H(div) conforming and DG methods for incompressible Euler s equations

Transcription

1 H(div) conforming and DG metods for incompressible Euler s equations Jonny Guzmán Filánder A. Sequeira Ci-Wang Su Abstract H(div) conforming and discontinuous Galerkin (DG) metods are designed for incompressible Euler s equation in two and tree dimension. Error estimates are proved for bot te semi-discrete metod and fully-discrete metod using backward Euler time stepping. Numerical examples exibiting te performance of te metods are given. Key words: Euler equation, discontinuous Galerkin metod 1 Introduction In tis paper we study H(div) conforming and DG finite element metods for te incompressible Euler equations in bot two and tree dimensions. Our metods are based on te velocity-pressure formulation. Let Ω be a bounded and simply connected polygonal domain in R d, d 2,3, wit boundary Γ. Te velocity u H 1 0 (Ω) := [H1 0 (Ω)]d, and te pressure p L 2 0 (Ω) satisfy u t + u u + p = 0 in (0,T) Ω, (1.1a) div(u) = 0 in (0,T) Ω, (1.1b) u n = 0 on (0,T) Γ, (1.1c) u(0,x) = u 0 (x) in Ω, were u t = t u is te time derivative, u is te tensor gradient of u, and T > 0. (1.1d) Te goal of tis paper is to define metods tat are L 2 stable, and, for DG metods, are also locally conservative. Te metods are inspired by te work [7] were tey developed locally conservative DG metods for te steady state Navier-Stokes equations. Tere tey take Newton iterations to solve numerically te equations and in eac step tey postprocess te DG approximation to get a new approximation tat belongs to H(div) and is divergence free. Here we apply tis idea to DG metods in eac time step for Euler s equations. However, we first consider H(div) conforming elements as tey seem natural for incompressible Euler s equations and are easier to analyze. In order to make te H(div) elements L 2 stable, one as to add numerical fluxes of te nonlinear term on te interfaces of Division of Applied Matematics, Brown University, Providence, Rode Island 02912, jonny guzman@brown.edu. Escuela de Matemática, Universidad Nacional de Costa Rica, Heredia, Costa Rica, filander.sequeira@una.cr. Present address: CI 2 MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Cile, fsequeira@ci2ma.udec.cl. Partially supported by te Becas-Cile Programme for foreign students and Centro de Investigación en Ingeniería Matemática (CI 2 MA). Division of Applied Matematics, Brown University, Providence, Rode Island 02912, su@cfm.brown.edu. Researc partially supported by NSF grant DMS and DOE grant DE-FG02-08ER

2 te triangulation. We start wit te semi-discrete metod, using bot central and upwind fluxes, and ten analyze a backward Euler time stepping metod. Once we ave developed H(div) conforming metods, it guides us in developing DG metods using te post-processing idea used in [7]. In [7] upwind fluxes are used, but it is important to note tat central numerical fluxes can also guarantee L 2 stability for Euler s equations. Te development and study of finite element metods for incompressible flows ave a long istory; seeforexampletebooksof Temam [14]andGiraultandRaviart [10]. More recently tereasbeenan interest in using H(div) conforming metods for tese problems [8] since tey produce divergence free approximations. However, to te best of our knowledge, an analysis of tese metods for te inviscid problem (i.e. Euler s equation) as not been considered. On te oter and, tere as been recent work on proving convergence rates for oter finite element metods for problems wit arbitrarily low viscosity [3]. We give an error analysis for bot te semi-discrete metods and te backward Euler time stepping metods. Te error estimate for te velocity in te L 2 norm converges wit rate O( k ) if te velocity space contains te polynomials of degree k. Notice tat tis is sub-optimal by one order. However, numerical experiments suggest tat tese results are not sarp for some polynomial orders and using a central numerical flux. In particular, te error estimate will not give an error estimate for te lowestorder Raviart-Tomas element. However, on structured grids our numerical experiments sow tat te lowest-order Raviart-Tomas elements seem to be converging. Moreover, wen using te upwind numerical flux numerical experiments suggest tat te metod is optimal. However, at te present time we are not able to prove tis result. Our estimates assume tat te velocity belongs to W 1,. Of course, tese a-priori estimates are not known (and migt not old) in tree dimensions for general smoot initial data. However, in two dimensions te a-priori estimates were proved by Kato [11] for smoot initial data. In addition to providing numerical experiments to ceck te order of convergence of our metods, we give numerical experiments to sow ow te metods beave in ig gradient flows. We see tat using upwind flux te metod seems to do very well and comparable to DG metods tat use te vorticity-potential formulation [12]. Te paper is organized as follows. In te next section we present te semi-discrete metods and prove error estimates. In section 3 we present te backward Euler metods. Finally, in section 4 we provide some numerical examples. 2 Semi-discrete metods We begin by introducing some preliminary notations. Let T be a sape-regular and quasi-uniform triangulation of Ω witout te presence of anging nodes, and let E be te set of edges/faces F of T. In addition, we denote by E i and E te set of interior and boundary faces, respectively, of E, and we set T := T : T T. Next, let (, ) U denote te usual L 2 and L 2 := [L 2 ] d inner product over te domain U R d, and similarly let, G be te L 2 and L 2 inner product over te surface G R d 1. Ten, we introduce te inner products: (, ) T := (, ) T and, T :=, T. T T T T On te oter and, let n + and n be te outward unit normal vectors on te boundaries of two neigboring elements T + and T, respectively. We use (τ ±,v ±,q ± ) to denote te traces of (τ,v,q) on F := T + T from te interior of T ±, were τ, v and q are second-order tensorial, vectorial and 2

3 scalar functions, respectively. Ten, we define te means and jumps [[ ]] for F E i, as follows τ := 1 ( τ + +τ ), 2 v := 1 ( v + +v ), 2 q := 1 ( q + +q ), 2 [[τn]] := τ + n + +τ n, [[v n]] := v + n + +v n, [[qn]] := q + n + +q n. Te metod is derived using te conservative or divergence form of te equation. To tis end, denoting as te usual dyadic or tensor product, tat is, (u v) ij = (u t v) ij = u i v j, we consider te formula div(u v) = v u + div(v)u, (2.1) togeter wit te divergence-free condition, to write te problem (1.1) in te form u t + div(u u) + p = 0 in (0,T) Ω, div(u) = 0 in (0,T) Ω, u n = 0 on (0,T) Γ, u(0,x) = u 0 (x) in Ω, (2.2) were div denotes te usual divergence operator div acting along eac row of te corresponding tensor. Finally, given an integer l 0 and a subset U of R d, we denote by P l (U) te space of polynomials defined in U of total degree at most l, wit P l (U) := [P l (U)] d. Furtermore, for eac T T, we define te local Raviart-Tomas space of order l (see, e.g. [2, 13]) were x = ( x1. x d RT l (T) := P l (T) + P l (T)x ) is a generic vector of R d. In addition, we set be te local Nédélec space of order l on T T. ND l (T) := P l (T) + P l (T) x 2.1 H(div) conforming metods In tis section, we define H(div) conforming finite element scemes associated wit te model problem (2.2). We start by introducing te metod using te central flux, but in a later section we present te metod using te upwind flux. For simplicity we only consider te Raviart-Tomas finite element spaces, but we note tat one can use instead te BDM finite elements (see, e.g. [2, 13]). Te globally defined Raviart-Tomas spaces are given by V for te velocity and Q for te pressure, given by V := v H(div;Ω) : v T RT k (T) T T and v n = 0 on Γ, Q := q L 2 0(Ω) : q T P k (T) T T. Now, te finite element metod is defined by: Find (u,p ) V Q, suc tat ( t u,v ) T (u u, v ) T (p,div(v )) T + σ(u,p )n,v T = 0, (q,div(u )) T = 0, (2.3) u (0,x) = u,0 (x) in Ω, for all (v,q ) V Q, were is te broken gradient, u,0 is some projection of u 0 on V, and σ(u,p ) represents te numerical flux of u u+pi on E. In particular, we take σ(u,p ) := u u +p I on E and for Ei we define σ(u,p ) := u u + p I. (2.4) 3

