Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements

Transcription

1 Noname manuscript No. will be inserted by te editor Grad-div stabilization for te evolutionary Oseen problem wit inf-sup stable finite elements Javier de Frutos Bosco García-Arcilla Volker Jon Julia Novo Received: date / Accepted: date Abstract Te approximation of te time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin metod wit grad-div stabilization is studied. Te main goal is to prove tat adding a grad-div stabilization term to te Galerkin approximation as a stabilizing effect for small viscosity. Bot te continuous-in-time and te fully discrete case backward Euler metod, te two-step BDF, and Crank Nicolson scemes are analyzed. In fact, error bounds are obtained tat do not depend on te inverse of te viscosity in te case were te solution is sufficiently smoot. Te bounds for te divergence of te velocity as well as for te pressure are optimal. Te analysis is based on te use of a specific Stokes projection. Numerical studies support te analytical results. Keywords. time-dependent Oseen equations, inf-sup stable pairs of finite element spaces, grad-div stabilization, backward Euler sceme, two-step back- Javier de Frutos Instituto de Investigación en Matemáticas IMUVA, Universidad de Valladolid, Spain. E- mail: frutos@mac.uva.es. Te researc was supported by Spanis MICINN under grant MTM P and European Cooperation in Science and Tecnology troug COST Action IS114. Bosco García-Arcilla Departamento de Matemática Aplicada II, Universidad de Sevilla, Spain. bosco@esi.us.es. Te researc was supported by Spanis MICINN under grant MTM Volker Jon Weierstrass Institute for Applied Analysis and Stocastics, Leibniz Institute in Forscungsverbund Berlin e.v. WIAS, Morenstr. 39, 1117 Berlin, Germany, and Free University of Berlin, Department of Matematics and Computer Science, Arnimallee 6, Berlin, Germany. jon@wias-berlin.de. Te researc was supported by Spanis MICINN under grant MTM P. Julia Novo Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain. julia.novo@uam.es. Te researc was supported by Spanis MICINN under grant MTM P.

2 Javier de Frutos et al. ward differentiation sceme BDF, Crank Nicolson sceme, uniform error estimates 1 Introduction A considerable amount of papers ave been recently written concerning te numerical approximation of te steady Oseen equations. Tese equations play a crucial role in te numerical simulation of te time-dependent incompressible Navier Stokes equations. After aving applied an implicit time discretization, a fixed point iteration can be used every time step to solve te resulting nonlinear equations. In tis context one as to approac a steady Oseen problem in every time step of tis iteration. It is well known tat te standard Galerkin metod suffers from instabilities for small values of te viscosity. Stabilized metods ave to be used to improve te numerical simulations. Te wellknown streamline upwind/petrov-galerkin SUPG metod combined wit te pressure-stabilization/petrov-galerkin PSPG metod allows to acieve bot stability and accuracy. A grad-div stabilization is usually included. Te SUPG/PSPG/grad-div stabilized metod applied to te steady Oseen equations was analyzed in [6], see also [4]. In [19], see also [4], te p version of te stabilized SUPG/PSPG/grad-div metod was analyzed for te same equations. Similar error bounds as in [6] were obtained. Te version of te metod was revisited in [13] using conforming inf-sup stable elements. In [], te reduced SUPG/grad-div stabilized sceme was studied again in te case of using inf-sup stable elements. A stabilized finite element formulation using ortogonal subscales was analyzed in [8]. All te results mentioned above concern te steady Oseen equations. Te analysis of te time-dependent Oseen equations can be seen as a first step towards te analysis of te evolutionary Navier Stokes equations. However, in te case of te evolutionary Oseen equations te literature is rater scarce. Te stabilized approac based on ortogonal subscales was described in [7] but not analyzed. Te analysis of te metod using time-dependent subscales can be found in [9]. Recently, in [1] te time-dependent Oseen problem was considered using Local Projection Stabilization LPS metods wit stabilization of te streamline derivative togeter wit grad-div stabilization. In te case of using metods of order k witout compatibility condition, error bounds are obtained under a restriction on te mes size: a certain measure for te mes size sould be of order of te square root of te viscosity, see [1, 35] for details. In order to avoid te restriction on te mes size for small viscosity, te autors of [1] consider pairs satisfying a certain element-wise compatibility condition between te discrete velocities on te fine mes and on te projection space. Even in tat case optimal error bounds for te pressure were not obtained in [1]. Several autors ave previously studied te effect of grad-div stabilization. In [] te autor considers te approximation of te steady incompressible Navier Stokes equations using bot SUPG and grad-div stabilization. Te

3 Grad-div stabilization for te evolutionary Oseen problem 3 grad-div stabilization is sown to enance te accuracy of te solution and to improve te convergence of preconditioned iterations for te linearized Navier Stokes problem if te corresponding stabilization parameter is not too large. In [3] te use of te grad-div term on te numerical solution of te Stokes equations is considered. Te autors sow tat tis stabilization improves te well-posedness of te continuous problem for small values of te viscosity. Tey also analyze te influence of tis stabilization on te accuracy of te approximation. A refined analysis was presented in [16]. In [1] te grad-div stabilization is considered as a subgrid pressure model in te framework of variational multiscale metods. Some error estimates for te steady Oseen problem wit grad-div stabilization are proved. In [1] a significant role of grad-div stabilization for inf-sup stable approximations is observed wile a SUPG-type stabilization seems to be muc less important. More precisely te autors of [1] say: it turned out tat te grad-div stabilization wit a globally constant parameter set is essential for an improvement of local mass balance and gives always good results in our numerical experiments. Neverteless, te teoretical foundation is not really convincing. Wit te teoretical results obtained in te present paper a teoretical foundation is added to te fact already observed about te improvement due to te grad-div stabilization. Even for te simulation of turbulent flows it was observed in [17, Fig. 3] tat te addition of only te grad-div stabilization to te Galerkin approximation was sufficient for performing stable simulations. Finally, te role of grad-div stabilization in preconditioning tecniques sould be mentioned. One of te advantages of suc a formulation is its positive effect in te solution of te Scur complement problem. As it is claimed in [], wit tis approac, te solution of te Scur complement is no longer te bottleneck of te iterative solution as it is te case in many block preconditioning approaces to te original system arising from te linearized Navier Stokes equations, see also [3, 15]. Te optimal value of te stabilization parameter is found in tese works to be small. Tis fact is in agreement wit te size O1 found to be optimal in te present paper. In te present paper, mixed finite element approximations to te timedependent Oseen problem are analyzed using inf-sup stable pairs of finite element spaces and a grad-div stabilization. It is sown tat te plain Galerkin approximations can be stabilized by adding only a grad-div stabilization term. Optimal error bounds wit constants tat do not depend on te viscosity parameter are obtained for te L norm of te divergence of te velocity and te L norm of te pressure, assuming te solution is smoot enoug. In addition, an error bound for ν 1/ times te gradient of te velocity is proved tat is optimal in te diffusion-dominated regime altoug it is a weak term in te convection-dominated regime. Te derived optimal error bounds for te L norm of te divergence of te velocity and te L norm of te pressure are global bounds tat can only be applied to globally smoot solutions. In [1] local error estimates are obtained for te SUPG metod applied to evolutionary convection-reaction-diffusion equations combined wit te backward Euler sceme. Te question of getting

