Analysis of the grad-div stabilization for the time-dependent Navier Stokes equations with inf-sup stable finite elements

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1 arxiv: v3 [mat.na] 2 May 217 Analysis of te grad-div stabilization for te time-dependent Navier Stokes equations wit inf-sup stable finite elements Javier de Frutos Bosco García-Arcilla Volker Jon Julia Novo May 4, 217 Abstract Tis paper studies inf-sup stable finite element discretizations of te evolutionary Navier Stokes equations wit a grad-div type stabilization. Te analysis covers bot te case in wic te solution is assumed to be smoot and consequently as to satisfy nonlocal compatibility conditions as well as te practically relevant situation in wic te nonlocal compatibility conditions are not satisfied. Te constants in te error bounds obtained do not depend on negative powers of te viscosity. Taking into account te loss of regularity suffered by te solution of te Navier Stokes equations at te initial time in te absence of nonlocal compatibility conditions of te data, error bounds of order O 2 in space are proved. Te analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. Bot te continuous-intime case and te fully discrete sceme wit te backward Euler metod as time integrator are analyzed. Keywords Incompressible Navier Stokes equations; inf-sup stable finite element metods; grad-div stabilization; error bounds independent of te viscosity; nonlocal compatibility condition; backward Euler metod Instituto de Investigación en Matemáticas IMUVA, Universidad de Valladolid, Spain. Researc supported by Spanis MINECO under grants MTM P MINECO, ES and MTM P AEI/FEDER, UE frutos@mac.uva.es Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain. Researc supported by Spanis MINECO under grant MTM P bosco@esi.us.es Weierstrass Institute for Applied Analysis and Stocastics, Leibniz Institute in Forscungsverbund Berlin e. V. WIAS, Morenstr. 39, 1117 Berlin, Germany and Freie Universität Berlin, Department of Matematics and Computer Science, Arnimallee 6, Berlin, Germany. Departamento de Matemáticas, Universidad Autónoma de Madrid. Researc supported by Spanis MINECO under grants MTM P MINECO, ES and MTM P AEI/FEDER, UE julia.novo@uam.es 1

2 1 Introduction Let Ω R d, d {2, 3}, be a bounded domain wit polyedral and Lipscitz boundary Ω. Te incompressible Navier Stokes equations model te conservation of linear momentum and te conservation of mass continuity equation by t u ν u + u u + p = f in, T Ω, u = in, T Ω, 1 u, = u in Ω, were u is te velocity field, p te pressure, ν > te viscosity coefficient, u a given initial velocity, and f represents external forces acting on te fluid. Te Navier Stokes equations 1 are equipped wit omogeneous Diriclet boundary conditions u = on Ω. Te interest of tis paper is te case of small viscosity or, equivalently, ig Reynolds number. To tis end, a Galerkin finite element metod augmented wit a grad-div stabilization term for 1 is considered. Grad-div stabilization adds a penalty term wit respect to te continuity equation to te momentum equation. It was originally proposed in [16] to improve te conservation of mass in finite element metods. Tere are a number of papers analyzing te grad-div stabilization for steady-state problems, e.g., [21, 27, 28]. On te one and, it is known tat wile grad-div stabilization improves mass conservation, te computed finite element velocities are by far not divergence-free [24]. On te oter and, it was observed in te simulation of turbulent flows tat using exclusively grad-div stabilization resulted in stable simulations, compare [23, Fig. 3] and [29, Fig. 7]. Tis observation is one of te motivations for te present paper: to derive error bounds for te Galerkin finite element metod wit grad-div stabilization wose constants do not depend on inverse powers of ν. Te analysis will be performed for pairs of finite element spaces tat satisfy a discrete inf-sup condition. Error bounds wit constants independent of ν were previously obtained in [15] for te evolutionary Oseen equations. Contrary to te present paper, te wind velocity in te convective term of te Oseen equations is divergence-free and tis property considerably simplifies te analysis. Besides extending te analysis from [15], more realistic conditions on te initial data are assumed in te present paper, conditions wic affect te regularity near te initial time. An analysis of inf-sup stable elements wit divergence-free approximations of te Navier Stokes equations is presented in [3]. Tere, error bounds independent of negative powers of ν were proved for te Galerkin metod witout any stabilization, utilizing ideas, e.g., from [15]. Adding a grad-div stabilization term as in te present paper allows te use of more general, not necessarily divergence-free, finite elements. Some related works analyzing stabilized finite element approximations to te Navier Stokes equations include [8], were te continuous interior penalty metod is studied and [4, 12], were te local projection stabilization LPS metod is studied. It is discussed in [26] tat te case of te Navier Stokes equations wit grad-div stabilization but witout LPS metod can be considered as a special case of te analysis presented in [4]. Notice owever tat te error bounds in [4] depend explicitly on inverse powers of te viscosity parameter ν, unless grids are taken sufficiently fine ν, were is te mes widt, wereas tis is not te case in 2

3 te present paper. In [7], error bounds for stabilized finite element approximations to te Navier Stokes equations are obtained depending on an exponential factor proportional to te L Ω norm of te gradient of te large eddies instead of te gradient of te full velocity u in te case tat Ω is te unit square and te boundary conditions are periodic. An analysis of a fully discrete metod based on LPS in space and te Euler metod in time is carried out in [3]. Te error bounds in [3] are not independent of negative powers of ν. In all tese papers, some stabilization terms are added to te Galerkin formulation. In particular, all tese metods, save te metod studied in [26], include a stabilization for te convective term. Te aim of te present paper consists in deriving error bounds tat are independent of inverse powers of te viscosity parameter for finite element approximations tat do not include a stabilization of te convective term. In te present paper, optimal error bounds wit constants tat do not depend explicitly on inverse powers of te viscosity parameter will be obtained for te L 2 Ω norm of te divergence of te velocity, wic measures te closeness of te velocity approximation of being divergence-free, and te L 2 Ω norm of te pressure, assuming tat te solution is sufficiently smoot. In addition, an error bound for ν 1/2 times te gradient of te velocity is proved. Tis error bound is optimal in te viscosity-dominated regime, altoug it is a weak term in te convection-dominated regime. Note tat all error bounds migt depend implicitly on te viscosity troug te dependency on iger order Sobolev norms of te solution of te continuous problem. In Section 3, it will be assumed tat te solution satisfies nonlocal compatibility conditions. Te analysis is valid for pairs of inf-sup stable mixed finite elements of any degree. In te case of first order mixed finite elements, te error bound for te pressure can be proved only in two spatial dimensions. Due to te increasing use of iger order metods in computational fluid dynamics, te question of optimal approximation of te Navier Stokes equations under realistic assumptions of te data as become important. Te regularity customarily ypotesized in te error analysis for parabolic problems generally cannot be expected for te Navier Stokes equations. No matter ow regular te initial data are, solutions of te Navier Stokes equations cannot be assumed to ave more tan second order spatial derivatives bounded in L 2 Ω up to te initial time t =. Higer regularity requires te solution to satisfy some nonlocal compatibility conditions tat are unlikely to be fulfilled in practical situations [18, 19]. Taking into account tis loss of regularity at t = locally in time, te optimal rate of convergence of te grad-div mixed finite element metod is studied in Section 4. Te analysis of [3, 4, 8, 7, 3] assumes tat te solution satisfies nonlocal compatibility conditions. To te best of our knowledge, te present paper is te first one were error bounds independent of te viscosity parameter are proved witout tose assumptions, and te best bounds tat we obtain are not better tan O 2. In te literature, [13, 14, 18, 19], error bounds up to O 5 log ave been obtained for bot standard and two-grid mixed finite element metods witout assuming nonlocal compatibility conditions. But contrary to te O 2 bounds in te present paper, te error constants in tose O 5 log bounds depend on ν 1. In Section 5, te analysis of te fully discrete case is presented. For te time integration, te implicit Euler metod is considered. Again, bot te regular case 3