4 Tis is te metod using te central flux. In a later section we introduce te metod using te upwind flux wic seems to do better numerically. Next, usingteabovedefinitionfor σ, togeter witteformula(2.1), tefacttatu isdivergence free (from te second equation in (2.3)), and integration by parts, we can rewrite (2.3) as: Find (u,p ) V Q, suc tat ( t u,v ) T + (u u,v ) T [[(u u )n]], v F (p,div(v )) T = 0, (q,div(u )) T = 0, (2.5) u (0,x) = u,0 (x) in Ω, for all (v,q ) V Q. It will be useful to rewrite te [[(u u )n]] F. Let F = T + T. Ten, [[(u u )n]] = [[(u n)u ]] = (u + n+ )u + + (u n )u. In addition, from te fact tat u + n+ = u n+, since u H(div;Ω), it follows tat [[(u u )n]] = (u + n+ )u + (u n+ )u = (u+ n+ )(u + u ). From now on we will use te notation (witout loss of generality) [[[v]]] := v + v. Also, we use te notation (u n) F = (u + n+ ) F. Hence, we write [[(u u )n]] = (u n)[[[u ]]]. Now from tis we see tat te tird term in te rigt-and side of first equation in (2.5) is consistent, since [[[u]]] = 0 on E i wen u is smoot. Lemma 2.1 (Conservation of energy). Given u V te solution of (2.5), we ave d dt u 2 L 2 (Ω) = 0. Proof. Taking v := u in te first equation of (2.5) and using tat u is divergence free, it follows Tus, note tat (u u,u ) T = d 2dt u 2 L 2 (Ω) + (u u,u ) T (u n)[[[u ]]], u F = 0. (2.6) = 1 2 = T T T T T u ( u 2 ) = 1 2 T(u n) u 2 = 1 2 wic, togeter wit (2.6) complete te proof. F T T F T div(u ) u 2 + u [[ u 2 n]] T (u n) u 2 (u n)[[[u ]]] u, (2.7) We remark ere tat, from te previous lemma, integrating in time over (0,t), we can deduce tat u (t, ) L 2 (Ω) = u,0 L 2 (Ω) for eac t (0,T). Tat is, we proved tat te sceme (2.5) is stable. 4

5 2.1.1 Error estimates Our next goal is to obtain error estimates for te sceme (2.5). In order to do tat, we now introduce teraviart-tomasinterpolationoperator(see[2,13])π k : H1 (Ω) V, wicsatisfiestefollowing approximation properties: for eac v H m (Ω), wit 1 m k +1, tere olds v Π k (v) L 2 (T) + T (v Π k (v)) L 2 (T) C m T v m,t T T. (2.8) Moreover, we also ave te following bounds v Π k (v) L (T) + T (v Π k (v)) L (T) C T v L (T) T T. (2.9) In addition, let P k : L2 (Ω) Q be te L 2 -ortogonal projector. Hence, for eac q H m (Ω), wit 0 m k+1, tere olds (see, e.g. [5]) q P k (q) L 2 (T) C m T q H m (T) T T. (2.10) We now aim to derive te a priori error estimates for te sceme (2.5). To tis end, tanks to te triangle inequality, we only need to provide estimates for te approximation errors, namely, E u := Π k (u) u and E p := P k (p) p. To do tis, we use te fact tat te exact solution satisfies te approximation metod (2.5), in order to obtain te error equations: ( t (u u ),v ) T + (u u u u,v ) T (u n)[[[u u ]]], v F (p p,div(v )) T = 0, (q,div(u u )) T = 0, for all (v,q ) V Q. In addition, from te property div(π k (u)) = Pk (div(u)) = 0, we can rewrite te error equations in te form ( t E u,v ) T + (u u u u,v ) T (u n)[[[e u ]]], v F (E p,div(v )) T = ( t (Π k (u) u),v ) T (u n)[[[π k (u) u]]], v F (P k (p) p,div(v )) T, (2.11) (q,div(e u )) T = 0, for all (v,q ) V Q, were it is important to remark ere tat E u is divergence free. Teorem 2.1. Assume tat u W 1, ([0,T] Ω) d is uniformly bounded. Also, suppose tat T is quasi-uniform. Ten, tere exists C > 0, independent of, suc tat were wit C u := u W 1, ([0,T] Ω). Also, (u u )(T, ) L 2 (Ω) C(u) k B(u), C(u) := (1+C(1+C u ))exp(c(1+c u )T), B(u) := u 0 H k+1 (Ω) + u L 2 (0,T;H k+1 (Ω)) + u t L 2 (0,T;H k+1 (Ω)). 5

6 Proof. We begin by coosing v := E u in (2.11). Tus, we ave 1 d 2dt Eu 2 L 2 (Ω) = (u u u u,e u ) T + (u n)[[[e u ]]], E u F I F E i 1 I 2 + (Π k (u t) u t,e u ) T (u n)[[[π k (u) u]]], Eu F, (2.12) were we ave used te fact tat t Π k (u) = Πk (u t). Next, note tat I 3 I 1 = (u u Π k (u),eu ) T ((u u ) Π k (u),eu ) T (u E u,e u ) T, = (u u Π k (u),eu ) T ((u u ) Π k (u),eu ) T I 2, were in te last term, we apply te same arguments of (2.7) by using E u instead of u in te last two functions. Furtermore, using (2.9) we deduce tat I 1 + I 2 C u Π k (u) u L 2 (Ω) E u L 2 (Ω) + CC u u u L 2 (Ω) E u L 2 (Ω) C u Π k (u) u L 2 (Ω) E u L 2 (Ω) + CC u Π k (u) u L 2 (Ω) + E u L 2 (Ω) Π k (u) u 2 L 2 (Ω) + Π k (u) u 2 L 2 (Ω) CC u E u 2 L 2 (Ω) + CC u E u L 2 (Ω). (2.13) On te oter and, for I 3 it follows I 3 = (E u n)[[[π k (u) u]]], Eu F + (Π k (u) n)[[[πk (u) u]]], Eu F C 1 Π k (u) u L (Ω) F E u 2 + C Π k (u) L (Ω) 1/2 F [[[Πk (u) u]]] 2 1 1/2 F E u 2. (2.14) In addition, given v H 1 (T ) and applying a discrete trace inequality, we observe tat 1 F [[[v]]] 2 2 ( ) v v F 2C tr T T F T Ĉ 2 v 2 L 2 (Ω) + v 2 L 2 (Ω) T T F T 1 F 1 T v 2 L 2 (T) + T v 2 L 2 (T) 1 F v 2, (2.15) and, in te same way togeter wit an inverse inequality we obtain F E u 2 Ĉ Eu 2 L 2 (Ω). (2.16) Hence, replacing (2.15) and (2.16) in (2.14) and using (2.9) we deduce tat I 3 CC u E u 2 L 2 (Ω) + CC u 2 Π k (u) u 2 L 2 (Ω) + Π k (u) u 2 L 2 (Ω). (2.17) 6