4 4 Javier de Frutos et al. local error bounds for te metod studied in tis paper following te tecniques in [1] will be te subject of future researc. Let Ω R d, d {, 3}, be a bounded polyedral domain wit Lipscitz boundary Ω and let, T be a time interval wit T <. Te timedependent Oseen problem, as a model problem for te linearized Navier Stokes equations, reads as follows t ũ ν ũ + b ũ + p = f in, T ] Ω, ũ = in [, T ] Ω, ũ = on [, T ] Ω, ũ, x = ũ x in Ω, 1 were ũ :, T Ω R d and p :, T Ω R are te unknown velocity and pressure, ν > is te viscosity, f :, T Ω R d te external forces, and b :, T Ω R d a solenoidal vector field, i.e., b =, wit b L, T ; L Ω d. For te numerical analysis it is of advantage to perform a cange of variables: u, p = e αt ũ, p wit α >. A direct calculation sows tat wit tis transformation one obtains a problem wit a positive zerot order term: t u ν u + b u + αu + p = f in, T ] Ω, u = in [, T ] Ω, u = on [, T ] Ω, u, x = u x in Ω. Te analysis can be applied to te new problem. Finally, one can transform back to te original variables. In te analysis, α = 1/T is cosen suc tat te error bounds only cange in a multiplicative constant of size e αt e. In tis paper a grad-div sceme to approac problem will be analyzed. Te outline of te paper is as follows. Section introduces te grad-div stabilization of te Galerkin approximation and some preliminaries are stated. Section 3 is devoted to te analysis of te continuous-in-time case. In Section 4 te fully discrete case is considered using te backward Euler metod, te two-step backward differentiation formula BDF, and te Crank Nicolson sceme as time integrators. Finally, some numerical studies wic support te analytical results are presented in Section 5. Preliminaries and notation Using te function spaces, see [6, Section 1.] V = H 1 Ω d, Q = L Ω = { q L Ω : q, 1 = }, te weak formulation of problem is: Find u, p V Q suc tat for all v, q V Q, t u, v + ν u, v + b u + αu, v v, p + u, q = f, v. 3

5 Grad-div stabilization for te evolutionary Oseen problem 5 Notice tat b v, v = v V. 4 Te Hilbert space H div = {u L Ω d : u =, u n Ω = } will be endowed wit te inner product of L Ω d and te space V div = {u H 1 Ω d : u = } wit te inner product of H 1 Ω d. Let Π : L Ω d H div be te Leray projector tat maps eac function in L Ω d onto its divergence-free part. Te Stokes operator in Ω is given by A : DA V div V div, A = Π, DA = H Ω d V div. Te norm in L Ω for scalar-, vector-, and tensor-valued functions is denoted by and te norm in H k Ω by k. In te error analysis, te Poincaré Friedrics inequality will be used and also te inequality v C P F v v V 5 v v, v H 1 Ω d, 6 wic follows from te identity, v = v + v, a relation tat can be obtained from te vector identity v = v + v after taking te inner product in L Ω d wit v and integrating by parts. Let V V and Q Q be two families of finite element spaces tat correspond to a family of partitions T of Ω into mes cells wit maximal diameter. In tis paper, only pairs of finite element spaces will be considered tat satisfy te discrete inf-sup condition inf q Q v, q sup β >. 7 v V v q It will be also assume tat V H 1 Ω d and Q L Ω comprise piecewise polynomials of degrees at most k and l, respectively. It will be assumed tat te meses are quasi-uniform and tat te following inverse inequality olds for eac v V, see, e.g., [6, Teorem 3..6], v W m,q K d C inv l m d 1 q 1 q K v W l,q K d, 8 were l m 1, 1 q q, K is te size diameter of te mes cell K T, and W m,q K d is te norm in W m,q K d. Te space of discretely divergence-free functions is denoted by Te linear operator A V div = {v V : v, q = q Q }. : V div V div is defined by A v, w = v, w v, w V div. 9

6 6 Javier de Frutos et al. Note tat from tis definition it follows tat A 1/ v = v, A 1/ v = v v V div. 1 Additionally, te linear operators B : V div, V div, given by B v, w = v, w v, w V div, 11 and D : L Ω V div, given by D q, v = v, q q L Ω, v V div 1 are defined. Finally, te so-called discrete Leray projection Π div : L Ω d V div is introduced, wic is te L ortogonal projection of L Ω d onto V div Π div v, w = v, w w V div. 13 By definition, it is clear tat te projection is stable in te L norm: Π divv v for all v L Ω d. Denote by π te L projection of te pressure p in onto Q. Ten, for l and m 1 one as p π m C l+1 m p l+1, p H l+1 Ω. 14 Tere exists an interpolation operator I : H 1 Ω V tat satisfies for all v H l Ω d and all mes cells K T v I v,k + K v I v 1,K C l K v l,ωk, 1 l r + 1, 15 were ωk denotes a certain local neigborood of K, see [5]. To carry out te analysis, te Stokes problem ν u + p = g in Ω, u = on Ω, 16 u = in Ω, will be considered. Te standard Galerkin approximation u, p V Q is te solution of te mixed finite element approximation to 16, given by ν u, v + p, v = g, v v V, 17 u, q = q Q. Following [14] one gets te estimates u u 1 C inf u v 1 + ν 1 inf p q, 18 v V q Q p p C ν inf u v 1 + inf p q, 19 v V q Q u u C inf u v 1 + ν 1 inf p q. v V q Q

7 Grad-div stabilization for te evolutionary Oseen problem 7 It can be observed tat te error bounds for te velocity depend on negative powers of ν. For te purpose of analysis, it is useful to ave a projection of u, p onto V Q were te bounds for te velocity are uniform in ν. Tis goal can be acieved for smoot functions by coosing a special rigt-and side in 16. For te Oseen problem, let u, p be te solution of wit u V H k+1 Ω d, p Q H k Ω, k 1, and define te rigt-and side of te Stokes problem 16 by g = f t u b u αu p. 1 Ten u, is te solution of 16. Denoting te corresponding Galerkin approximation in V Q by s, l, one obtains from 18 u s + u s 1 C k+1 u k+1, were te constant C does not depend on ν. l Cν k u k+1, 3 Remark 1 Assuming te necessary smootness in time and considering 16 wit g = t f t u b u αu p, ten one can derive an error bound of form also for t u s. In te same way, one can proceed for iger order derivatives in time. Te metod tat will be studied for solving te Oseen problem is obtained by adding to te Galerkin metod a control of te divergence constraint, te so-called grad-div stabilization: Find u, p :, T ] V Q suc tat for all v, q V Q one as t u, v + A µ u, p, v, q = f, v, 4 were te initial discrete velocity is an appropriate approximation of u in V, A µ w, r, v, q = ν w, v + b w + αw, v v, r + w, q + µ w, v, and µ is a stabilization parameter, wose optimal asymptotic coice will be determined by te results of te numerical analysis. 3 Error analysis of te metod in te continuous-in-time case Te proof of te error estimates is based on te comparison of te Galerkin approximation u, p in 4 wit te approximation s, l of te Stokes

8 8 Javier de Frutos et al. equations wit rigt-and side 1. Let e = u s V div, ten a straigtforward calculation, using u =, yields t e, v + A µ e, p l, v, q = t u s, v + b u s + αu s, v 5 +µ u s, v + p, v, v V, q Q. Taking v, q = e, p l in 5, observing tat p, e = p, e = p π, e since e V div, using 4, and applying te Caucy Scwarz inequality gives 1 d dt e + ν e + α e + µ e t u s e + b u s e +α u s e + µ u s + p π e. Wit Young s inequality and iding terms on te left-and side, one obtains Assuming now d dt e + ν e + α e + µ e 3 t u s + b α u s + α u s +µ u s + µ 1 p π. 6 u, p L, t; H k+1 Ω d L, t; H l+1 Ω, t u, t p L, t; H k Ω d L, t; H l Ω, 7 applying te estimates and 14, integrating 6 on, t, and recalling tat α = 1/T, one gets e t + ν e L,t;L + α e L,t;L + µ e L,t;L e + C k T + µ u L,t;H k+1 + T tu L,t;H k +Cµ 1 l+1 p L,t;H l+1, 8 were C = C b L,t;L. Remark It will be assumed tat u H k Ω d and tat te projection s is well defined at t =, wic implies tat compatibility conditions at t = are assumed. Ten, if, for example, u = I u, te error e can be bounded by e I u u + u s, and ten 15 and can be applied.