4 and te case in wic nonlocal compatibility conditions are not assumed are analyzed. In tis last case, te errors are sown to be O 2 log t 1/2 + t 1/2, were t is te size of te time step. Section 6 provides numerical studies supporting te analytical results and a summary finises te paper. 2 Preliminaries and notation Trougout te paper, W s,p D will denote te Sobolev space of real-valued functions defined on te domain D R d wit distributional derivatives of order up to s in L p D. Tese spaces are endowed wit te usual norm denoted by W s,p D. If s is not a positive integer, W s,p D is defined by interpolation [1]. In te case s =, it is W,p D = L p D. As it is standard, W s,p D d will be endowed wit te product norm and, since no confusion can arise, it will be denoted again by W s,p D. Te case p = 2 will be distinguised by using H s D to denote te space W s,2 D. Te space H 1 D is te closure in H1 D of te set of infinitely differentiable functions wit compact support in D. For simplicity, s resp. s is used to denote te norm resp. seminorm bot in H s Ω or H s Ω d. Te exact meaning will be clear by te context. Te inner product of L 2 Ω or L 2 Ω d will be denoted by, and te corresponding norm by. Te norm of te space of essentially bounded functions L Ω will be denoted by. For vector-valued functions, te same conventions will be used as before. Te norm of te dual space H 1 Ω of H 1 Ω is denoted by 1. As usual, L 2 Ω is always identified wit its dual, so one as H 1 Ω L2 Ω H 1 Ω wit compact injection. Using te function spaces V = H 1 Ωd, and Q = L 2 Ω = { q L 2 Ω : q, 1 = }, te weak formulation of problem 1 is as follows: Find u, p V Q suc tat for all v, q V Q, t u, v + ν u, v + u u, v v, p + u, q = f, v. 2 Te Hilbert space H div = {u L 2 Ω d u =, u n Ω = } will be endowed wit te inner product of L 2 Ω d and te space V div = {u V u = } wit te inner product of V. Let Π : L 2 Ω d H div be te Leray projector tat maps eac function in L 2 Ω d onto its divergence-free part see e.g. [11, Capter IV]. Te Stokes operator in Ω is given by A : DA H div H div, A = Π, DA = H 2 Ω d V div. Te following Sobolev s embedding [1] will be used in te analysis: For 1 p < d/s let q be suc tat 1 q = 1 p s d. Tere exists a positive constant C, independent of s, suc tat v L q Ω C v 1 W s,p Ω, q 1 q, v W s,p Ω. 3 4

5 If p > d/s te above relation is valid for q =. A similar embedding inequality olds for vector-valued functions. Let V V and Q Q be two families of finite element spaces composed of piecewise polynomials of degrees at most k and l, respectively, tat correspond to a family of partitions T of Ω into mes cells wit maximal diameter. In tis paper, we will only consider pairs of finite element spaces satisfying te discrete inf-sup condition, v, q inf sup β, 4 q Q v V v q wit β >, a constant independent of te mes size. For example, for te MINI element it is k = l = 1 and for te Hood Taylor element one as l = k 1. Since te error bounds for te pressure depend bot on te mixed finite element used and on te regularity of te solution, and in general it will be assumed tat p Q H k Ω wit l k 1, in te sequel te error bounds will be written depending only on k. It will be assumed tat te family of meses is quasi-uniform and tat te following inverse inequality olds for eac v V, see e.g., [1, Teorem 3.2.6], v W m,p K C inv n m d 1 q 1 p K v W n,q K, 5 were n m 1, 1 q p, and K is te size diameter of te mes cell K T. Te space of discrete divergence-free functions is denoted by V div = {v V v, q = q Q }, and by A : V div V div is denoted te following linear operator A v, w = v, w v, w V div. 6 Note tat from tis definition, it follows for v V div tat A 1/2 v = v, A 1/2 v = v. Additionally, two linear operators C defined by : V div V div and D : L 2 Ω V div are C v, w = v, w v, w V div, 7 D p, v = p, v v V div. 8 In wat follows, Π div : L 2 Ω d V div will denote te so-called discrete Leray projection, wic is te ortogonal projection of L 2 Ω d onto V div Π div v, w = v, w w V div. 9 By definition, it is clear tat te projection is stable in te L 2 Ω d norm: Π div v v for all v L 2 Ω d. Te following well-known bound will be used I Π div v + I Π div v 1 C j+1 v j+1 v V div H j+1 Ω d, 1 5

6 for j =,..., k. Denoting by π te L 2 Ω projection onto Q, one as tat for m 1 q π q m C j+1 m q j+1 q H j+1 Ω, j =,..., l. 11 For simplicity of presentation, te notation π will be used instead of π p for te pressure p in 1. In te error analysis, te Poincaré Friedrics inequality v C v v H 1 Ω d, 12 will be used. In te sequel, I u V will denote te Lagrange interpolant of a continuous function u. Te following bound can be found in [6, Teorem 4.4.4] u I u W m,p K c int n m u W n,p K, m n k + 1, 13 were n > d/p wen 1 < p and n d wen p = 1. In te analysis, te Stokes problem ν u + p = g in Ω, u = on Ω, 14 u = in Ω, will be considered. Let us denote by u, p V Q te mixed finite element approximation to 14, given by ν u, v v, p = g, v v V, u, q = q Q. Following [17, 22], one gets te estimates u u 1 C inf u v 1 + ν 1 inf p q, 15 v V q Q p p C ν inf u v 1 + inf p q, 16 v V q Q u u C inf u v 1 + ν 1 inf p q. 17 v V q Q It can be observed tat te error bounds for te velocity depend on negative powers of ν. For te analysis, it will be advantageous to use a projection of u, p into V Q wit uniform in ν, optimal, bounds for te velocity. In [15] a projection wit tis property was introduced. Let u, p be te solution of te Navier Stokes equations 1 wit u V H k+1 Ω d, p Q H k Ω, k 1, and observe tat u, is te solution of te Stokes problem 14 wit rigt-and side g = f t u u u p. 18 6