7 Now, we return to (2.12), wic satisfies tat 1 d 2dt Eu 2 L 2 (Ω) 1 2 Eu 2 L 2 (Ω) Πk (u t) u t 2 L 2 (Ω) + (I 1 +I 2 ) + I 3, were, replacing (2.13) and (2.17), we obtain tat d dt Eu 2 L 2 (Ω) C(1+C u ) E u 2 L 2 (Ω) + C Πk (u t) u t 2 L 2 (Ω) +CC u 2 Π k (u) u 2 L 2 (Ω) + Π k (u) u 2 L 2 (Ω). (2.18) Hence, applying (2.8) we get d dt Eu 2 L 2 (Ω) C(1+C u ) E u 2 L 2 (Ω) + C(1+C u ) 2k( ) 2 u t 2 H k+1 (Ω) + u 2 H k+1 (Ω). wic, applying te Gronwall s inequality (see, e.g. [9]), yields E u (T, ) 2 L 2 (Ω) exp(c(1+c u )T) E u (0, ) 2 L 2 (Ω) + C(1+C u ) ( 2 u t 2 L 2 (0,T;H k+1 (Ω)) + u 2 L 2 (0,T;H k+1 (Ω)) ). Finally, we use tat E u (0, ) 2 L 2 (Ω) C2(k+1) u 0 2 H k+1 (Ω) to complete te proof. Te next goal is to establis error estimates for te pressure variable. To do tis, we first obtain an estimate for t (u u ), wic is te subject of te next result. Lemma 2.2. Assume te same ypoteses of Teorem 2.1. Ten, tere exists C > 0, independent of, suc tat t E u (T, ) L 2 (Ω) (C(u) k d/2 B(u)+C u ) k 1 C(u)B(u)+ u(t, ) H k+1 (Ω) + C k+1 u t (T, ) H k+1 (Ω). Proof. First, we take v := t E u in (2.11) and using tat div( t E u ) = t div(e u ) = 0, we obtain t E u 2 L 2 (Ω) = (u u u u, t E u ) T + (u n)[[[e u ]]], t E u F + (Π k (u t) u t, t E u ) T (u n)[[[π k (u) u]]], t E u F u u u u L 2 (Ω) t E u L 2 (Ω) + Π k (u t) u t L 2 (Ω) t E u L 2 (Ω) + C u L 1/2 (Ω) 1 F [[[Eu ]]] 2 1/2 F t E u 2 + C u L (Ω) F [[[Πk (u) u]]] 2 1 F E i 1/2 1/2 F t E u 2. Next, using (2.15) and (2.16), we deduce after some algebraic manipulation tat t E u L 2 (Ω) C 1 u L (Ω) E u L 2 (Ω) + u u u u L 2 (Ω) + Π k (u t) u t L 2 (Ω) + u L (Ω)( 1 Π k (u) u L 2 (Ω) + (Π k (u) u) L 2 (Ω)). (2.19) 7

8 To bound te nonlinear term we add and subtract terms to get u u u u L 2 (Ω) = (u u ) u+u (u u ) L 2 (Ω) C u u u L 2 (Ω) + u L (Ω) (u u ) L 2 (Ω) C u u u L 2 (Ω) + u L (Ω)( (u Π k u) L 2 (Ω) +C 1 E u L 2 (Ω)) C u E u L 2 (Ω) +C u Π k (u) u L 2 (Ω) + u L (Ω)( (u Π k u) L 2 (Ω) +C 1 E u L 2 (Ω)), were we ave used an inverse estimate. Terefore, t E u L 2 (Ω) C ( 1 u L (Ω) +C u ) E u L 2 (Ω) + Π k (u t) u t L 2 (Ω) + u L (Ω) (Π k (u) u) L 2 (Ω) + ( 1 u L (Ω) +C u ) Π k (u) u L 2 (Ω). We can bound u L (Ω) using an inverse estimate u L (Ω) E u L (Ω) + Π k (u) L (Ω) C d/2 E u L 2 (Ω) + CC u. (2.20) Hence, t E u (t) L 2 (Ω) C 1 ( d/2 E u L 2 (Ω) +C u ) E u L 2 (Ω) + Π k (u t) u t L 2 (Ω) + ( d/2 E u L 2 (Ω) +C u ) Finally, using Teorem 2.1 and (2.8) establises te result. Note tat in te above proof we ave also proved ( (Π k (u) u) L 2 (Ω) + 1 Π k (u) u L 2 (Ω) (u u u u )(T, ) L 2 (Ω) (C(u) k d/2 B(u)+C u ) k 1 C(u)B(u)+ u(t, ) H k+1 (Ω) ). + C k+1 t u(t, ) H k+1 (Ω). (2.21) We end tis section wit te a-priori error estimate for te pressure, wic is establised next. Teorem 2.2. Assume te ypotesis of Teorem 2.1. Ten, tere exists C > 0, independent of, suc tat (p p )(T, ) L 2 (Ω) (C(u) k d/2 B(u)+C u +C) k 1 C(u)B(u)+ u(t, ) H k+1 (Ω) + C k+1 u t (T, ) H k+1 (Ω) + p(t, ) H k+1 (Ω). Proof. We begin by recalling ere te discrete inf-sup given by β q L 2 (Ω) sup v V v 0 (q,div(v )) T v H(div;Ω) q Q, (2.22) wic, in particular for q := E p, it follows E p L 2 (Ω) 1 β sup v V v 0 (E p,div(v )) T v H(div;Ω). (2.23) 8

9 Now, from te error equation (2.11) and proceeding as in te proof of Lemma 2.2, we ave (E p,div(v )) T = ( t E u,v ) T + (u u u u,v ) T (u n)[[[e u ]]], v F (Π k (u t) u t,v ) T + (u n)[[[π k (u) u]]], v F + (P k (p) p,div(v )) T t E u L 2 (Ω) v L 2 (Ω) + u u u u L 2 (Ω) v L 2 (Ω) + C 1 E u L 2 (Ω) v L 2 (Ω) + C 1 Π k (u) u L 2 (Ω) + (Π k (u) u) L 2 (Ω) v L 2 (Ω) + Π k (u t) u t L 2 (Ω) v L 2 (Ω) + P k (p) p L 2 (Ω) div(v ) L 2 (Ω). Te above result togeter wit (2.23) establises E p L 2 (Ω) C t E u L 2 (Ω) + u u u u L 2 (Ω) + 1 E u L 2 (Ω) + 1 Π k (u) u L 2 (Ω) + (Π k (u) u) L 2 (Ω) + Π k (u t) u t L 2 (Ω) + P k (p) p L 2 (Ω) Terefore, tanks to p p L 2 (Ω) E p L 2 (Ω) + P k (p) p L 2 (Ω), (2.21), Lemma 2.2 and te approximation properties (2.8) and (2.10), we can easily complete te proof. Notice te te error estimate for te pressure predicts O( k 1 ) (for k 2) in two and tree dimensions Using an upwind flux Here, we introduce an alternative version of te conforming metod(2.5), analyzed in previous sections. In order to do tat, we begin by redefining te numerical flux σ (cf. (2.4)) in a new general form, given by: σ(u,p ) := û w u + p I, were û w is a new numerical trace for u related wit te convective term. In particular, taking û w := u = 1 ( ) 2 u int +uext we arrive exactly to te sceme (2.5). Tat is, te metod (2.5) correspond to a central sceme. On te oter and, for some problems wit ig gradients, it is more natural to use an upwind sceme, in order to get better accuracy and order of convergence. In Section 4 we will present some examples of tis. In fact, we see numerically tat using upwind flux gives optimal convergence rates for bot te velocity and pressure variables. According to above, we consider te following upwind flux û w := u int if u n 0, u ext if u n < 0. Tis definition is given in te same way of tat presented in [12] for te vorticity, and it is not difficult to ceck tat we can obtain again te metod (2.5), wit an extra term given by a weigted full jumps 9