9 Grad-div stabilization for te evolutionary Oseen problem 9 Teorem 1 Let u, p V Q be te solution of 3 and let u, p V Q be te solution of 4. Assume tat 7 and te conditions from Remark old. Ten, te following error estimate olds for all t, T ] u u t + ν u u L,t;L + α u u L,t;L +µ u u L,t;L C k T + µ u L,t;H k+1 + T tu L,t;H k + u k + ut k +Cµ 1 l+ p L,t;H l+1, 9 were C = C b L,t;L. Proof Te result is obtained by applying te triangle inequality to te leftand side of 9 and using 8 and. Remark 3 Te most common situation for te coice of te finite element spaces consists in coosing te polynomial degree of te pressure one degree smaller tan for te velocity, i.e., l = k 1. Ten, it follows from 9 tat te optimal error bound O k is obtained for µ = O1. In te case k = l, e.g., for te MINI element were k = l = 1, one can coose µ = O or even µ = O. Wic coice is te better one depends on te concrete situation, see te discussion in [16]. To facilitate te presentation of te furter analysis, it will be assumed encefort tat l = k 1. Lemma 1 Te following stability estimate olds for te discrete velocity u t + ν u L,t;L + α u L,t;L + µ u L,t;L u + 1 α f L,t;L t [, T ]. 3 Proof Taking v, q = u, p in 4 and using 4, it follows immediately tat 1 d dt u + ν u + α u + µ u f u. Applying Young s inequality on te rigt-and side and integrating on, t gives 3. Remark 4 From 3 it follows in particular tat lim u µ L,t;L lim µ 1 µ u + 1 α f L,t;L =. Tis beavior is in agreement wit te result of [5] were te autors sow tat te grad-div stabilized Taylor Hood approximation to te evolutionary Navier Stokes equations converges to te Scott Vogelius solution as te stabilization parameter tends to infinity. Te Scott Vogelius element pair provides point-wise mass conservation. In te numerical tests of [5] te grad-div stabilized Taylor Hood approximation wit large stabilization parameter is sown to provide excellent mass conservation for te Navier Stokes approximation.

10 1 Javier de Frutos et al. Te next step in te error analysis consists in obtaining a bound for te pressure error. Tis bound is derived as usual on te basis of te discrete inf-sup condition 7. As a first step, a bound for t e 1 is needed. By definition, it is t e 1 = t e, ϕ sup, ϕ H 1Ωd,ϕ ϕ were, denotes te corresponding duality pairing. To start wit, te bound of t e 1 is reduced to a bound of A 1/ t e. From [1, Lemma 3.11] it is known tat t e 1 C t e + C A 1/ Π t e, 31 were Π is te Leray projector introduced in Section. Applying [1,.15], one obtains A 1/ Π t e C t e + A 1/ t e, 3 wit A defined in 9. From 31, 3, using te symmetry of A, 1, and te inverse inequality 8, it follows tat t e 1 C t e + C A 1/ t e = C A 1/ A 1/ t e + C A 1/ t e = C A 1/ t e + C A 1/ t e C A 1/ t e. 33 Next, a bound for A 1/ t e will be derived. Projecting te error equation 5 onto te discretely divergence-free space V div and using integration by parts, one gets t e, v + ν e, v + b e + αe, v + µ e, v = t u s, v + b u s + αu s, v 34 +µ u s, v p π, v, were π is te L projection of p onto Q. Recalling definition 1 one as p π, v = D p π, v, suc tat t e = νa e Π div b e + αe µb e + Π div t u s b u s + αu s + µb u s 35 +Π div D p π.

11 Grad-div stabilization for te evolutionary Oseen problem 11 Wit 11, te Caucy Scwarz inequality, 6, and 1, one obtains for all v V div A 1/ B v = sup w V div = sup w V div sup w V div sup w V div = sup w V div,w,w,w,w,w B v, A 1/ w w v, A 1/ w w v A 1/ w w v A 1/ w w Te same argument applied to A 1/ D p π gives v w w = v. 36 A 1/ D p π p π. 37 For g L Ω d, te definition 13 and te symmetry of A allows to write A 1/ Π divg, v = g, A 1/ v for all v V div. Taking v = A 1/ Π divg in tis relation and applying 1 yields V div A 1/ Π div g g 1 A 1/ and, ence, Next, A 1/ A 1/ t e A 1/ Π div g = g 1 A 1/ Π div g A 1/ Π div g g 1 g L Ω d. 38 is applied to 35. Using 36, 37 and 38, one gets ν A 1/ e + b e + αe 1 + µ e + t u s b u s + αu s 1 + µ u s + p π. Taking te square of 39 and integrating on, t yields A 1/ s e s ds C ν A 1/ e s ds + +µ e s ds + b e + αe s 1 ds s u s s 1 ds + p π s ds + b u s + αu s s 1 ds +µ u s s ds. 4

12 1 Javier de Frutos et al. It will be proved tat all te terms on te rigt-and-side of 4 are O k. To tis end, it will be assumed as before tat te initial error e is of order O k. Ten, te desired asymptotic beavior is obtained for te first and tird terms directly from 8. Comparing wit te bound in 8, an extra factor µ = O1 multiplies te tird term. For te second term in 4, te definition of te H 1 Ω norm, integrating by parts and Poincaré s inequality leads to b e + αe s 1 ds C1 + α e s ds, suc tat te desired order of convergence can be again deduced from 8. Concerning t u s 1, te definition of te H 1 Ω norm and Poincaré s inequality are applied to bound tis term by C t u s. Now, is applied see Remark 1 and wit ypotesis 7, te estimate for t u s 1 is O k. Once tis term is bounded, it is clear tat te integral of its square is also bounded s u s s 1 ds C k t u L,T ;H k. To bound u s, estimate is used again. Arguing as for te second term, one obtains b u s + αu s s 1 ds C1 + α u s s ds. Te bound for tis term is concluded by applying. Combining te estimates for 4 wit 33, and recalling tat α = 1/T, it follows tat s e s 1 ds = O k. 41 Teorem Let te assumptions of Teorem 1 old, let ν 1 and l = k 1, ten p p L,t;L C 1 + µ k u k + ut k + C k 1 + µ T + µ u L,t;H k+1 + T 1 + µ tu L,t;H k + C 1 + µ 1 l+ p L,t;H l+1, 4 were C = C β 1, b L,t;L.