7 Denoting te corresponding Galerkin approximation in V Q by s, l, one obtains from u s + u s 1 C j+1 u j+1, j k, 19 l Cν j u j+1, j k, 2 were te constant C does not depend on ν. Remark 1 Assuming te necessary smootness in time and considering 14 wit g = t f t u u u p, one can derive an error bound of te form 19 also for t u s. One can proceed similarly for iger order derivatives in time. In Section 4, were boundedness of derivatives up to t = is not assumed, te bound 19 is also valid, but ten te quantities assumed to be bounded up to t = are t j 1/2 ut j+1, t j+1/2 t ut j+1, t j+3/2 tt ut j+1, etc. Note tat for a given t >, te assumptions in te present section old for t t, and tose of Section 4 for t t. Following [9], one can also obtain te following bound for s u s C u, 21 were C does not depend on ν. Te metod tat will be studied for te approximation of te solution of te Navier Stokes equations 1 is obtained by adding to te Galerkin equations a control of te divergence constraint grad-div stabilization. More precisely, te following grad-div metod will be considered: Find u, p :, T ] V Q suc tat t u, v + ν u, v + bu, u, v p, v, + u, q + µ u, v = f, v, 22 for all v, q V Q, wit u = I u. Here, and in te rest of te paper, bu, v, w = Bu, v, w u, v, w H 1 Ω d, were, Bu, v = u v + 1 uv 2 u, v H1 Ω d Notice te well-known property bu, v, w = bu, w, v u, v, w V, 23 suc tat, in particular, bu, w, w = for all u, w V. 3 Te regular continuous-in-time case In tis section, error bounds for te continuous-in-time discretization are derived for te case in wic regularity up to time t = is assumed. Some of te lemmas are written in suc a way tat can also be applied in Section 4 for te analysis of te situation witout compatibility assumptions. 7

8 3.1 Error bound for te velocity Using test functions in V div and applying definitions 6 9, one finds tat 22 implies tat u satisfies were t u + νa u + B u, u + µc u = Π div f, 24 B u, v = Π div Bu, v, u, v H1 Ω d, and C is defined in 7. Notice tat Π div can be extended from L 2 Ω d to H 1 Ω in suc a way tat B u, v is well defined for u, v H 1Ωd. Te following two lemmas will be used in Sections 3 and 4. Lemma 1 Let w : [, T ] V div be an arbitrary function piecewise differentiable wit respect time. Let u be te mixed finite element approximation to te velocity defined in 24. Define te truncation errors t 1, : [, T ] V div and t 2 : [, T ] L 2 Ω suc tat te following equation is satisfied t w + νa w + B w, w + µc w = Π div f + t 1, D t 2, 25 were D as been defined in 8. Ten, if te function gt = w t + w t 2 2µ 26 is integrable in, T, i.e., w L 1, T ; L and w L 2, T ; L, te error e = u w can be bounded as follows e t 2 + e Kt,s 2ν e s 2 + µ e s 2 ds e Kt, e 2 + e Kt,s t 1, µ t 2 2 ds, were Kt, s = s w + w 2 dr. 2µ Proof Subtracting 24 from 25, taking te inner product wit e V div, and performing some standard computations yields 1 d 2 dt e 2 + ν e 2 + µ e 2 + bw, w, e bu, u, e = t 1,, e + t 2, e. 27 Observe tat bw, w, e bu, u, e = be, w, e bu, e, e = be, w, e, 28 8

9 were in te last step it was used tat, due to 23, bu, e, e =. Applying Hölder s inequality one finds be, w, e w e e w e w e 2 + µ 4 e 2 + w 2 4µ e Tus, from 27, using te Caucy Scwarz and Young s inequalities, taking into account te definition of function g in 26, and rearranging terms, it follows tat d dt e 2 + 2ν e 2 + µ e 2 g e 2 + t 1, µ t 2 2. Multiplying by te integrating factor exp Kt, and integrating in time, te result follows in a standard way. Te following lemma will be used in te proof of te main results of te paper. Lemma 2 Tere exists a positive constant C suc tat for any w V and v V H 2 Ω d te following bound olds Bw, w Bv, v C w + w L 2d/d 1 Ω + v 2 w v 1. Proof Since V L Ω and by te well known Sobolev embedding H 2 Ω L Ω see e.g., [1], it follows tat Bw, w, Bv, v L 2 Ω. Ten, te application of te Hölder inequality yields Bw, w Bv, v = Bw, w v + Bw v, v w w v w L 2d/d 1 Ω w v L 2d Ω + w v L 2d Ω v L 2d/d 1 Ω w v v. Te statement of te lemma follows from 3. Te proof of te error estimate is based on te comparison of te Galerkin approximation to te velocity u in 22 wit te approximation s defined at te end of Section 2. Te pair s, l V Q solves ν s, v l, v s, q = f, v t u, v bu, u, v + p, v. 3 Adding and subtracting terms gives t s + νa s + B s, s + µc s = Π div f Πdiv tu t s B u, u B s, s D p π + µc s. Taking into account 7 and u =, one can see tat Lemma 1 can be applied wit w = s, t 1, = Π div τ 1 + τ 2, and t 2 = τ 3 + τ 4, were τ 1 = t u t s, τ 2 = Bu, u Bs, s, τ 3 = p π, τ 4 = µ s u. 9 31

10 Let u satisfy te ypotesis in Teorem 1 below. In order to apply Lemma 1, te integrability in, T of te function g defined in 26, wit w = s, as to be proved. To tis end, it will be sown tat te two terms s 2 and s are bounded by an integrable function in, T. For te latter, one can simply apply 21. For te former term, one first observes tat from te assumed regularity of u it follows tat u is continuous and, ence, I u C u for some C >. Ten, one can write s s I u + I u C inv d/2 s I u + u, were in te last inequality inverse inequality 5 as been applied. Applying 13, 19, and 3, one gets s C u 2, s C u, 32 were te constants are independent of ν. Tus, by applying Lemma 1 wit e = u s, one obtains e t 2 + e Kt,s 2ν e s 2 + µ e s 2 ds e Kt, e 2 + From 11 and 19 see also Remark 1 one gets e Kt,s τ 1 + τ µ τ 3 + τ 4 2 ds. 33 and τ 3 + τ 4 2 C2k p 2 k + µ2 u 2 k+1, 34 τ 1 + τ 2 2 C2k t u 2 k + 2 τ For τ 2, te application of Lemma 2 gives τ 2 C s + s L 2d/d 1 Ω + u 2 u s To bound s L 2d/d 1 Ω, one finds wit te inverse inequality 5 tat and wit 19 it follows tat s L 2d/d 1 Ω C 1/2 s, s L 2d/d 1 Ω C 1/2 u s C 1/2 u Altogeter, from 36, using also 32 and tat is bounded at least from te diameter of Ω, one obtains τ 2 C u 2 u s 1 C k u 2 u k In view of 32, one as 1 expkt, s C explt wit LT = T us + us 2 2 2µ ds, 39 were C is independent of ν. From 33, and taking into account tat e C k u k, one derives te following error estimate for te velocity. 1