10 onto E i. Tat is, we seek u V and p Q, suc tat ( t u,v ) T + (u u,v ) T (u n)[[[u ]]], v F + u n [[[u ]]],[[[v ]]] F (p,div(v )) T = 0 v V, (2.24) (q,div(u )) T = 0 q Q, u (0,x) = u,0 (x) in Ω. It is important to remark ere, tat te introduction of tis new term does not pose any difficulty in order to prove stability and convergence. In fact, bot follow te same arguments, using tat wen v = u tis term is positive. In particular, te error estimates are basically te same and te stability, see remark after te proof of Lemma 2.1, now is given by u (t, ) L 2 (Ω) u,0 L 2 (Ω) for eac t (0,T). 2.2 DG scemes In tis section, we introduce a discontinuous Galerkin metod for te model problem (2.2). Te velocity space will consist of polynomials of degree k + 1 for te fully discontinuous subspace V dg := v L 2 (Ω) : v T P k+1 (T) T T and v n = 0 on Γ, wereas, te pressure space remains uncanged. Tat is, Q := q L 2 0(Ω) : q T P k (T) T T. In te previous section we only defined te jumps and averages on te interior faces/edges. Here we also define tem on boundary faces. Tat is, for F E, as is usual, we set v := v, [[v n]] := v n and q := q. Tus, in order to define te approximation sceme, we first introduce a postprocessed flux. For eac v H 1 (T ), we find v P k+1 (T ) suc tat (v n)q = (v n)q q P k+1 (F), F T, (2.25) F T v p = F T v p p ND k 1 (T), (2.26) for eac T T. Note tat if v V dg ten v BDM 0 k+1 (Ω) were, BDM k+1 (Ω) := v H(div;Ω) : v T P k+1 (T) T T BDM 0 k+1 (Ω) := v BDM k+1(ω) : v n = 0 on Γ. For tis postprocessed flux, we ave te following result. Lemma 2.3. Given T T and v P k+1 (T), tere is exists a constants C > 0, independent of T, suc tat v v L 2 (T) C 1/2 T [[v n]]. F T 10

11 Proof. We proceed as in [6, Lemma 4.2]. Indeed, if we set δ := v v P k+1 (T) we ave tat δ satisfying te equations (δ n)q = (v v ) nq q P k+1 (F), F T, F F δ p = 0 p ND k 1 (T). T Te result togeter wit a scaling argument (see [2]), imply tat δ L 2 (T) C 1/2 T (v v ) n 0, T, wic, using te fact tat (v v ) n = ±[[v n]], we complete te proof. Now, similar as in (2.3), we consider te Galerkin sceme: Find (u,p ) V dg Q, suc tat ( t u,v ) T (u u, v ) T (p,div (v )) T + σ(u,p )n,v T = 0, for all (v,q ) V dg Q, were ( q,u ) T + û n,q T = 0, (2.27) u (0,x) = u,0 (x) in Ω, σ(u,p ) := u u + p I + α 1 F [[u n]]i, (2.28) and α > 0 is stabilization parameter. In addition, we define te numerical flux û as û := u on E. Tus, from te second equation of (2.27) and te definition of u tat (cf. (2.25) and (2.26)), we note 0 = ( q,u ) T + û n,q T = ( q,u ) T + u n,q T = (q,div(u )) T for all q Q. Te above identity and te fact tat div(u ) T P k (T) for eac T T, imply tat u is divergence-free. Tis conclusion and te fact tat u as a continuous normal component are te main reasons tat wile we consider u instead of u in te metod (2.27). Ten, using integration by parts, te fact tat div(u u ) = u u (cf. (2.1)), and te definition of te numerical fluxes, it is not difficult to ceck tat te above DG sceme is as follows: Find u V dg and p Q suc tat ( t u,v ) T + (u u,v ) T + α 1 F [[u n]],[[v n]] F (u n)[[[u ]]], v F (p,div (v )) T + [[v n]], p F = 0, (2.29) (q,div (u )) T [[u n]], q F = 0, u (0,x) = u,0 (x) in Ω, for all (v,q ) V dg Q. It is important to note ere, tat u is not necessarily divergence-free as in te metod of Section 2.1. In addition, unlike te metods in te previous section, te DG metod (2.29) is locally conservative. 11

12 Lemma 2.4 (Stability). Given u V dg te solution of (2.29). Ten, we ave d dt u 2 L 2 (Ω) 0. Proof. We take v := u and q := p in (2.29), and ten we deduce 1 d 2dt u 2 L 2 (Ω) + (u u,u ) T + α 1 F [[u n]] 2 (u n)[[[u ]]], u F = 0. Next, wit tat same arguments of (2.7), we ave (u u,u ) T (u n)[[[u ]]], u F = 0, wic establis tat 1 d 2dt u 2 L 2 (Ω) + α Finally, from te fact tat α > 0, we complete te proof. 1 F [[u n]] 2 = Error estimates for DG metod Now we are ready to provide error estimates for te DG sceme (2.29). We will need to define te BDM/Nédélec projection. ((Π BDM v v) n)q = 0 q P k+1 (F), F T, (2.30) F T (Π BDM (v) v) p = 0 p ND k 1 (T). (2.31) We ave te following approximation results for 1 m k+2. v Π BDM (v) L 2 (T) + T (v Π BDM (v)) L 2 (T) C m T v m,t T T. (2.32) Moreover, we also ave te following bounds v Π BDM (v) L (T) + T (v Π BDM (v)) L (T) C T v L (T) T T. (2.33) Let now E u = Π BDM (u) u and E p = P k(p) p. Ten, we follow (2.11) and consider te error equations: ( t E u,v ) T + (u u u u,v ) T + α 1 F [[Eu n]],[[v n]] F (u n)[[[eu ]]], v F (E p,div (v )) T + [[v n]], E p F = ( t (Π BDM (u) u),v ) T + α 1 F [[(ΠBDM (u) u) n]],[[v n]] F F E i (u n)[[[πbdm (u) u]]], v F (P k (p) p,div (v )) T + [[v n]], P k (p) p F, (2.34) (q,div (E u )) T [[E u n]], q F = 0, 12

13 for all (v,q ) V dg Q. Teorem 2.3. Assume tat u W 1, ([0,T] Ω) d. Also, suppose tat T is quasi-uniform. Ten, tere exists C > 0, independent of, suc tat were wit C u := u W 1, ([0,T] Ω). Also, (u u )(T, ) L 2 (Ω) C(u) k+1 B(u) C(u) := (1+C(1+C u ))exp(c(1+c u )T), B(u) := u 0 H k+2 (Ω) + u L 2 (0,T;H k+2 (Ω)) + u t L 2 (0,T;H k+2 (Ω)) + p L 2 (0,T;H k+1 (Ω)). Proof. We begin by coosing v := E u and q := E p in te error equations (2.34). Ten, we ave tat 1 d 2dt Eu 2 L 2 (Ω) +α + (u n)[[[eu ]]], E u F 1 F [[Eu n]] 2 = (u u u u,e u ) T I 1 + (Π BDM (u t ) u t,e u ) T I 2 + α 1 F [[(ΠBDM (u) u) n]],[[e u n]] F (P k (p) p,div (E u )) T + I 5 (u n)[[[πbdm (u) u]]], E u F I 3 I 4 [[E u n]], P k (p) p F. (2.35) I 6 Next, we want to find bounds for I i, i = 1,...,6. First since div (E u ) is a piecewise polynomial of degree k we ave I 5 = 0. Also, note tat by (2.30) I 3 = 0. Before we bound te rest of te terms. We note tat by Lemma 2.3 and [[Π BDM (u) n]] = 0, we know u u 2 L 2 (Ω) C F [[u n]] 2 = C F [[E u n]] 2 C E u 2 L 2 (Ω). (2.36) F E F E Now we bound I 1, using tat I 1 = (u u Π BDM (u),e u ) T ((u u ) Π BDM (u),e u ) T (u E u,e u ) T, = (u u Π BDM (u),e u ) T ((u u ) Π BDM (u),e u ) T I 2, were in last term, we apply te same argument of (2.7) as in te proof of Teorem 2.1. Furtermore, using tat u L (Ω) C u and Π BDM (u) L (Ω) CC u (see (2.33)), we deduce tat I 1 + I 2 C u Π BDM CC u Π BDM (u) u L 2 (Ω) + Π BDM + CC u u u L 2 (Ω) E u L 2 (Ω) (u) u L 2 (Ω) E u L 2 (Ω) + CC u u u L 2 (Ω) E u L 2 (Ω) (u) u L 2 (Ω) + E u L 2 (Ω) E u L 2 (Ω) CC u E u 2 L 2 (Ω) + CC u Π BDM (u) u 2 L 2 (Ω) + CC u Π BDM (u) u 2 L 2 (Ω), (2.37) were we also used (2.36). 13