13 Grad-div stabilization for te evolutionary Oseen problem 13 Proof Using te discrete inf-sup condition 7 and 5, one obtains β p π ν e + b e + αe 1 + t e 1 + µ e + t u s 1 + b u s + αu s 1 +µ u s + p π + l. Squaring bot sides of tis inequality and integrating on, t one as β p π s ds C ν e s ds b e + αe s 1 ds s e s 1 ds + µ e s ds s u s s 1 ds + +µ u s s ds + b u s + αu s s 1 ds p π s ds + l s. Arguing exactly as for te estimates of te rigt-and side of 4, using estimates 41 for se s 1 ds, 3 to bound te last term, and finally te triangle inequality, ten 4 is proved. Remark 5 Neiter te analysis of tis section nor te analysis of te fully discrete case in next section require te assumption Q H 1 Ω. In addition, te analysis works for inf-sup stable divergence-free pairs of finite element spaces, like te Scott Vogelius pair on barycenter-refined grids, witout grad-div stabilization. Proving te error bound for te velocity wit suc elements, te term p, e = p, e in 5 vanises since e =. To get te error bound for te pressure, te term p π, v vanises wen 34 is projected onto V div. Tus, in te case of stable divergence-free elements estimates wit ν-independent constant for a sufficiently smoot solution are also acieved. 4 Fully discrete cases Tis section studies fully discrete cases. First, in Section 4.1, te backward Euler sceme is considered as temporal discretization and ten BDF in Section 4.. Finally, te Crank Nicolson sceme is analyzed in Section 4.3. Error estimates for bot velocity and pressure errors are derived. Consider a decomposition of te time interval [, T ] wit equidistant steps suc tat = t < t 1... < t N = T and t n = t n 1 +, n = 1,..., N.

14 14 Javier de Frutos et al. 4.1 Backward Euler metod Te backward Euler metod, togeter wit an inf-sup stable finite element discretization, applied to 1 reads as follows: Find Ũ n, P n V Q suc tat for all v, q V Q Ũn Ũ n 1, v + ν Ũ n, v + b Ũ n, v n v, P n + Ũ n, q + µ Ũ n, v = f n, v, n = 1,..., N, 43 were Ũ is a finite element approximation of ũ. Using te notation U n, P n = e αtn Ũ n, P n, a direct calculation yields e αtn Ũ n Ũ n 1 = U n U n 1 + α U n 1 = U n U n 1 + α U n α U n U n 1, were α = 1 e α / = α + C. Using tis relation in 43 it follows tat U n, P n satisfies U n U n 1, v + νu n, v + b U n, v n + α U n, v α U n U n 1, v v, P n + U n, q + µ U n, v = f n, v, n = 1,..., N, 44 for all v, q V Q. Let s n = s t n be te solution of te discrete Stokes problem 17 wit rigt-and side 1 corresponding to t n and denote e n = U n s n V div. Subtracting te equation for sn from 44 gives e n e n 1, v + ν e n, v + b e n, v + α e n, v v, P n l n + e n, q + µ e n, v α e n e n 1, v = t s n sn sn 1, v + t u n s n, v 45 +b u n s n + αu n α s n 1, v +µ u n s n, v + p n, v, v V, q Q. A direct calculation sows tat e n e n 1 n, e n = 1 en 1 en en e n 1.

15 Grad-div stabilization for te evolutionary Oseen problem 15 Using tis relation, taking v = e n in 45, using 4, en V div, integration by parts, te Caucy Scwarz and Young s inequality yields 1 e n e n 1 + e n e n 1 + ν e n + α e n + µ e n α e n e n 1, e n + 1 α t s n sn sn 1 + α 4 en + 1 α tu n s n + α 4 en α b un s n + αu n α s n 1 + α 4 en +µ u n s n + µ 4 en + 1 µ pn π n + µ 4 en. Wit te notations T1 n = 1 α tu n s n, T n = 1 α t s n sn sn 1, T n 3 = 1 α b un s n + αu n α s n 1, T n 4 = µ u n s n + 1 µ pn π n, estimate 46 can be written in te form 1 e n e n 1 + e n e n 1 + ν e n + µ en + α 4 en α e n e n 1, e n + T1 n + T n + T3 n + T4 n. Observing tat and assuming α e n e n 1, e n en en 1 + α e n, α < α 8 < 1 4α = 41 e α, 47 one gets 1 e n e n 1 + ν e n + µ en + α 8 en T1 n + T n + T3 n + T4 n. 48 Observe tat 47 olds if < log4/3t, since α = 1/T.

16 16 Javier de Frutos et al. After summation of te discrete times j = 1,..., n, one obtains e n + ν e j + µ e j + α 4 ej e + T j 1 + T j + T j 3 + T j Te terms on te rigt-and side ave to be bounded. Applying togeter wit Remark 1 and recalling tat α α = 1/T yields T j 1 CT k t ut j k, j = 1,..., N. 5 To bound T j, a sequence of inequalities is used applying a Taylor series expansion wit reminder in integral form, te Caucy Scwarz inequality, te triangle inequality, Remark 1 for te second derivative, and te Poincaré Friedrics inequality 5 T j CT t s j sj sj 1 = CT 1 tj t t j 1 tt s dt t j 1 CT 1 tj j t t j 1 dt tt s dt t j 1 t j 1 CT j t j 1 tt s dt CT j t j 1 tt u 1 dt, j = 1,..., N. 51 Te first part of T j 3 is bounded by applying b u j s j C k ut j k+1, j = 1,..., N. 5 To bound αu n α s n 1 one can proceed as follows. Denoting s t = e αt s t, a direct calculation leads to te representation α s n 1 = e αtn sn s n 1 = αs n + T5 n + T6 n, sn sn 1 = e αtn t s n t s n + T n 5 + T n 6 were T n 5 s n = e αtn s n 1 t s n, T6 n = t s n sn sn 1. Since T n 5 + T n 6 can be bounded similarly to T n, for n = 1,..., N, one gets αu n α s n 1 αu n s n + C e αtn n t n 1 tt ũ 1 dt + n t n 1 tt u 1 dt.53

17 Grad-div stabilization for te evolutionary Oseen problem 17 Togeter wit 5 it follows tat, for j = 1,..., N, T j 3 CT k ut j k+1 + CT Finally, applying gives Likewise, using 14 yields suc tat j tt ũ 1 + tt u 1 dt. t j 1 µ u j s j Cµ k ut j k+1. 1 µ pj π j C k µ pt j k, T j 4 Ck µ ut j k+1 + µ 1 pt j k, j = 1,..., N. 54 Collecting all estimates, one gets from 49 for n = 1,..., N, e n + ν e j + µ e j + α 4 ej e + CT tt ũ L,t n;h 1 + ttu L,t n;h 1 +C k T + µ ut j k+1 +C k µ 1 pt j k + T k=1 t ut j k. 55 Teorem 3 Let ũ, p V Q be te solution of 1 and let Ũ n, P n V Q be te backward Euler approximation solving 43 for n = 1,..., N. Assume tat e = O k, l = k 1, ν 1, < log4/3t so tat 47 olds and tat te solution is sufficiently regular suc tat all norms appearing in te formulation of tis teorem are well defined. Ten, te following error estimate olds ũt n Ũ n + ν ũt j Ũ j + µ ũt j Ũ j C k ũ k + ũt n k + CT ũ H,t n;h 1 + C k T + µ ũt j k C k T t ũt j k + µ 1 pt j k.