11 Teorem 1 For T > let us assume for te solution u, p of 2 tat u L 2, T, H k+1 Ω L 2, T, W 1, Ω L, T, H 2 Ω, u H max{2,k} Ω, t u L 2, T, H k Ω, and p L 2, T, H k Ω/R wit k 1. Ten tere exists a positive constant C depending on T pt 2 u 2 k + H k /R + t ut 2 k µ + µ + ut 2 2 ut 2 k+1 dt, 4 but not directly on inverse powers of ν, suc tat te following bound olds for e = u s and t [, T ] e t 2 + were LT is defined in 39. ν e s 2 + µ e s 2 ds C explt 2k, 41 Remark 2 Note tat Teorem 1 is formulated for te most common coice of infsup stable finite element spaces were te polynomial degree of te velocity space is larger by one tan te degree of te pressure space. In tis situation, te constant C in 41 depends on µ 1 and on µ, see 4. Tus, te asymptotic optimal coice of te stabilization parameter is µ 1, wic is a well-known result for tis situation. For pairs of inf-sup stable spaces wit te same polynomial degree, like te MINI element, te same regularity wit respect to te polynomial degree for velocity and pressure is usually assumed and te estimates for proving te error bound can be adapted accordingly. In particular, one gets instead of 34 τ 3 + τ 4 2 C2k 2 p 2 k+1 + µ2 u 2 k+1, suc tat equilibrating te two terms containing µ gives te coice µ, wic is known from te literature [21]. However, also coosing µ 1 or µ 2 leads for te MINI element to optimal error bounds wit constants independent of ν. Altogeter, tere is some freedom for te coice of µ and coosing tis parameter to be a constant is a valid option also for te MINI element. Remark 3 By writing u u = e + s u, applying te triangle inequality, Teorem 1, and 19, it follows tat te bound 41 olds true canging e by u u. 3.2 Error bound for te pressure Te error bound for te pressure will be obtained now using te same arguments as used in [15]. Applying te inf-sup condition 4, substituting in te numerator 22 and 3, adding and subtracting terms, and using te Caucy Scwarz inequality, it follows tat β p π ν e + Bu, u Bs, s 1 + t e 1 + µ e + τ τ τ 3 + τ 4 + l. 11

12 Note tat, due to 41, te presence of te terms ν e and µ e on te rigt-and side of 42 limits te maximum convergence rate to O k. Te same convergence rate is obtained for te term τ 4, wic is estimated wit 19. Te fift term is bounded wit 35 and te sixt term, using 38, by τ 1 1 C τ 1 C k t u k, τ 2 1 C τ 2 C k u 2 u k For te second term on te rigt-and side of 42, te skew-symmetry of b,, gives Bu, u Bs, s 1 = sup bu, e, φ + be, s, φ φ 1 =1 = sup bu, φ, e + be, φ, s. φ 1 =1 44 Using now Hölder s inequality and te Sobolev embedding 3, one finds te bound bu, φ, e e u φ 1 + e u L 2d/d 1 Ω φ L 2d Ω C u + u L 2d/d 1 Ω e φ 1. For te second term on te rigt-and of 44, arguing similarly, one gets suc tat be, φ, s e s φ 1 + C e s L 2d/d 1 Ω φ 1, Bu, u Bs, s 1 C u + u L 2d/d 1 Ω + s e 45 + C s L 2d/d 1 Ω e. Next, te terms between parenteses will be bounded. Applying 5 and 32, one finds u e + s C d/2 e + s C d/2 e + u Recalling 5 and 37 yields u L 2d/d 1 Ω e L 2d/d 1 Ω + s L 2d/d 1 Ω C 1/2 e + C 1/2 u Remark 4 Te rigt-and side of 46 is bounded for always for d = 2. It follows from 41 tat for d = 3 te term is bounded for k 2. Note tat most inf-sup stable pairs of finite element spaces ave velocity spaces wic are at least of second order so tat tis is not a big restriction. On te oter and, one can deduce from 47 and 41 tat te term u s 2 ds is bounded. L 2d/d 1 Ω 12

13 Wit 45 and using in addition 5, 32, and 37, one obtains Bu, u Bs, s 1 C u 2 e + e + C d/2 e + 1/2 e e. 48 Next, te tird term on te rigt-and side of 42 will be bounded. Arguing as in [15], it will be sown first tat t e 1 can be estimated by bounding A 1/2 t e. From [5, Lemma 3.11] it is known tat t e 1 C t e + C A 1/2 Π t e, 49 were Π is te Leray projector defined in Section 2. Applying [5, 2.15] one gets A 1/2 Π t e C t e + A 1/2 t e, 5 wit A defined in 6. Wit 49, 5, te symmetry of A, and te inverse inequality 5, one obtains t e 1 C t e + C A 1/2 t e = C A 1/2 A 1/2 t e + C A 1/2 t e = C A 1/2 t e + C A 1/2 t e C A 1/2 t e. Taking into account tat A 1/2 Π div g g 1, for all g L 2 Ω d, see [5, 2.16], and arguing as in [15], te following estimate for A 1/2 t e can be derived A 1/2 t e ν A 1/2 e + Bu, u Bs, s 1 + Cµ e + t u s 1 + Bu, u Bs, s 1 + Cµ u s + C p π. All velocity-related terms on te rigt-and side were already estimated in tis section. Te pressure terms in 42 and 51 are estimated wit 11 and 2. Ten, arguing exactly as in [15], one concludes te following estimate. Teorem 2 Under te assumptions of Teorem 1 tere exists a positive constant C suc tat te following bound olds β 2 T were in te case d = 3 te bound is valid for k 2. Remark 5 By splitting 51 p π t 2 dt C 2k, 52 p p = p π + π p, applying te triangle inequality, Teorem 2, and 11, it follows tat te bound 52 olds true replacing p π by p p. 13