14 In te case of I 4, from Π BDM (u) L (Ω) C u, note tat I 4 = (u BDM n)[[[π (u) u]]], E u F = + (u u n)[[[π BDM (u) u]]], E u F (Π BDM (u) n)[[[π BDM (u) u]]], E u F (E u n)[[[π BDM (u) u]]], E u F Π BDM (u) u L (Ω) 1/2 F u u 2 1 F E u 2 1/2 + Π BDM (u) u L (Ω) E u 2 + C u 1 F [[[ΠBDM (u) u]]] 2 1/2 1/2 F E u 2, and from (2.15), (2.16), and (2.36) wit an inverse inequality, we deduce tat I 4 u u ( L 2 (Ω) C 1 Π BDM (u) u L (Ω)) E u L 2 (Ω) + ( C 1 Π BDM (u) u L (Ω)) E u 2 L 2 (Ω) + CC u 2 Π BDM (u) u 2 L 2 (Ω) + Π BDM (u) u 2 L 2 (Ω) + CC u E u 2 L 2 (Ω) C ( 1+ 1 Π BDM (u) u L (Ω)) E u 2 L 2 (Ω) + C 2 Π BDM (u) u 2 L 2 (Ω) + C Π BDM (u) u 2 L 2 (Ω). In addition, applying (2.33), we conclude tat I 4 CC u E u 2 L 2 (Ω) + CC u 2 Π BDM (u) u 2 L 2 (Ω) + Π BDM (u) u 2 L 2 (Ω). (2.38) Now, in similar way to (2.15), given q H 1 (T ) we ave tat F q 2 Ĉ q 2 L 2 (Ω) + 2 q 2 L 2 (Ω), wic allows us to deduce I 6 = F [[E u n]], 1/2 F Pk (p) p F α 2 α 2 F E 1/2 + C F P k (p) p 2 F E 1 F [[Eu n]] 2 F E 1 F [[Eu n]] 2 + C Pk (p) p 2 L 2 (Ω) + C2 P k (p) p 2 L 2 (Ω). (2.39) 14

15 On te oter and, replacing (2.37) (2.39) in (2.35), we obtain tat 1 d 2dt Eu 2 L 2 (Ω) + α 1 F 2 [[Eu n]] 2 CC u E u 2 L 2 (Ω) ΠBDM (u t ) u t 2 L 2 (Ω) F E i + CC u 2 Π BDM (u) u 2 L 2 (Ω) + Π BDM u u 2 L 2 (Ω) + C P k (p) p 2 L 2 (Ω) + C 2 P k (p) p 2 L 2 (Ω). Hence, using (2.32) we ave d dt Eu 2 L 2 (Ω) C(1+C u ) E u 2 L 2 (Ω) + C(1+C u ) 2(k+1) 2 u t 2 H k+2 (Ω) + u 2 H k+2 (Ω) + p H k+1 (Ω) Finally, applying Gronwall s inequality gives te result. Teorem 2.4. Assuming te ypotesis of te previous teorem we ave te existence of a C > 0, independent of, suc tat (p p )(T, ) L 2 (Ω) (C(u) k+1 d/2 B(u)+C u +C) k C(u)B(u)+ u(t, ) H k+2 (Ω) + C k+1 u t (T, ) H k+2 (Ω) + p(t, ) H k+1 (Ω). Proof. Similar to te proof of Teorem Upwind flux for DG metod Similarly as Section 2.1.2, we now introduce a DG metod using an upwind flux. Indeed, as before, we redefine te numerical flux σ (see (2.28)) in te form were we take û w as σ(u,p ) := û w u + p I + α 1 F [[u n]]i, û w := u int if u n 0, u ext if u n < 0. Once again, wit tis definition we can obtain again te metod (2.29), wit an extra consistent term given by u n [[[u ]]], [[[v ]]] F, wic, allow us to prove stability and convergence in te same way of before, using te fact tat wen v = u te above term is positive. Summarizing, we find u V dg and p Q suc tat ( t u,v ) T + (u u,v ) T + α 1 F [[u n]],[[v n]] F (u n)[[[u ]]], v F + u n [[[u ]]], [[[v ]]] F (p,div (v )) T + [[v n]], p F = 0, (2.40) (q,div (u )) T [[u n]], q F = 0, u (0,x) = u,0 (x) in Ω, 15

16 for all (v,q ) V dg Q. 3 Fully-discrete metods In tis section we define fully-discrete versions of bot approaces introduced in Section 2. In order to do tat, for te time discretization we consider te backward Euler metod, tat is, we write u t (t n+1, ) = 1 t u(tn+1, ) u(t n, ) + E 0 (t n+1 ), (3.1) were t > 0 is te time step, t n := n t, 0 n N, and E 0 (t n+1 ) is te truncation error. We know tat E 0 (t n+1 ) L 2 (Ω) C tn+1 t n u tt (s, ) L 2 (Ω)ds. (3.2) For simplicity of te following analysis we denote u n := u(t n, ) for te exact value and u n := u (t n, ) for te approximation. Also, given Π k te corresponding projection used before in eac case, respectively, we define e n u := Πk (un ) u n as te discrete error. Similar convention is used for te pressure variable. On te oter and, using (3.1) we ave tat te exact solution of (1.1) satisfies tat (u n+1,v ) T + t(u n+1 u n+1,v ) T t(p n+1,div(v )) T = (u n,v ) T t(e 0 (t n+1 ),v ) T, or equivalently, (q,div(u n+1 )) T = 0, (u n+1,v ) T + t(u n u n+1,v ) T t(p n+1,div(v )) T = (u n,v ) T (q,div(u n+1 )) T = 0, for all (v,q ) V dg Q. We recall ere tat V V dg. + t((u n u n+1 ) u n+1,v ) T t(e 0 (t n+1 ),v ) T, (3.3) 3.1 H(div) conforming metods Next, using (3.1) in te semi-discrete metod (2.5), we introduce te fully-discrete approximation as: Find (u n+1,p n+1 ) V Q suc tat (u n+1,v ) T + t(u n u n+1,v ) T t (u n n)[[[un+1 ]]], v F t(p n+1,div(v )) T = (u n,v ) T, (3.4) (q,div(u n+1 )) T = 0, for all (v,q ) V Q. Note tat we eliminated te nonlinearity of te problem using te previous approximation. Also, it follows from te proof of Lemma 2.1 tat wen we take v := u n+1 in (3.4), we ave u n+1 2 L 2 (Ω) = (u n,un+1 ) T, wic establis tat u n+1 L 2 (Ω) u n L 2 (Ω), tat is, te metod (3.4) is stable. Our next goal is establis an error estimate for te velocity. 16