18 18 Javier de Frutos et al. Proof Estimate 56 is obtained applying te triangle inequality and using and 55. Te bound for te error in te pressure can be obtained along te lines of te proof for te time-continuous case. Considering 45 and simplifying te terms α e n, v α e n en 1, v to α e n 1, v, ten one obtains wit te te discrete inf-sup condition 7 β P n π n ν e n + b e n + α e n e n en 1 + µ e n 1 + t u n s n 1 + ts n sn sn 1 + b u n s n αu n α s n µ u n s n + p n π n + l n, 57 and, consequently β P j πj C [ ν e j + b e j α e j 1 e j ej 1 + µ e j + t u j s j 1 1 ts j sj sj 1 + b u j s j 1 αu j α s j µ p j π j + l j ] u n s n. 58 Te first, tird, and fift terms on te rigt-and side above are already bounded in 55. Since te bound b e j 1 C e j is satisfied, one can also apply 55 to estimate tis term. For terms ranging from te sixt to te nint on te rigt-and side of 58, one can first bound 1 by and ten apply te bounds 5 to 53. To bound te tent term on te rigt-and side above, 54 is used. For te last two terms one can apply 14 and 3. Altogeter, to conclude te estimate, it only remains to bound te fourt term on te rigt-and side of 58. Te estimate of tis term follows wit te same arguments as in te time-continuous case. First, one applies e j ej 1 C 1 A 1/ e j ej 1

19 Grad-div stabilization for te evolutionary Oseen problem 19 and ten one derives te estimate e j A 1/ ej 1 ν A 1/ ej + b e j + α e j µ e j + t u j s j 1 + ts n sn sn 1 + b u j s j αu j α s j µ s j + p j π j. From tis inequality, one gets te bound for te fourt term on te rigt-and side of 58 in te same way as for te oter terms. Teorem 4 Let te assumptions of Teorem 3 old, ten te following error estimate for te pressure is valid P j pt j C k ũ k + CT 1 + µ ũ H,t n;h 1 + C k T + µ1 + µ + C k T 1 + µ ũt j k+1 59 t ũt j k µ 1 pt j k. 4. BDF As in te case of te backward Euler metod, Ũ n, P n denotes te fully discrete approximation at time t n, n =,..., N, wic satisfies for BDF 1 D + 1 D Ũ n, v + ν Ũ n, v + b Ũ n, v v, P n + Ũ n, q + µ Ũ n, v = f n, v. 6 Here, D is te backward difference for a sequence y n N n=1, i.e., Dy n = y n y n 1, n = 1,..., N. Note tat D y n = y n y n 1 + y n, for n =,..., N, suc tat D + 1 D Ũ n = 3 Ũn Ũ n Ũn, n =,..., N. It will be assumed tat Ũ 1 is obtained by one step wit te backward Euler metod and tat Ũ is an appropriate finite element approximation of ũ.

20 Javier de Frutos et al. Recall tat U n, P n = e αtn Ũ n, P n, suc tat a straigtforward calculation yields e αtn 3 Ũn Ũ n Ũn 3 = U n U n 1 + α U n U n 1 α U n, 61 were α = 1 e α /, as in te case of te backward Euler metod, and α = 1 e α /. For te last two terms on te rigt-and side one can write α U n 1 1 α U n = α U n α U n U n 1 = α U n α D U n + + U n + α 1 α U n α 1 α U n. Now, observe tat α α / = 1/ e α + 1/e α tat is, a second backward difference of e αt at t =, and, tus α 1 α = β, 6 were β = e αξ α / for some ξ,. Tus, 61 can be written in te form 3 e Ũn αtn Ũ n Ũn 3 = U n U n U n 63 + α U n α D U n + βu n Ten, arguing as in te case of te backward Euler metod, one finds for BDF tat e n = U n s n V div satisfies 1 D + 1 D e n, v + ν e n, v + b e n, v + α e n, v v, P n l n + e n, q + µ e n, v α D e n, v + βe n, v = t s n 1 D + 1 D s n, v + t u n s n, v 64 +b u n s n + αu n α s n D s n βs n, v +µ u n s n, v + p n, v, v V, q Q. Observe tat tere are te following differences between 64 and 45. In te first term on bot left- and rigt-and sides, te first divided differences appearing in 45 are replaced by D + D //. Also, te last term on te left-and side in 45, e n en 1, v, is replaced by te last two terms on te left-and side in 64. Te rest of te terms in 45 is te same as in.

21 Grad-div stabilization for te evolutionary Oseen problem Finally, in te fourt term on te rigt-and side in 45, te difference αu n α s n 1, is replaced by αu n α s n D s n βsn in 64. Due to te similarities between bot expressions, now only te terms wic are different to 45 will be considered in detail. Taking v = e n in 64, a direct calculation reveals tat D + 1 D e n, e n = 1 4 en en + De n D e n 1 4 en en 1 + De n Te last two terms on te left-and side of 64 can be estimated from below as follows α D e n, e n + βe n, e n D e n α + β e n 4 β en D e n α 4 16 en α 16 en, 66 were te last inequality is true only if is sufficiently small, i.e., if α + β α olds. Since in view of 6, β/α = 1 α /α = e α / = 1 e α /, one finds tat 67 olds if 51 e α 1/4, wic, since α = 1/T is assumed, olds if < log/19t. For te fourt term on te rigt-and side of 64, using 63 wit U j and Ũ j replaced by s j and sj = s j eαtj, respectively, for j = n, n 1, n, one can write α s n D s n + βs n Tus, it is = e αtn 1 D + 1 D s n 1 D + 1 D s n. αu n α s n D s n βs n = αu n e αtn t s n + t s n + T n 5 + T n 6 = αu n s n + T n 5 + T n 6, were T5 n = e αtn t s n e 1 αtn D + 1 D s n, T6 n = 1 D + 1 D s n t s n. 68 Consequently, wen taking v = e n in 64 te fourt term on te rigt-and side can be bounded in te following way αu n α s n D s n βs n, e n α 16 en +8 α α un s n +T n 56, 69

22 Javier de Frutos et al. were T n 56 = 8 α T n 5 + T n 6. 7 Ten, arguing as in te case of te backward Euler metod and using 65, 66, and 69, one gets for BDF, instead of 48, 1 e n 4 + e n + De n e n 1 e n 1 + De n 1 + ν e n + µ en + α 8 en α 16 en + T n 1 + ˆT n + ˆT n 3 + T n 4 + T n 56, were T n 1 and T n 4 are as in te case of te backward Euler metod, ˆT n ˆT n 3 = 1 α t s n 1 D + 1 D s n, = 1 α b un s n + 8 α α un s n, and T n 56 is defined in 68 and 7. Multiplying 71 by 4, summing up, and rearranging terms gives 71 e n + 4ν e j + α 4 ej + µ e j j= e 1 + e 1 + De 1 + α e 4 + e T j 1 + ˆT j + ˆT j 3 + T j 4 + T j 56. j= 7 A direct calculation sows tat e 1 + e 1 + De 1 + α e 4 + e α e α e 4 C e 1 + e. 73 In view of 5 and one as tat ˆT j 3 CT k ut j k+1, j =,..., N. 74 A Taylor series expansion yields for ˆT j ˆT j C T j t j t t j t t j ttt s t dt,