14 4 Te continuous-in-time case: analysis witout nonlocal compatibility conditions It is well known tat, no matter ow regular te data are, solutions of te Navier Stokes equations cannot be assumed to ave more tan second order spatial derivatives bounded in L 2 Ω up to initial time t =, since iger regularity requires te data to satisfy nonlocal compatibility conditions wic are not likely to appen in practical situations [18, 19]. Te analysis of tis section takes into account te lack of regularity at t =. Along te section it is assumed tat inf-sup stable mixed finite elements of second order are used, for example te Hood Taylor element consisting of continuous piecewise quadratic polynomials for te velocity and continuous piecewise linear polynomials for te pressure. It sall be assumed tat for some T > M 1 = max t T ut 1 < +, M 2 = max ut 2 < t T Also, according to [18, Teorems 2.4 and 2.5], and assuming te rigt-and side f in 1 is smoot enoug, it sall be assumed tat, for k 2, M k = max t T tk 2/2 ut k + t ut k 2 + pt H k 1 /R < +, 54 and, for k 3 T Kk 2 = t k 3 ut 2 k + sut 2 k 2 + pt 2 H k 1 /R + spt 2 H k 3 /R ds < Remark 6 Observe tat in view of Remark 2, for te case k = 1 in Teorem 1 wic covers te case of te so-called MINI element te constant C in 4 and te function LT from 39 depend on M T µ 1 + µ + M2 2 + K2 3 and T + 2T 1/2 K 3 + T M2 2µ 1 /2, respectively, were no negative powers of t appear. Tus, in te absence of nonlocal compatibility conditions at t =, te analysis of te previous section applies to te case k = 1, but it does not apply to te case k 2 since t ut 2 k is not integrable near t =. 4.1 An auxiliary function For te analysis, te auxiliary function û : [, T ] V satisfying t û + νa û = Π div f p B u, u, û = Π div u, 56 will be considered and te following notations will be used z = Π div u û, θ = Π div u s. Notice tat in view of te triangle inequality, 19, te approximation property of te L 2 Ω projection, and it follows tat for < t T, θ t j M 3 3 j, j =, 1, 57 t1/2 θ 2 j + s2 s θ 2 j ds C K K j, j =, 1, 58 14

15 for some positive constant C independent of ν. Observe also tat projecting 1 onto V div, using te definition of s wit 14 and te rigt-and side given in 18, yields Π div tu + νa s + B u, u + Π div p = Πdiv f, so tat Π div tu + νa Π div u = Πdiv f B u, u Π div p + νa θ. Subtracting now 56 and applying te commutation of te Leray projection and te temporal derivative, one finds tat te error z satisfies t z + νa z = νa θ, z =. 59 Lemma 3 Tere exists a positive constant C independent of ν suc tat te error z = Π div u û of te discrete velocity û defined by 56 satisfies te following bounds for < t T : z t 2 + z s 2 1 ds CK2 3 4, 6 A 1/2 z t 2 + z s 2 ds CK2 3 6, 61 z t j C t 1/2 K 3 + K 5 + M 3 3 j, j =, Proof Multiplying 59 by z, integrating on Ω, applying te Caucy Scwarz inequality, and Young s inequality gives 1 d 2 dt z 2 + ν z 2 = νa1/2 z, A 1/2 θ ν 2 z 2 + ν 2 θ 2. Using integration in time and taking into account tat z =, it follows tat z t 2 + ν z s 2 ds ν θ s 2 ds. Now, applying 58 and te Poincaré Friedrics inequality 12, te bound 6 follows directly. Repeating tese arguments but multiplying by A 1 z instead of z gives 61. To prove 62, multiply 59 by ta 1 tz and integrate in Ω to get t A 1/2 2 t z + ν d t z 2 = νt t z, θ + ν 2 dt 2 z 2. Integrating between and t and integrating te term arising from νt t z, θ by parts, one gets s A 1/2 s z 2 = νtz t, θ t ν ds + ν 2 t z t 2 z, θ + s s θ ds + ν 2 z 2 ds. 15

16 Applying te Caucy Scwarz inequality and Young s inequality to te first two terms on te rigt-and side and rearranging terms it follows tat s A 1/2 2 s z ds + ν 4 t z t 2 νt θ t 2 + ν θ 2 + s2 s θ 2 ds + ν z 2 ds. Te bound 62 for j = now follows by applying and 61. Wit te same arguments, but multiplying by t t z instead of ta 1 tz te bound 62 for j = 1 is obtained. Remark 7 For piecewise polynomials of degree iger tan two, it is possible to obtain iger order bounds, but wit negative powers of ν. For example, for piecewise cubics, by repeating te arguments in te proof of Lemma 3, it is possible to sow tat z t j C ν 1/2 t K 4 + K 6 + M 4 4 j, j =, 1, t T, using as test function t 2 A 1 tz. Similar negative powers of ν are obtained also wit some oter tecniques like tose in [13]. At te moment, it is an open question weter different tecniques could be applied to get iger order bounds wit constants independent of inverse powers of ν. For tis reason, only piecewise quadratics for te velocity are considered in tis section. 4.2 Error bound for te velocity Observe tat te first equation in 56 can be rewritten as t û + νa û + B û, û + µc û = Π div f + B û, û B u, u D p + µc û. Lemma 1 will be applied wit w = û, t 1, = τ 2, and t 2 = τ 3 + ˆτ 4, were τ 2 = B û, û B u, u, ˆτ 4 = µ û u, and were τ 3 is defined in 31. Te application of tis lemma requires to sow tat bot û 2 and û are integrable in, T. To bound û, apply te triangle inequality and te inverse inequality 5 to get û C inv d/2 û I u + I u C inv d/2 û Π div u + Π div u u + u I u + I u. Since I u C u, utilizing 3 gives I u C u 2 CM 2. Applying 13, 1, and 6 yields û C K 3 + M 2 4 d/2 + M

17 Te bound of û will be sown for te more difficult and practically more relevant case d = 3. Wit te triangle inequality and te inverse inequality 5, one obtains û C inv 5/2 û s + s 64 C inv 5/2 û Π div u + Π div u u + u s + s. Applying 21, 3, and 54 yields s CM 3 t 1/2. Arguing as before and applying 62, 1, and 19 gives û Ct 1/2 K 3 + K 5 + M 3 1/2 + M Tus, from 63 and 65 it follows tat gt = û t + û t 2 2µ K3 + K 5 + M 3 C τ 1/2 + K 3 + M 2 2 2µ L 1 t 1/2, were Ten Kt, s = L 1 = C K 3 + K 5 + M 3 + K 3 + M µ s gr dr L 1 s 1 r 1/2 dr 2L 1t 1/2 s 1/2. 66 Lemma 1 wit w = û gives for ê = u û ê t 2 + e Kt,s 2ν ê s 2 + µ ê s 2 ds e Kt, e 2 + e τ Kt,s µ τ 3 + ˆτ 4 2 ds. To estimate te truncation errors first apply Lemma 2 to get τ 2 C û + û L 2d/d 1 + u 2 u û Using te triangle inequality, te inverse inequality 5, 37, 6, 1, 19, and 53 one gets û L 2d/d 1 C inv 3/2 û s + s L 2d/d 1 C 3/2 û Π div u + Π div u u + u s + C 1/2 u 2 C 1/2 K 3 + u 2 CK 3 + M