17 Teorem 3.1. Assume tat u W 1, ([0,T] Ω) d is uniformly bounded. Also, suppose tat T is quasi-uniform. Ten, tere exists C > 0, independent of, suc tat u n u n L 2 (Ω) C exp(cc u T)( k + t)a(u), for all 0 n N, wit C u := u W 1, ([0,T] Ω). Also, were A(u) := ( T +C u T 3/2 ) u t L 2 (0,T;H k+1 (Ω)) + C u T ut L 2 (0,T;L 2 (Ω)) + T u tt L 2 (0,T;L 2 (Ω)) + (C u T +) u 0 H k+1 (Ω). Proof. We begin by subtracting equation (3.3) from equation (3.4) togeter wit te fact tat [[[u n+1 ]]] = 0 on E i, in order to obtain te error equation (e n+1 u,v ) T + t(u n u n+1 u n u n+1,v ) T t (u n n)[[[en+1 u ]]], v F t(p n+1 p n+1,div(v )) T = (u n u n,v ) T + (Π k (un+1 ) u n+1,v ) T + t((u n u n+1 ) u n+1,v ) T t (u n n)[[[πk (un+1 ) u n+1 ]]], v F t(e 0 (t n+1 ),v ) T. Now, we take v := e n+1 u and using tat div(e n+1 u ) = 0 in Ω, it follows tat u 2 L 2 (Ω) = t(u n u n+1 u n u n+1,e n+1 u ) T + t I 1 e n+1 (u n n)[[[en+1 u ]]], e n+1 u F I 2 + (u n u n,en+1 u ) T + (Π k (un+1 ) u n+1,e n+1 u ) T + t((u n u n+1 ) u n+1,e n+1 u ) T t (u n n)[[[πk (un+1 ) u n+1 ]]], e n+1 u F t(e 0 (t n+1 ),e n+1 u ) T, (3.6) I 3 wic, in similar way to (2.13), we note tat I 1 + I 2 = t(u n u n+1 Π k (un+1 ),e n+1 u ) T t((u n u n ) Π k (un+1 ),e n+1 u ) T t C u Π k (un+1 ) u n+1 L 2 (Ω) +CC u Π k (un ) u n L 2 (Ω) + CC u e n u L 2 (Ω) e n+1 u L 2 (Ω), (3.7) were, we used tat u n L (Ω) C u and Π k (un+1 ) L (Ω) CC u. Also, follows (2.14) and using (2.15), (2.16) and (2.9), we ave 1 I 3 C t 1 Π k (un+1 ) u n+1 L 2 (Ω) F e n u 2 + Π k (un ) L (Ω) CC u t 1 F [[[Πk (un+1 ) u n+1 ]]] 2 e n u L 2 (Ω) + 1 Π k (un+1 ) u n+1 L 2 (Ω) + Π k (un+1 ) u n+1 L 2 (Ω) F e n u 2 (3.5) e n+1 u L 2 (Ω). (3.8)

18 On te oter and, we return to (3.6), and observe e n+1 u 2 L 2 (Ω) e n u L 2 (Ω) + Π k (un+1 u n ) (u n+1 u n ) L 2 (Ω) + C u t u n+1 u n L 2 (Ω) + t E 0 (t n+1 ) L 2 (Ω) e n+1 u L 2 (Ω) + (I 1 +I 2 ) + I 3, wic, replacing (3.7) and (3.8), we deduce tat e n+1 u L 2 (Ω) (1+CC u t) e n u L 2 (Ω) + Π k (un+1 u n ) (u n+1 u n ) L 2 (Ω) + CC u t Π k (un ) u n L 2 (Ω) + 1 Π k (un+1 ) u n+1 L 2 (Ω). + Π k (un+1 ) u n+1 L 2 (Ω) + t C u u n+1 u n L 2 (Ω) + E 0 (t n+1 ) L 2 (Ω). Next, using tat (u n+1 u n )(x) = tn+1 t n u t (s,x)ds, (3.9) togeter wit (2.8), it follows tat Similarly, we can sow and, from (3.2), tn+1 Π k (un+1 u n ) (u n+1 u n ) L 2 (Ω) C k+1 u t (s, ) H k+1 (Ω) ds. In addition, using tat tn+1 tc u u n+1 u n L 2 (Ω) tcc u u t (s, ) L 2 (Ω)ds tn+1 t E 0 (t n+1 ) L 2 (Ω) C t u tt (s, ) L 2 (Ω)ds. t n t n t n u n+1 (x) = u 0 (x) + tn+1 0 u t (s,x)ds, (3.10) and (2.8), we ave 1 Π k (un+1 ) u n+1 L 2 (Ω) C k u 0 H k+1 (Ω) + tn+1 0 u t (s, ) H k+1 (Ω)ds. Analogously, we can sow CC u t Π k (un ) u n L 2 (Ω) + 1 Π k (un+1 ) u n+1 L 2 (Ω) + Π k (un+1 ) u n+1 L 2 (Ω) tn+1 CC u t u k 0 H k+1 (Ω) + u t (s, ) H k+1 (Ω)ds. 0 Terefore, gatering togeter all te above equations, we deduce tat e n+1 u L 2 (Ω) (1+CC u t) e n u L 2 (Ω) + C( t+ k )B(u,n), (3.11) 18

19 were tn+1 tn+1 B(u,n) := u t (s, ) H k+1 (Ω)ds + C u u t (s, ) L 2 (Ω)ds + t n t n tn+1 + tc u u 0 H k+1 (Ω) + u t (s, ) H k+1 (Ω)ds. 0 tn+1 t n u tt (s, ) L 2 (Ω)ds Now, from te recurrence relation (3.11), we obtain tat n 1 e n u L 2 (Ω) (1+CC u t) n e 0 u L 2 (Ω) + C (1+C u t) i B(u,n 1 i) ( k + t) Finally, noting tat i=0 n 1 C(1+CC u t) n ( k + t) u 0 H k+1 (Ω) + B(u,n 1 i) ( = C 1+C C ) n n 1 ut ( k + t) u 0 n H k+1 (Ω) + B(u,n 1 i). n 1 tn tn B(u,n 1 i) u t (s, ) H k+1 (Ω)ds + C u u t (s, ) L 2 (Ω)ds + tn 0 0 i=0 u tt (s, ) L 2 (Ω)ds + C u t n u 0 H k+1 (Ω) + te result now follows by using Caucy-Scwarz inequality. i=0 0 tn 0 i=0 u t (s, ) H k+1 (Ω)ds, Now, we establis te a-priori error estimate for te pressure, and for tat we first consider te next result. Lemma 3.1. Assuming te ypotesis of te previous teorem we ave te existence of a C > 0, independent of, suc tat for all 0 n N u n+1 u n+1 un u n ( CC, t (u)exp(cc u T) k 1 + t ) A(u) t t L 2 (Ω) + C 1+C, t (u) ( k + t)d n (u), were and C, t (u) := exp(cc u T) d/2 ( k + t)a(u) + C u D n (u) := u t L (t n,t n+1 ;H k+1 (Ω)) + u t L (t n,t n+1 ;L 2 (Ω)) + u tt L (t n,t n+1 ;L 2 (Ω)) + u L (t n,t n+1 ;H k+1 (Ω)). Proof. From te error equation (3.5) we ave (δ,v ) T = (u n u n+1 u n u n+1,v ) T + (u n n)[[[un+1 u n+1 ]]], v F + (p n+1 p n+1,div(v )) T + 1 t (Πk (un+1 u n ) (u n+1 u n ),v ) T + ((u n u n+1 ) u n+1,v ) T (E 0 (t n+1 ),v ) T v V, 19