23 Grad-div stabilization for te evolutionary Oseen problem 3 j =,..., N, were x + = maxx, for x R. Wit te Caucy Scwarz inequality, one obtains for j =,..., N, ˆT j C T tj t t j j t j t t j dt ttt s t dt t j j j CT 3 ttt s t 3 dt CT ttt ut 1 dt. 75 t j t j Since a similar bound is valid for T j 56, and te bounds 5 and 54 on T 1 n and T4 n computed in te case of backward Euler metod also apply in te present case, from 7 75, it follows for n =,..., N, tat e n + 4ν e j + α 4 ej + µ e j j= C e 1 + e + CT 4 ttt u L,t n;h 1 + tttũ L,t n;h 1 +C k T + µ ut j k+1 + T t ut j k 76 +C k µ pt j k. Finally, one as to sow tat e 1 = Ok + 4. Te first step is performed wit te backward Euler sceme. To prove te needed order wit respect to time, it will be exploited tat te lengt of te first time interval is just. Starting from 45 wit n = 1 and taking v = e 1, te first and tird term on te rigt-and side of 45 are estimated wit te Caucy Scwarz and Young s inequality and t s 1 s1 s, e 1 b u 1 s 1 + αu 1 α s, e 1 ts 1 s1 s e1, CT b u 1 s 1 + α 4 e1 + αu 1 α s e1. All oter terms in 45 are bounded as in 46. Arguing in te same way as after 45, now instead of 48, one arrives at te estimate 1 1 e1 e + ν e 1 + µ e1 + α 8 e1 T1 1 + α T 1 + T T3 1 + T4 1, were T 1 31 = 1 α b u1 s 1, T 1 3 = αu 1 α s.

24 4 Javier de Frutos et al. Taking into account te bounds on T1 1, T 1, T4 1 given in 5, 51, and 54, respectively, as well as 5 and 53, one obtains e 1 + ν e 1 + α 4 e1 + µ e 1 e + CT k ut 1 k+1 + t ut 1 k + pt 1 k + CT 3 tt ũ 1 + tt u 1 dt. Te estimate of te last term on te rigt-and side provides an additional factor from te lengt of te first interval 3 tt u 1 dt 4 tt u L,,H 1 C 4 tt u L,,H 1 + ttt u L,,H 1 In te second step L,,H 1 C H1,,H 1 was used, wic is a consequence of te Sobolev inequality L, C H1,. Analogously, one obtains 3 tt ũ 1 dt 4 tt ũ L,,H 1 C 4 tt ũ L,,H 1 + ttt ũ L,,H 1 Inserting te estimates for te first step into 76 yields for n = 1,..., N, e n + 4ν e j + α 4 ej + µ e j j= C e + CT 4 ttt u L,t n;h 1 + tttũ L,t n;h 1 + CT 4 u H 3,,H 1 + ũ H 3,,H 1 + C k T + µ ut j k+1 + T t ut j k + C k µ 1 pt j k. 77 From tis estimate, arguing as for te backward Euler sceme, one gets te following teorem. Teorem 5 Let ũ, p V Q be te solution of 1, wic sould be sufficiently smoot suc tat all norms appearing in te formulation of tis teorem are well defined, and let Ũ n, P n V Q be te BDF approximation solving 6 for n =,..., N. Assume tat e = Ok, l = k 1, ν 1,..

25 Grad-div stabilization for te evolutionary Oseen problem 5 log/19t so tat 67 olds and tat te backward Euler metod is used for te first step. Ten, te following error estimate olds Ũ n ũt n n + ν Ũ j ũt j + µ Ũ j ũt j C k ũ k + ũt n k + CT 4 ũ H 3,t n;h 1 + C k T + µ ũt j k+1 + T t ũt j k + C k µ 1 pt j k. 78 Te estimate for te pressure error is performed also along te same lines as for te backward Euler sceme. It starts from 64. For convenience, te term α e n, v α D e n, v +βe n, v in 64 is expressed in te form α e n, v α D e n, v + βe n, v = α e n 1 1 α e n, v. Ten te arguments used for te backward Euler metod are valid for BDF if one replaces te occurrences of e n en 1 /, s n sn 1 /, α e n 1, and α s n 1 by D + D //e n, D + D /s n /, α e n 1 α e n /, and α s n D s n βsn, respectively. Tus, instead of 57, for te BDF one gets β P n π n 1 1 D + D e n + ν e n 1 + b e n 1 + α e n 1 1 α e n 1 + µ e n + t u n s n 1 + t s n 1 1 D + D s n + b u n s n αu n α s n D s n βs n µ u n s n + p n π n + l n. Since b e n 1 C en and α = α 1 + e α α, it follows tat b e n 1 + α e n 1 1 α e n C e n + e n 1 + e n. 1 Using also te fact tat 1 C, te truncation errors, i.e., te terms ranging from te sixt to te tent on te rigt-and side of 79, can be bounded as in te estimate for te velocity. Applying in addition 77 yields β P j j= C πj 1 1 D + D e j + C 1 j= p n π n + l n + T n 7, j=

26 6 Javier de Frutos et al. were T7 n can be bounded by te rigt-and side of 77. Since te second sum on te rigt-and side can be bounded by applying 14 and 3, it only remains to bound te first sum. Again, arguing as in te case of te backward Euler metod, 1 D + D /e n 1 can be bounded in terms of te square of tose terms on te rigt-and side in 79 ranging from te second until te elevent. It follows tat, for n =,..., N, one finds β P j πj j= C e + CT 1 + µ 4 u H 3,,H 1 + ũ H 3,,H 1 +CT 1 + µ 4 ttt u L,t n;h 1 + tttũ L,t n;h 1 +C k T + µ1 + µ ũt j k+1 + T 1 + µ t ut j k +C k 1 + µ 1 pt j k. Now, te estimate for te pressure follows wit te same arguments as before. Teorem 6 Let te assumptions of Teorem 5 old. Ten, te following bound olds for te approximation to te pressure using BDF P j pt j j= C k ũ k + CT 1 + µ 4 ũ H 3,t n;h 1 +C k T + µ1 + µ ũt j k+1 + T 1 + µ +C k 1 + µ 1 t ut j k pt j k Crank Nicolson sceme Te Crank Nicolson sceme is peraps te most popular time integrator among te scemes considered ere. Te analysis of tis sceme is more involved tan te analysis of te two oter metods. For instance, it is more difficult to estimate te convective term, wic will be performed ere wit a special family of test functions, see 95. Te Crank Nicolson metod, togeter wit an inf-sup stable finite element discretization, applied to 1 reads as follows: Find Ũ n, P n 1/ V Q