18 By inserting 68 and 63 in 67 it follows tat As in 34 one also gets τ 2 CK 3 + M 2 u û τ 3 + ˆτ 4 2 C p 2 H 2 /R 4 + µ 2 u û To bound u û 1 in 69 and 7 one adds and subtracts Π div u. Applying ten 55, 1, and 6 leads to te estimate τ µ τ 3 + ˆτ 4 2 ds CK3 2 µ + µ 1 + K 3 + M = C Collecting all estimates and applying at te initial time te triangle inequality, te interpolation estimate 13 and 1, te following teorem is proved. Teorem 3 For T >, assuming te solution u, p of 1 satisfies 53, 54 and 55 te following bound olds for te error ê = u û, t [, T ]: ê t 2 + 2ν ê s 2 + µ ê s 2 ds e Kt, M2 2 + C 4, 72 were Kt, s is defined in 66 and C is te constant in 71. Remark 8 By decomposing u u = ê + û u = ê z + Π div u u, and applying te triangle inequality, Teorem 3, 6, and 1, it follows tat te bound 72 olds true canging ê by u u. Remark 9 Observe tat it is te factor 5/2 in 64 tat prevents te analysis in te present section to apply to te case k = 1. On te oter and, te analysis in Section 3 applies to k = 1 since one compares u wit s for wic te bounds 32 are available. Te comparison wit s in Section 3, owever, induces te truncation error τ 1, wic, as commented in Remark 6, prevents te extension of te analysis in tat section to te case k > Error bound for te pressure Te analysis follows closely tat of Section 3.2. Again, using te inf-sup condition 4, a straigtforward calculation leads to β p π ν ê + Bu, u Bû, û 1 + t ê 1 + µ ê + Bu, u Bû, û 1 + µ u û + p π + l, 18

19 were l denotes ere te discrete pressure corresponding to te formulation of 56 in V. Repeating te arguments used wen obtaining 43 and 69, one gets Bu, u Bû, û 1 CK 3 + M 2 u û 1. In te same way as for 45, one obtains Bu, u Bû, û 1 u + u L 2d/d 1 + û ê + û L 2d/d 1 ê. Using te inverse inequality 5, 68, 46, and 63 leads to Bu, u Bû, û 1 CK 3 + M 2 ê + ê + C d/2 ê + 1/2 ê ê. Arguing as in Section 3.2, one gets for t ê 1 t ê 1 ν A 1/2 ê + Bu, u Bû, û 1 + Cµ ê + Bu, u Bû, û 1 + Cµ u û + C p π. All terms on te rigt-and side of tis estimate ave already been bounded. Arguing like at te end of Section 3.2, one derives te following estimate. Teorem 4 Under te assumptions of Teorem 3 tere exists a positive constant C suc tat te following bound olds Remark 1 By writing β 2 p π s 2 C p p = p π + π p, applying te triangle inequality, Teorem 4, and 11, it follows tat te bound 73 olds true replacing p π by p p. 5 A fully discrete metod We now analyze te discretization of 24 by te implicit Euler metod. For tis purpose, we consider a partition = t < t 1 <... < t N = T of te interval [, T ], and for eac time level we look for approximations U n ut n in V div and p n pt n in Q, satisfying D t U n, v + ν U n, v + bu, U n, v p n, vn + u n, q + µ U n, v = ft n, v, 74 for all v, q V Q, for n = 1,..., N, were U V is given, and D t U n = U n U n 1. t n t n 1 19

20 In wat follows we will take U = s and consider for simplicity constant step sizes, tat is t n t n 1 = t, n = 1,..., N. Te canges for variable step sizes as well as for oter consistent initial approximations are straigtforward. Also, oter time integrators can be considered and te analysis can be carried out arguing essentially as in te next lines. Te existence of te approximation can be proved wit te elp of Brouwer s fixed point teorem as in [31]. Te approximations U n satisfy D t U n + νa U n + B U, U n + µc U n = Πdiv ft n, 75 were we keep te notation of previous sections. To obtain error bounds, we will use te following discrete Gronwall lemma tat can be found in [2]. Lemma 4 Let k, B, and a n, b n, c n, γ n be nonnegative numbers suc tat a n + k b n k j= γ n a n + k j= c n + B, n 1. Suppose tat kγ n < 1, for all n, and set σ n = 1 kγ n 1. Ten, te following bound olds a n + k b n exp k σ j γ j k c n + B, n 1. j= Lemma 5 Fix γ, 1, and let w n n=, tn, n=1 and tn,1 n=1 and let t n,2 n=1 a series in L2 Ω satisfying j= j= j= be series in V div D t w n +νa w n +B w n, wn +µc w n = Πdiv ft n+νa t n, +tn,1 +D t n,2. 76 Assume tg n < γ were g n = w n + w n 2 2µ, n = 1, 2, Tere exists a positive constant C depending only on γ suc tat te following bound olds for te differences e n = wn U n : e n 2 + e j ej 1 n 1 C exp t g j 2 + t e 2 + t ν e j 2 + µ e j 2 ν t j, 2 + t j, µ tj,2 2. Proof A direct calculation sows tat e n en 1 n, e n = 1 2 en en en en

21 so tat subtracting 75 from 76, taking te inner product in L 2 Ω wit 2e n, adding = 2µ w n, en 2µ wn ut n, e n and after some rearrangements we ave 1 t e n 2 e n e n en ν e n µ en 2 ν t n, 2 2be n, wn, en + tn, µ tn,2 2 + e n 2, 78 were te product e n, B w n, wn B U n, U n = bwn, wn, en bu n, U n, en as been treated as in 28. Arguing as in 29 we may write be n, wn, en w n + wn 2 e n 4µ 2 + µ 4 en 2. Tus, multiplying by t in 78 it follows tat e n 2 e n e n en 1 2 +ν t e n 2 +µ t e n 2 c n + tg n en 2, 79 were c n = t ν t n, 2 + t n, µ tn,2 2. Adding te expression in 79 to tose corresponding to n 1, n 2, etc, down to 1, we ave 1 tg n en 2 + e j ej t ν e j 2 + µ e j 2 e 2 + n 1 c j + tg j ej 2. Since we are assuming tat tg n γ, we ave 1 tgn > 1 γ > and te proof is finised by applying Lemma Error analysis in te regular case We apply Lemma 5 wit w n = s t n, so tat t n, =, tn,1 = Πdiv τ n 1,1 + τ n 1,2 + τ 2 t n, t n,2 = τ 3t n + τ 4 t n, were τ 2, τ 3 and τ 4 are tose defined in 31, and τ n 1,1 = s t n ut s t n 1 ut n 1 t τ n 1,2 = ut n ut n 1 t t ut n = 1 t We notice tat in view of 32 we ave tat g j ˆL = 1 + C max t T n = 1 t 2 ut + ut 2 2 2µ 21 n t n 1 t s t ut dt, 8 t n 1 s t n 1 tt us ds. 81, j = 1,..., N, 82

