMINIK SCHÖTZAU Here we ave used te triangle inequality, te approximation result in Lemma 6.1, and te norm equivalence property in (6.1). Since, on eac edge e E, [ v n] 2 0,e = [ v n] 2 0,e + [ v n] 2 0,e = [ v n] 2 0,e + [ v n] 2 0,e, we ave 1 e [ v n] 2 0,e v 2. e E Ten, by te triangle inequality, te trivial bound C 2 1,, te approximation property in Lemma 6.1, and te previous estimate, A v v c 2 C [ v A v 2 + v v c 2 [ ] C v A v 2 1, + v 2 ] C [ e [ v n] 2 0,e + v 2 ] C v 2. e E 1 Gatering tese bounds and using te ortogonality of te decomposition (6.2) wit respect to,weobtain v 2 1, C[ v 2 + v c ] 2 C v 2. Tis completes te proof Te inf-sup condition for B. To complete te proof of te error estimate for te pressure, it only remains to prove te inf-sup condition for B in (4.21). To do tat, we first note tat B ( w,q)=b ( w,p Q k q), for any w V k H 0(div; Ω) and q L 2 (Ω). Hence, to prove (4.21) it is enoug to construct a velocity field v V k H 0(div; Ω) suc tat (6.3) P Q k q L2 (Ω)/R B ( v,p Q k q), v 1, C. To do tat, we first use te continuous inf-sup condition to find a field v H0 1(Ω)2 suc tat P Q k q L2 (Ω)/R B ( v, P Q k q), v 1 C; see, e.g., [9] or [22]. Ten let v = I k v be te BDM projection of v of degree k; see[9, Section III.3.3] for details. Clearly, v V k H 0(div; Ω), and, by te definition of I k, ( ) ( P Q k q, div v = grad(p Q k T q), v + )T v n K,P Q k q K K T ( = grad(p Q k q), v + )T v n K,P Q k q K T ( ) = P Q k q, div v, T wic sows tat B ( v, P Q k q)=b ( v,p Q k q). Furtermore, it can be readily seen tat v 1, C v 1, wic yields te result in (6.3) Approximation results. Next, we prove te approximation results in (4.27) and (4.26). K License or copyrigt restrictions may apply to redistribution; see ttp://