27 Grad-div stabilization for te evolutionary Oseen problem 7 suc tat for all v, q V Q Ũn Ũ n 1, v + ν + + b Ũn n 1/ + Ũ n 1 Ũ n + Ũ n 1 n n 1 f + f =, v, Ũn + Ũ n 1, v, v, q + µ v v, P n 1/ 81 n = 1,..., N, were Ũ is a finite element approximation of ũ, t n 1/ = t n /, n = 1,..., N, and for a function v = vt, te notation v j = vt j is used. To simplify notation, in te sequel te superscript in te convection field will be omitted. It will be assumed tat Ũ, q = for all q Q, so tat taking v = in 81, it follows tat Ũ n, q =, q Q, n =,..., N. 8 For g = f t ũ b ũ p, let ũ, be te solution of 16, and let s, l be te corresponding Galerkin approximation in V Q. Using ũ n = and integration by parts, one finds wit a straigtforward calculation tat te error ẽ n = Ũ n s n V div, n =,..., N, satisfies for all v, q V Q, ẽn ẽ n 1, v + ν v, P n 1/ ẽn + ẽ n 1 π n 1/ +, v + b ẽn + ẽ n 1 ẽn + ẽ n 1, q + µ v, v = r n 1/ 13, v + r n 1/ 45, v, n = 1,..., N, 83 were π n 1/ denotes te ortogonal projection of pt n 1/ onto Q, r n 1/ 13 = r n 1/ 1 + r n 1/ + r n 1/ 3, r n 1/ 45 = r n 1/ 4 + r n 1/ 5,

28 8 Javier de Frutos et al. and r n 1/ 1 = tũ n + t ũ n 1 ũn ũ n 1, r n 1/ =ũn ũ n 1 sn s n 1, r n 1/ 3 = b sn + s n 1 ũn + ũ n 1 + b, r n 1/ 4 = µ sn + s n 1 + µ ũn + ũ n 1, r n 1/ 5 = l n n 1 + l pn + p n 1 + π n 1/. Since s n, q = for all q Q, and in view of 8, it follows tat ẽ n, q =, q Q, n =,..., N. 84 Using similar arguments to tose applied in 51 and 75, one sows tat r n 1/ 1 = 3/ ttt ũ L t n 1,t n,l. 85 Wit and Remark 1, one obtains r n 1/ 1 tn = t ũ t s dt t n 1 C k tũ 1/ L t n 1,t n,h k 86 and r n 1/ 3 + r n 1/ 4 C1 + µ k ũ n k+1 + ũ n 1 k Applying a Taylor series expansion and 3 gives 1 r n 1/ 5 = C k ũ n j k+1 + p n j k + 3/ tt p L t n 1,t n,l. j= Te following formulae are valid for a, b, u, v R au bv = a + b au + bv = a + b + v u v + a bu, v u + v + a bu. By applying tem, one obtains wit straigtforward calculations 88 e αt ẽn n 1/ ẽ n 1 e αt ẽn n 1/ + ẽ n 1 = 1 + β en en 1 = 1 + β en + en 1 + α en + en 1, + α en en 1, 89

29 Grad-div stabilization for te evolutionary Oseen problem 9 were α = sinα/, β = cosα/ 1, and, as before, e n = e αtn ẽ n, n =,..., N. Observe tat α α and β α /8, as. Now, te terms on te left-and side of 83 will be considered. Setting γ n = e αt n 1/, 9 it follows tat ẽn γ ẽ n 1 n, e n + e n 1 = 1 + β e n e n 1 ẽn + ẽ n 1 γ n ν γ n µ ẽn + ẽ n 1 ẽn + ẽ n 1 γ n b + α e n + e n 1, 91, e n + e n 1 = 1 + β ν e n + e n α ν e n e n 1,, e n + e n 1 = 1 + β µ e n + e n 1 9, e n + e n α µ e n e n 1, = α b e n, e n Te last term as to be canceled by an appropriate coice of te test function. To tis end, inner products wit e n en 1 will be considered. One as ẽn + ẽ n 1 γ n b, e n e n 1 = 1 + β b e n, e n Now, te test function v V div in 83 is cosen as a linear combination of e n + en 1 and e n en 1 suc tat te terms on te rigt-and sides of 93 and 94 cancel eac oter. For tis purpose, use q = and define v = γ n e n + e n 1 + ρe n e n 1 α, ρ = 1 + β. 95

30 3 Javier de Frutos et al. Direct calculations sow ẽn ργ ẽ n 1 n, e n e n 1 = α e n e n 1 + α ρ e n e n 1, 96 ẽn + ẽ n 1 ργ n ν, e n e n 1 = 4 α ν e n e n 1 ργ n µ ẽn + ẽ n 1, e n e n α ρν e n e n 1, 97 = 4 α µ e n e n α ρµ e n e n Wit te Caucy Scwarz inequality and Young s inequality, one obtains for v defined in 95 13, v 4 α e n + e n α e n e n 1 r n 1/ +C 1 e αt n 1/ r n 1/ 13, 99 were C 1 = 1 α + α 41 + β, 1 suc tat te first two terms on te rigt-and side in 99 cancel wit one alf of te corresponding terms in 91 and 96. In te same way, one gets r n 1/ 45, v 1 + β 4 µ en + e n 1 + α β µ en e n 1 + C e αt n 1/ r n 1/ 45, 11 were C = µ1 + β, 1 so tat te first two terms on te rigt-and side in 11 cancel wit one alf of te sum of te corresponding terms in 9 and 98. Te pressure term on te left-and side of 83 vanises since v V div because of 84.

31 Grad-div stabilization for te evolutionary Oseen problem 31 Taking v, q as specified in 95, summing from 1 to n, and ignoring some non-negative terms on te left-and side yields 1 + β e n + ν e j + ej 1 + µ ej + ej β e + E 13 m + C 1 e αt j 1/ r j 1/ 13 + C e αt j 1/ r j 1/ 45, n=1 were C 1 and C are te constants in 1 and 1 and E = α ν e 4 + µ e + α 1 + β e. For α = 1/T and T, it is a simple calculation to ceck tat C 1 CT and C Cµ 1 were C. In view of and since e αt j 1/ 1, one obtains C 1 e αt j 1/ r j 1/ 13 + C e αt j 1/ r j 1/ 45 C { 4 T ttt ũ L,t n,l + µ 1 tt p L,t n,l 14 +T k t ũ L,t n,h k + [ 1 + µ T + µ 1 + µ 1] k ũ j k+1 + µ 1 k j= } p j k. For α = 1/T and T, one as α α sin1/ C and 1 + β cos1/ C. Ten, it follows from 13 and 14 tat j= e n + ν e j + ej 1 + µ ej + ej 1 e + C { ν e + µ e + e + 4 T ttt ũ L,t n,l + µ 1 tt p L,t n,l 15 +T k t ũ L,t n,h k + [ } 1 + µ T + µ 1 + µ 1] k ũ j k+1 + µ 1 k p j k. j= j= Applying te triangle inequality and, te following result is obtained.