22 so tat n 1 exp t g j exp ˆLtn. 83 We also ave tat τ 2 t n, τ 3 t n, and τ 4 t n ave already been bounded in 34 and 35. Furtermore, applying Caucy-Scwarz inequality in 8 and te bound 19 wit j = k 1 applied to t s t ut we ave τ n 1,1 2 C 2k tn t u 2 k t dt t n 1 and applying te Caucy-Scwarz inequality in 81 Tus, we ave te following result. n τ n 1,2 2 C t tt u 2 dt. t n 1 Teorem 5 For T > let us assume for te solution u, p of 2 tat u L, T, H k+1 Ω L, T, W 1, Ω, u H max{2,k} Ω, t u L 2, T, H k Ω, tt u L 2, T, L 2 Ω and p L 2, T, H k Ω/R wit k 1. Ten, tere exist positive constants C 1 and C 2 depending on and T u 2 k + t ut 2 k dt + max t T T pt 2 H k /R + µ + ut 2 2 ut 2 µ k+1 tt ut 2 dt, respectively, but none of tem depending on inverse powers of ν, suc tat te following bound olds for e = U n s t n and 1 n N e n 2 + t were ˆL is defined in 82. ν e n 2 + µ e n 2 exp ˆLtn C 1 2k + C 2 t 2, 84 For te pressure, we can obtain error bounds by repeating te analysis in Section 3.2 wit t e replaced by D t e n and te truncation error τ 1 by τ n 1,1 + τ n 1,2. We observe, owever, tat instead of 48 we now ave BU n, U n Bs t n, s t n 1 C ut n 2 e n + e n + C d/2 e n + 1/2 e n e n. Now, te errors e n and en, as sown in 84 are bounded in terms of powers of and t. Tus, we ave te following result. 22

23 Teorem 6 In te conditions of Teorem 5, tere exists a positive constant C suc tat te following bound olds β 2 t N p n π t n 2 C 1 + t2 2k + t were in te case d = 3 te bound is valid for k 2. Remark 11 Observe tat in Teorem 6 te presence of negative powers of in te error bound 85 does not affect te convergence rate in te pressure wenever t C d/2 for any positive constant C. Tis condition will be automatically satisfied if we try to balance spatial and temporal discretization errors, since in tat case we would ave to take t k. 5.2 Error analysis witout compatibility conditions We now assume tat 53 olds and tat M 3 < + and K 3, = T d t tt ut 2 dt < +. In te case k = 1 and d = 2 we will also assume K 3 < +. Te cases k = 1 and k 2 will be analyzed separately. For k = 1, and taking into account tat ut ut 2 and ut C ut 3 CM 3 t 1/2, te analysis of te previous section is still valid wit te following two canges. First we must replace 82 and 83 by g j ˆL 1 + 2M 3 t 1/2 j, for j = 1,..., N were so tat n 1 exp t g j ˆL 1 = 1 + M 2 2 2µ, 86 exp ˆL1 t n + 2M 3 tn. 87 Te second and more relevant cange is te estimation of τ 1,2, wic now is τ n 1,2 2 1 tn tn t 2 t t n dt t t n tt ut 2 dt t n 1 t n n t n 1 t tt ut 2 dt. Tus, we ave te following result. Teorem 7 Fix T > and assume tat te solution u, p of 1 satisfies 53, and tat M 3, K 3 and K 3, are finite. Assume linear finite element approximations in te velocity are used. Ten, 23

24 i Tere exists a positive constant C 1 depending on M 2 and K 3 but not on inverse powers of ν, suc tat te following bound olds for te error e n = s t n U n, 1 n N: e n 2 + e j ej 1 were ˆL 1 is defined in t ν e n 2 + µ e n 2 exp ˆL1 t n + 2M 3 tn C1 2 + K 3, t, ii In te case d = 2 ten tere exists a positive constant C 2 depending on M 2, M 3, ˆL 1 and K 3 but not on inverse powers of ν, suc tat te following bound olds: N β t 2 p n π t n 2 C t 2 + t. Remark 12 Let us observe tat contrary to Teorem 5 we ave found a limitation in te rate of convergence of order O t 1/2 in te temporal error. To our knowledge tis paper is te first one in wic error bounds wit constants independent on ν are obtained for te fully discrete case witout assuming nonlocal compatibility conditions for te Navier-Stokes equations. At present it is an open problem to find out if tis O t 1/2 limitation could be avoided using a different tecnique of analysis. For k 2, we will apply Lemma 5 to te differences were e n = un U n, u n = Πdiv ut n n =, 1,..., N. By projecting 1 onto V div and adding and subtracting terms, it is easy to ceck tat te projections u n satisfy for n = 1,..., N, D t u n + νa u n + B u n, un + µc u n = νa t n, t n 1, D t n 2, were, t n, = u n s t n, t n 1, = Π div τ n 1,2 + τ n 2, and t n 2 = τ 3t n + τ n 4, τ n 1,2 and τ 3 being tose defined in 81 and 31, and were τ 2 = B u n, un B ut n, ut n, τ n 4 = µ u n ut n. We will need estimates in L Ω of u n, wic are given by te following result. Lemma 6 Tere is a constant C D > suc tat te following bounds old: Π div u C D u 2, Π div u C D u 3. Proof We prove te second inequality for d = 3, te case d = 2 and te first inequality being proved similarly. By adding ±I u, using 5 and 13 we ave Π div u C 2+d/2 Π div I u + u. 24

25 For te first term, by writing Π div I = Π div I + I I and using 1 wit j = 2, 3 and 13 wit m = and n = 2, 3 we ave 2+d/2 Π div I ut C u 2 u 3 1/2 C u 3. Ten, te result follows by estimating ut wit Sobolev s inequality 3. Tus, as wit te case w n = s t n, we also ave tat for w n = un te value gn defined in 77 satisfies te bound 87. We also observe tat τ 2 and τ 4 can be estimated similarly to 38 so tat we can write τ n 2 C ut n 2 ut n u n 1 C u 2 u 3 2 M 3 CM 2 2, and, similarly to 34, t 1/2 n τ n 3 + τ n 4 2 C pt n µ 2 ut n C1 + µ 2 M 2 3 t n 4, and, applying 1 and 19 t n, 2 C ut n C M 2 3 t n 4. Ten, noticing tat t n = 1 + t j j 1 + log tn+1. t j=2 we ave t C ν t j, 2 + t j, µ tj,2 2 tn+1 log M3 ν 2 + M µ t + µ 4 + K 3, t. We conclude wit te following result. Teorem 8 For T >, assuming te solution u, p of 1 satisfies 53, and tat M 3, K 3 and K 3, are finite. Assume tat piecewise approximations in te velocity of degree k 2 are used. Ten, te following bound olds for te error e n = Πdiv ut n U n, 1 n N e n 2 + e j ej t ν e n 2 + µ e n 2 exp ˆL1 t n + 2M 3 tn C 3 log tn K 3, t, 88 t were ˆL 1 is defined in 86, and C 3 depends on M 2 and M 3 but not on inverse powers of ν. 25