20 were δ := 1 t (en+1 u e n u). Ten, taking v := δ and using tat div(δ ) = 0, we deduce tat δ 2 L 2 (Ω) u n u n+1 u n u n+1 L 2 (Ω) δ L 2 (Ω) + C u n 1/2 L (Ω) 1 F [[[un+1 u n+1 ]]] 2 F δ 2 + Πk ( u n+1 u n ) ( u n+1 u n ) δ t t L 2 (Ω) L 2 (Ω) + C u u n+1 u n L 2 (Ω) δ L 2 (Ω) + E 0 (t n+1 ) L 2 (Ω) δ L 2 (Ω). Now, we follow te proof of Lemma 2.2, to obtain tat u n u n+1 u n u n+1 L 2 (Ω) C u u n u n L 2 (Ω) + C( d/2 e n u L 2 (Ω) +C u ) and from (2.15) and (2.20) we ave u n L (Ω) F [[[un+1 u n+1 ]]] e n+1 u L 2 (Ω) + Π k (un+1 ) u n+1 L 2 (Ω) + 1 Π k (un+1 ) u n+1 L 2 (Ω) + Π k (un+1 ) u n+1 L 2 (Ω) Next, applying (3.12) and (3.13), togeter wit (2.16), it follows tat 1 2 1/2, (3.12) C( d/2 e n u L 2 (Ω) +C u ) 1 u n+1 u n+1 L 2 (Ω). (3.13) δ L 2 (Ω) C u u n u n L 2 (Ω) + C 1 ( d/2 e n u L 2 (Ω) +C u ) u n+1 u n+1 L 2 (Ω) + C( d/2 e n u L 2 (Ω) +C u ) 1 Π k (un+1 ) u n+1 L 2 (Ω) ( + Π k (un+1 ) u n+1 L 2 (Ω) + u n+1 u n ) ( u n+1 u n ) Πk t t + C u u n+1 u n L 2 (Ω) + E 0 (t n+1 ) L 2 (Ω). L 2 (Ω) On te oter and, using te fact tat u n+1 u n+1 un u n t t L 2 (Ω) Πk ( u n+1 u n t ) ( u n+1 u n ) t L 2 (Ω) + δ L 2 (Ω), we ave u n+1 u n+1 t un u n t L 2 (Ω) C u u n u n L 2 (Ω) + C 1 ( d/2 e n u L 2 (Ω) +C u ) u n+1 u n+1 L 2 (Ω) + C( d/2 e n u L 2 (Ω) +C u ) 1 Π k (un+1 ) u n+1 L 2 (Ω) ( + Π k (un+1 ) u n+1 L 2 (Ω) + 2 u n+1 u n ) ( u n+1 u n ) Πk t t L 2 (Ω) + C u u n+1 u n L 2 (Ω) + E 0 (t n+1 ) L 2 (Ω). (3.14) 20

21 Next, we proceed as in te last part of te proof of Teorem 3.1. Indeed, from (2.8), we obtain tat 1 Π k (un+1 ) u n+1 L 2 (Ω) + Π k (un+1 ) u n+1 L 2 (Ω) C k u n+1 H k+1 (Ω) C k u L (t n,t n+1 ;H k+1 (Ω)) Similarly, from (3.9) and (2.8), we ave ( u n+1 u n ) ( u n+1 u n ) Πk t t L 2 (Ω) C k+1 1 t tn+1 t n u t (s, ) H k+1 (Ω)ds C k+1 u t L (t n,t n+1 ;H k+1 (Ω)). In addition, using again (3.9) and (3.2), we deduce, respectively, tat u n+1 u n L 2 (Ω) tn+1 t n u t (s, ) L 2 (Ω)ds t u t L (t n,t n+1 ;L 2 (Ω)), and E 0 (t n+1 ) L 2 (Ω) C tn+1 t n u tt (s, ) L 2 (Ω)ds C t u tt L (t n,t n+1 ;L 2 (Ω)). Te result now follows after applying te previous teorem and te last four estimates into (3.14). Teorem 3.2. Assume te ypotesis of Teorem 3.1. Ten, tere exists C > 0, independent of, suc tat for all 0 n N te following estimate olds ( p n p n L 2 (Ω) CC, t (u)exp(cc u T) k 1 + t ) A(u) + C 1+C, t (u) ( k + t)d n (u), + C k+1 p(t n, ) H k+1 (Ω). Proof. We proceed as in te proof of Teorem 2.2. Indeed, from error equation (3.5), we deduce tat (e n+1 p,div(v )) T = t 1 ((u n+1 u n+1 ) (u n u n ),v ) T + (u n u n+1 u n u n+1,v ) T ((u n u n+1 ) u n+1,v ) T (u n n)[[[un+1 u n+1 ]]], v F + (P k (pn+1 ) p n+1,div(v )) T + (E 0 (t n+1 ),v ) T u n+1 u n+1 un u n v t t L 2 (Ω) + u n u n+1 u n u n+1 L 2 (Ω) v L 2 (Ω) + C u n L (Ω) 1 L 2 (Ω) 1/2 F [[[un+1 u n+1 ]]] 2 F v 2 1/2 + C u u n+1 u n L 2 (Ω) v L 2 (Ω) + P k (pn+1 ) p n+1 L 2 (Ω) div(v ) L 2 (Ω) + E 0 (t n+1 ) L 2 (Ω) v L 2 (Ω). 21

22 Tus, using (3.12), (3.13) and (2.16), we obtain tat (e n+1 p,div(v )) T C u n+1 u n+1 t un u n t L 2 (Ω) + C u u n u n L 2 (Ω) + C 1 ( d/2 e n u L 2 (Ω) +C u ) u n+1 u n+1 L 2 (Ω) + C( d/2 e n u L 2 (Ω) +C u ) 1 Π k (un+1 ) u n+1 L 2 (Ω) + Π k (un+1 ) u n+1 L 2 (Ω) + C u Π k (un+1 ) u n+1 L 2 (Ω) + C u u n+1 u n L 2 (Ω) + P k (pn+1 ) p n+1 L 2 (Ω) + E 0 (t n+1 ) L 2 (Ω) v H(div;Ω), wic, togeter wit te inf-sup condition (2.22), Lemma 3.1, Teorem 3.1, (2.10), and te last estimates obtained in te proof of Lemma 3.1, we can complete te proof. We end tis section by remarking tat we can extend te previous analysis for te upwind version of te metod (cf. (2.24)) given by: Find (u n+1,p n+1 ) V Q suc tat (u n+1,v ) T + t(u n u n+1,v ) T t (u n n)[[[un+1 ]]], v F + t u n n [[[un+1 ]]],[[[v ]]] F t(p n+1,div(v )) T = (u n,v ) T, (3.15) (q,div(u n+1 )) T = 0, for all (v,q ) V Q. 3.2 DG scemes Here we only mention tat wen we combine te tecniques used in sections 2.2 and 3.1 we can also obtain te same error estimates for DG scemes (2.29) and (2.40). Te fully-discrete versions of bot metods, using (3.1), are given by: Find u V dg and p Q suc tat (u n+1,v ) T + t((u )n u n+1,v ) T + α t t 1 F [[un+1 n]],[[v n]] F ((u )n n)[[[u n+1 ]]], v F t(p n+1,div (v )) T + t (q,div (u n+1 )) T [[v n]], p n+1 F = (u n,v ) T, [[u n+1 n]], q F = 0, u (0,x) = u,0 (x) in Ω, (3.16) 22