32 3 Javier de Frutos et al. Teorem 7 Let ũ, p V Q be te solution of 1, wic sould be sufficiently smoot suc tat all norms appearing in te formulation of tis teorem are well defined, and let Ũ n, P n V Q be te Crank Nicolson approximation solving 81 for n = 1,..., N. Let α = 1/T and µ >, assume tat Ũ, q = for all q Q, and tat for some positive constant C it is e + e Ck+1 u k+1, l = k 1, ν 1, and T. Ten, tere exist a positive constant C > independent of ν, µ,, and T suc tat te following error estimate olds for n = 1,..., N, ũt n Ũ n + + µ n ν ũt j + ũt j 1 ũt j + ũt j 1 Ũ j + Ũ j 1 Ũ j + Ũ j 1 C k {1 + µ ũ k+1 + ũt n k + T t ũ L,t n,h k + [ T 1 + µ + µ + µ 1] ũ j k+1 + µ 1 j= +C 4 { T ttt ũ L,t n,l + µ 1 tt p L,t n,l } p j k j= }. 16 Since µ and µ 1 appear on te rigt-and side of 16, one finds again tat µ = O1 is te appropriate asymptotic coice. For estimating te errors in te pressure, denote P n 1/ = e αt n 1/ n 1/ π = e αt n 1/ n 1/ P. Some of te terms tat ave been ignored on and π n 1/ te left-and side in 13 are tose arising from 97 and 98, togeter wit one alf of te sum of tose in 91 and 96. If tey are included in te previous computation, after dividing by 1 + β, one gets also tat α n 41 + β + ρ 4 ν e j e j + ej 1 ej 1 + e j ej 1 + µ e j ej 1 R, 17 were R is te rigt-and side of 15 and ρ is defined in 95. Multiplying 83 by γ n from 9 and using te inf-sup condition 7 yields wit a straigtforward calculation β P n 1/ π n 1/ γ n ẽ n ẽ n 1 + R, 18 1

33 Grad-div stabilization for te evolutionary Oseen problem 33 were R =ν n ẽn γ + ẽ n 1 ẽ n + ẽ n 1 + C b γ n + µ γ n ẽn + ẽ n 1 + γ n r n 1/ γ n r n 1/ Since ẽ n ẽ n 1 V div, te first term on te rigt-and side of 18 can be bounded, applying te same arguments as in 33 and 1, by C A 1/ γ n ẽ n ẽ n 1 /. Taking v = e n en 1 in 83, applying te same estimates tat led to 18, and applying Poincaré s inequality it is easy to sow tat te first term on te rigt-and side of 18 is bounded by CR, so tat P n 1/ π n 1/ β 1 CR. Squaring bot sides of tis inequality, dividing by, and using 89 to express te first tree terms on te rigt-and side of 19 in terms of e n and en 1 gives wit a direct calculation P n 1/ { π n 1/ C 1 + β ν e n + e n 1 + b en + e n 1 + µ e n + e n 1 + e αt n 1/ r n 1/ 13 + e αt n 1/ r n 1/ α ν e n e n 1 + b en e n 1 } +µ e n e n 1, 11 were 1 C as been used. Terms of te same form as on te rigtand side of 11 were estimated in 14, 15, and 17. Applying tese estimates and summing from 1 to n gives P j 1/ π j 1/ = O k + 4. Using te triangle inequality and te estimate for te L projection leads to te following teorem. Teorem 8 Let te assumptions of Teorem 7 old. Ten, tere exists a constant C > suc tat te following bound olds for te approximation to

34 34 Javier de Frutos et al. te pressure computed wit te Crank Nicolson metod 81 j 1/ P pt j 1/ n C k pt j 1/ k µ ũ k+1 + T 1 + µ t ũ L,t n,h k + µ 1 + µ p j k j= + [ T 1 + µ 3 + µ 1 + µ1 + µ ] ũ j k j= { } + C µ T ttt ũ L,t n,l + µ 1 tt p L,t n,l. 5 Numerical Studies Tis section will present a few numerical results wic support te error estimates from Section 4. Comparisons of numerical results wit and witout grad-div stabilization can be already found in te literature, e.g., see [1], and will be omitted ere for te sake of brevity. To tis end, te Oseen problem was considered in Ω =, 1 and in te time interval [, 5] wit different parameters ν and α and wit te prescribed solution sinπx.7 sinπy +. u = cost, cosπx.7 cosπy +. p = costsinx cosy + cos1 1 sin1. Te rigt-and side, te Diriclet boundary condition, and te initial condition were cosen in accordance to te prescribed solution. For all simulations, te Taylor Hood pair of finite elements P /P 1 on uniform grids was used triangles wit diagonals from bottom left to top rigt. Te coarsest grid in space level 3 consisted of 18 mes cells 578 velocity degrees of freedom, 81 pressure degrees of freedom. On tis grid, te lengt of te time step was used. Refining te spatial grid once uniformly, te lengt of te time step was reduced by te factor of four for te backward Euler sceme and te factor of two for BDF and te Crank Nicolson sceme. Wit tis approac, one expects second order convergence for te terms on te leftand side of te error estimates 56, 59, 78, 8, 16, and 111. Results will be presented for =.5 for te backward Euler sceme and =.5 for BDF and te Crank Nicolson metod. Wit tese coices two situations are illustrated: in te results of te backward Euler sceme te spatial error is dominant and in te results of te oter scemes, te temporal error is of more importance. Qualitatively te same results were obtained for all scemes in te respective oter situation. Numerical studies concerning te coice of te grad-div stabilization parameter for te steady-state Oseen equations and te Taylor Hood element

35 Grad-div stabilization for te evolutionary Oseen problem 35 ν =1, α = ν =1 3, α = error error level level ν =1 3, α =1 ν =1 3, α = error error level level ν =1 6, α = 1 - error 1-3 u u 5 L ν 1/ u u L,5;L 1-4 p p L,5;L µ 1/ u u L,5;L level Fig. 1 Backward Euler sceme, error reduction of te errors from estimates 56 and 59, several parameters. Q /Q 1 can be found in []. In all studies, te best coice of tis parameter was below 1, approximately one order. Because of tese results and also based on our own experience, te value of te stabilization parameter was set to be µ =.5 in te simulations. All simulations were performed wit te code MooNMD [18]. Results for te backward Euler sceme are presented in Figure 1. Te second order convergence can be seen for all errors but ν 1/ u u L,5;L in te case ν = 1 6 and α =, were te order of error reduction increases wit increasing level but it is not yet two. Note tat te error bound 56 is for te linear combination of tree errors for te velocity. For small ν, tis combination is dominated by µ 1/ u u L,5;L suc tat tis linear combination converges wit te predicted order. Te independence of te error of te divergence and te pressure on ν can be observed very well.

36 36 Javier de Frutos et al. ν =1, α = ν =1 3, α = error 1-4 error level level ν =1 5, α = 1 - error 1-3 u u 5 L ν 1/ u u L,5;L 1-4 p p L t,5;l µ 1/ u u L,5;L level Fig. BDF, error reduction of te errors from estimates 78 and 8, several parameters. Numerical results for te BDF sceme can be seen in Figure. Te beavior for α > compared wit α = is very similar to te backward Euler sceme suc tat te presentation of te corresponding results is omitted. To illustrate te beavior for small ν, results for ν = 1 5, α = are presented ere. One can observe clearly te reduced order convergence of ν 1/ u u L,5;L on coarser grids and te tendency to become second order on finer grids. All oter errors converge at least of second order already on coarser grids. Again, te pressure error and te error of te divergence are independent of ν. For te results obtained wit te Crank Nicolson sceme, see Figure 3, te same comments apply as for te results computed wit BDF. 6 Conclusions and future researc Tis paper studied te effect of grad-div stabilization added to te Galerkin metod for te transient Oseen equations. Te error analysis was performed for te continuous-in-time case and several fully discrete cases backward Euler metod, BDF formula, Crank Nicolson scemes. Optimal convergence of te L norms of te divergence of te velocity and te pressure were proved for sufficiently smoot solutions wit error constants independent of te viscosity. A cange of variable allowed to transform te original equations into new ones aving a non-vanising reaction term. Tanks to tis cange, te analysis could be performed equally well for dissipative metods, suc as te backward Euler metod and BDF, and for te non-dissipative Crank Nicolson sceme.