26 For te pressure, we take inner product of te first equation in 1 wit v V, subtract 74, add ±π t n, v and use te inf-sup condition, to obtain after some rearrangements β p n π t n ν e n + Bu n, un BU n, U n 1 + D t e n 1 +µ e n + Π div τ n 1,2 1 + I Π div tut n 1 + τ τ 3 + τ 4 + l t n, 89 were e n = un U n. As in Section 3.2, we estimate Πdiv τ n 1,2 1 C Π div τ n 1,2 and τ 2 1 C τ 2. Using 2 wit j = 1 we ave l t n ν 2 ut n 2. Also, repeating te arguments tat lead from 45 to 48 wit s replaced by Π div u, and in view of Lemma 6, we ave Bu n, un BU n, U n 1 C ut n 2 e n + e n +C d/2 e n + 1/2 e n e n. We now estimate I Π div tut n 1. For φ H 1Ωd, we use te Leray decomposition φ = Πφ + q, and recall tat Πφ 1 C φ 1 and q 2 C φ 1. We notice tat I Π div tu, q = I Π div tu, q = I Π div tu, q π q = I Π div tu, q π q C 2 t u 1 φ 1, 9 were in te last inequality we ave applied 1 wit j = and 11 wit m = 1 and j = 1. We also ave, I Π div tu, Πφ = I Π div tu, I Π div Πφ, so tat applying 1 wit j =, and togeter wit 9, it easily follows tat I Π div tut n 1 C 2 t ut n Finally, arguing as Section 3.2 te term D t e n 1 can be bounded by te terms on te te rigt-and side of 89 except itself, so tat we can conclude wit te following result. Teorem 9 In te conditions of Teorem 8, tere exists a constant C > not depending on inverse powers of ν suc te following bound olds, β 2 t N p n π t n 2 Crt n,, t 1 + rt n,, t d, were rt n,, t is te rigt-and side of 88. Remark 13 As in Remark 11, we observe tat if te two sources of error temporal and spatial are to be balanced in 88 at te final time t N = T, ten t 4 logn. Tus, d t = O 4 d logn, so tat te presence of negative powers of in te error bound in Teorem 9 does not alter te convergence rate, and te error is O 4 logn + t. 26

27 Figure 1: Left: numerical results for te velocity error, left-and side of estimate 41, and te pressure error, integral term on te left-and side of estimate 52 wit p p : different values of ν for a fixed spatial grid. Rigt: individual contributions of te left-and side of Numerical studies In tis section, numerical studies will be presented tat support te analytical results wit respect to te order of convergence and te independence of te errors of ν. As usual for suc purposes, an example wit a known solution is considered. Let Ω =, 1 2 and T = 5, ten te Navier Stokes equations 1 were considered wit te prescribed solution sinπx.7 sinπy +.2 u = cost, cosπx.7 cosπy +.2 p = costsinx cosy + cos1 1 sin1. It is clear tat examples constructed in tis way satisfy te nonlocal compatibility condition. Te simulations were performed for te P 2 /P 1 pair of finite element spaces on a regular triangular grid consisting on te coarsest level of two mes cells diagonal from lower left to upper rigt. Te number of degrees of freedom for velocity/pressure on level 3 is 578/81 and on level 8 it is /6649. As temporal discretization, te Crank Nicolson sceme was used. Te grad-div stabilization parameter was cosen to be µ =.25 in all simulations, see [15] for a motivation of tis specific coice. In eac discrete time, te fully nonlinear problem was solved. Te simulations were performed wit te code MooNMD [25]. Results of te numerical studies are presented in Figs. 1 and 2. For te simulations on level 6 wit different values of ν, Fig. 1, te equidistant time step.625 was used in te Crank Nicolson sceme. It can be clearly seen tat te velocity and pressure errors, wic were bounded in te analysis, are independent of ν. Considering te individual contributions of te velocity error, one can observe tat in particular te norm of te divergence is almost te same for all values of ν. For te simulations wit constant ν on a sequence of grids, te smaller time steps.2 and.1 were used. Because te curves for bot time steps are almost on top 27

28 Figure 2: Numerical results for te velocity error, left-and side of estimate 41, and te pressure error, integral term on te left-and side of estimate 52 wit p p. Left: different grid levels and different lengts of te time step for fixed ν te lines for te time steps.1 and.2 are almost on top of eac oter. Rigt: te individual terms of te left-and side of 41. of eac oter, see Fig. 2, it can be concluded tat te temporal error is negligible. Te pressure error decreases somewat faster tan predicted by te teory wit an order of nearly 2.9. Also te velocity error decreases faster on coarse grids because on tese grids te contribution e 5 dominates wic is reduced by a iger order tan two, compare te rigt picture of Fig. 2. But on finer grids, te predicted second order convergence can be seen. 7 Summary Inf-sup stable finite element discretizations are considered to approximate te evolutionary Navier Stokes equations. Te Galerkin finite element metod is augmented wit a grad-div stabilization term. It ad been reported in te literature [23, 29] tat stable simulations were obtained in te computation of turbulent flows using exclusively grad-div stabilization. Tis observation is te motivation of te present paper. Error bounds for te Galerkin plus grad-div stabilization metod were derived, bot for te continuous-in-time case and a fully discrete sceme. Te error constants do not depend on inverse powers of ν, altoug tey depend on norms of te solution tat are assumed to be bounded. Te paper extends a previous work by te same autors [15], were te evolutionary Oseen equations were considered. Te analysis covers bot te case in wic te solution is assumed to be smoot and te practically relevant situation in wic nonlocal compatibility conditions are not satisfied and, ence, te derivatives of te solution cannot be assumed to be bounded up to t =. To te best of our knowledge, tis paper is te first one were tis breakdown of regularity at t = as been taking into account to analyze te effect of te grad-div stabilization. Related works like [4, 8, 7] assume tat te solution satisfies nonlocal compatibility conditions. Te present paper also seems to be te first one were error bounds wit constants independent of ν are obtained for 28

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