23 for all (v,q ) V dg Q for te central flux, and: Find u V dg and p Q suc tat (u n+1,v ) T + t((u )n u n+1,v ) T + α t t ((u )n n)[[[u n+1 ]]], v F + t t(p n+1,div (v )) T + t for all (v,q ) V dg Q for te upwind flux. 1 F [[un+1 n]],[[v n]] F (u )n n [[[u n+1 ]]],[[[v ]]] F (q,div (u n+1 )) T [[v n]], p n+1 F = (u n,v ) T, [[u n+1 n]], q F = 0, u (0,x) = u,0 (x) in Ω, (3.17) Teorem 3.3. Assume tat u W 1, ([0,T] Ω) d and p L ([0,T] Ω) are uniformly bounded. Also, suppose tat T is quasi-uniform. Ten, tere exists C > 0, independent of, suc tat u n u n L 2 (Ω) C exp(cc u T)( k+1 + t)a(u,p), for all 0 n N, were C u := u W 1, ([0,T] Ω) and A(u,p) := ( T +C u T 3/2 ) u t L 2 (0,T;H k+2 (Ω)) + C u T ut L 2 (0,T;L 2 (Ω)) + T u tt L 2 (0,T;L 2 (Ω)) + (C u T +) u 0 H k+2 (Ω) + T p L (0,T;H k+1 (Ω)). Proof. It follows straigtforwardly from te proof of Teorems 2.3 and 3.1. Teorem 3.4. Assume te ypotesis of Teorem 3.3. In addition, assume tat te parameter α lies in (0,α 0 t), for some α 0 > 0 independent of. Ten, tere exists C > 0, independent of, suc tat for all 0 n N te following estimate olds ( p n p n L 2 (Ω) CC, t (u,p)exp(cc u T) k + t ) A(u, p) + C 1+C, t (u,p) ( k + t)d n (u,p), + C k+1 p(t n, ) H k+1 (Ω) were and C, t (u,p) := exp(cc u T) d/2 ( k+1 + t)a(u,p) + C u D n (u,p) := 2 u t L (t n,t n+1 ;H k+2 (Ω)) + u t L (t n,t n+1 ;L 2 (Ω)) + u tt L (t n,t n+1 ;L 2 (Ω)) + u L (t n,t n+1 ;H k+2 (Ω)) + p L (t n,t n+1 ;H k+1 (Ω)). Proof. Similar as te proof of Teorem

24 4 Numerical results In tis section, we present some numerical results for two dimensional problem (i.e. d = 2), illustrating te performance of te fully discrete scemes analyzed in Sections 3.1 and 3.2. In all te computations we consider four uniform meses tat are Cartesian refinements of a domain defined in terms of squares, and ten we split eac square into two congruent triangles. Also, we consider polynomial degree k 0,1,2 and for te DG scemes, we use only α = 1. In addition, te numerical results presented below were obtained using a MATLAB code, were te zero integral mean condition for te pressure is imposed via a real Lagrange multiplier. In Example 1 we follow [12] and consider te Taylor-Green vortex (see [4]). Tat is, we set Ω := [0,2π] 2, and te exact solution is given by ( u(t,x) = sin(x 1 )cos(x 2 )e 2t/Re, cos(x 1 )sin(x 2 )e 2t/Re) t, p(t,x) = 1 4( cos(2x1 ) + cos(2x 2 ) ) e 4t/Re, for all x := (x 1,x 2 ) t Ω and t (0,1), were Re = 100 is te Reynolds number. It is easy to ceck tat u is divergence free and Ωp = 0. Here, we compute te approximation of u at t = 1, were we consider t = 1/160 = In Table 4.1 we present te results obtained for Raviart-Tomas scemes (3.4) and (3.15), wereas in Table 4.2 we use DG scemes (3.16) and (3.17). We see tat te estimates we obtained using te Raviart-Tomas spaces and te central flux are sarp for te velocity wen k = 1. However, for k = 0, k = 2 te convergence rates are iger tan predicted teoretically. In particular, we could not prove convergence for k = 0, owever numerically te metod seems to be converging wit order 1. Similarly, for DG metod using te central flux te estimate we gave seem to be sarp for te velocity for k = 0 and k = 2 (notice tat te velocity space contains polynomials of degree k +1 for te DG space), but numerically te case k = 1 does better tan te teory predicts. Finally, using te upwind flux for bot te Raviart-Tomas metod or te DG metod one observes numerically optimal convergence rates for bot te velocity and pressure variables. Unfortunately, we cannot prove tese optimal error estimates. Central flux Upwind flux k d.o.f u u L 2 (Ω) p p L 2 (Ω) u u L 2 (Ω) p p L 2 (Ω) error order error order error order error order e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Table 4.1: History of convergence for Example 1, Raviart-Tomas sceme wit t = 1. For Example 2 we consider te double sear layer problem taken from [1] (see also [12]). We solve te Euler equation (1.1) in te domain Ω := [0,2π] 2 wit a periodic boundary condition and an initial 24

25 Central flux Upwind flux k d.o.f u u L 2 (Ω) p p L 2 (Ω) u u L 2 (Ω) p p L 2 (Ω) error order error order error order error order e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Table 4.2: History of convergence for Example 1, DG sceme wit t = 1. data given by u 0 (x) = (u 0 1 (x),u0 2 (x))t, wit tan((x2 π/2)/ρ) x u 0 2 π 1(x) = tan((3π/2 x 2 )/ρ) x 2 > π, and u0 2(x) = δsin(x 1 ), for all x := (x 1,x 2 ) t Ω, were we take ρ = π/15 and δ = In Figures , we present some contours of te vorticity ω := curl(u ) = x1 u 2 x2 u 1 at t = 6 and t = 8 to sow te resolution. We use 99 contours between 4.9 and 4.9, using te previous four meses, were , , , For tis Figures, we use te DG sceme wit te central flux (cf. (3.16)). Analogously, in Figures we use te DG sceme wit te upwind flux (cf. (3.17)). In all tis figures, we take t = 8/200 = We see tat te metod using te upwind flux seems to do muc better tan te metod using te central flux. In particular, wen using k = 2 and using te upwind flux te metod seems to do quite well. In fact, te metod seems to be comparable to DG metods using te vorticity-potential formulation and ig-order time integrators developed by Liu and Su in [12]. 5 Conclusions and future directions In tis paper we ave developed finite element metods for incompressible Euler equations. We prove error estimates, owever, numerical experiments suggest tat our analysis is not sarp, at least for te upwind metods. It would be interesting to see if a new analysis can prove te optimal estimates for te upwind scemes. Our fully discrete metods are implicit. In te future we would like to consider numerical metods tat treat te nonlinear part explicitly in order to make te metod more efficient. Acknowledgements. Te first two autors would like to tank Gabriel N. Gatica for facilitating teir collaboration, and Antonio Márquez for providing valuable recomendations concerning te computational implementation. 25

26 Figure 4.1: Example 2 (DG + central flux), contours for te vorticity wit k = 0 and t = 6. Figure 4.2: Example 2 (DG + central flux), contours for te vorticity wit k = 1 and t = 6. 26

27 Figure 4.3: Example 2 (DG + central flux), contours for te vorticity wit k = 2 and t = 6. Figure 4.4: Example 2 (DG + central flux), contours for te vorticity wit k = 0 and t = 8. 27

28 Figure 4.5: Example 2 (DG + central flux), contours for te vorticity wit k = 1 and t = 8. Figure 4.6: Example 2 (DG + central flux), contours for te vorticity wit k = 2 and t = 8. 28

29 Figure 4.7: Example 2 (DG + upwind flux), contours for te vorticity wit k = 0 and t = 6. Figure 4.8: Example 2 (DG + upwind flux), contours for te vorticity wit k = 1 and t = 6. 29

30 Figure 4.9: Example 2 (DG + upwind flux), contours for te vorticity wit k = 2 and t = 6. Figure 4.10: Example 2 (DG + upwind flux), contours for te vorticity wit k = 0 and t = 8. 30

31 Figure 4.11: Example 2 (DG + upwind flux), contours for te vorticity wit k = 1 and t = 8. Figure 4.12: Example 2 (DG + upwind flux), contours for te vorticity wit k = 2 and t = 8. 31

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