Stabilized finite element methods for the generalized Oseen problem

Transcription

1 Comput. Metods Appl. Mec. Engrg. 196 (007) Stabilized finite element metods for te generalized Oseen problem M. Braack a, E. Burman b, V. Jon c, G. Lube d, * a Institute of Applied Matematics, University of Heidelberg, Germany b Institut d Analyse et de Calcul Scientifique (CMCS/IACS), Ecole Polytecnique Federale de Lausanne, CH-1005 Lausanne, Switzerland c FR 6.1 Matematics, University of te Saarland, Saarbrücken, Germany d Institute of Numerical and Applied Matematics, University of Göttingen, Germany Received 1 October 005 received in revised form 15 Marc 006 accepted 6 July 006 Abstract e numerical solution of te non-stationary, incompressible Navier Stokes model can be split into linearized auxiliary problems of Oseen type. We present in a unique way different stabilization tecniques of finite element scemes on isotropic meses. First we describe te state-of-te-art for te classical residual-based SUPG/PSPG metod. en we discuss recent symmetric stabilization tecniques wic avoid some drawbacks of te classical metod. ese metods are closely related to te concept of variational multiscale metods wic seems to provide a new approac to large eddy simulation. Finally, we give a critical comparison of tese metods. Ó 006 Elsevier B.V. All rigts reserved. Keywords: Incompressible flow Navier Stokes equations Variational multiscale metods Stabilized finite elements 1. Introduction e motivation of te present paper stems from te finite element simulation of te incompressible Navier Stokes problem o t u mdu þðu rþu þrp ¼ ~ f ru ¼ 0 for te velocity u and te pressure p in a polyedral domain R d, d 6 3, wit a given source term ~ f. A standard algoritmic treatment of (1) and () is to semidiscretize in time (wit possible step lengt control) using an A-stable metod and to apply a fixed point or Newton-type iteration per time step. is leads to te following auxiliary problem of Oseen type in eac step of tis iteration: L Os ðb u pþ :¼ mdu þðb rþu þ cu þrp ¼ f in ð3þ ru ¼ 0 in : ð4þ * Corresponding autor. addresses: braack@iwr.uni-eidelberg.de (M. Braack), erik. burman@epfl.c (E. Burman), jon@mat.uni-sb.de (V. Jon), lube@ mat.uni-goettingen.de (G. Lube). ð1þ ðþ Also te iterative solution of te steady state Navier Stokes equations using a fixed point iteration leads to problems of type (3) and (4) wit c =0. e standard Galerkin finite element metod (FEM) for (3) and (4) may suffer from two problems: dominating advection (and reaction) in te case of 0 < m kbk L 1 ðþ, violation of te discrete inf sup (or Babuška Brezzi) stability condition for te velocity and pressure approximations. e well-known streamline upwind/petrov Galerkin (SUPG) metod, introduced in [5], and te pressure-stabilization/petrov Galerkin (PSPG) metod, introduced in [31,6], opened te possibility to treat bot problems in a unique framework using rater arbitrary FE approximations of velocity and pressure, including equal-order pairs. Additionally to te Galerkin part, te elementwise residual L Os (bu,p) f is tested against te (weigted) non-symmetric part (b Æ $)v + $q of L Os (bv,q). Moreover, it was sown in [18,3,40] tat an additional element-wise stabilization of te divergence constraint (4), encefort denoted /$ - see front matter Ó 006 Elsevier B.V. All rigts reserved. doi: /j.cma

2 854 M. Braack et al. / Comput. Metods Appl. Mec. Engrg. 196 (007) as grad div stabilization, is important for te robustness if 0 < m 1. Due to its construction, we will classify te SUPG/PSPG/grad div approac as an (element-wise) residual-based stabilization tecnique. Despite te success of tis classical stabilization approac to incompressible flows over te last 0 years, one can find in recent papers a critical evaluation of tis approac, see e.g. [0,1]. Drawbacks are basically due to te strong coupling between velocity and pressure in te stabilizing terms. (For a more detailed discussion, cf. Section 7.) Several attempts ave been made to relax te strong coupling of velocity and pressure and to introduce symmetric versions of te stabilization terms: Recently, te interior penalty tecnique of te discontinuous Galerkin (DG) metod was applied in te framework of continuous approximation spaces as proposed in [17] leading to te edge/face oriented stabilization introduced in [1]. It can be classified as well as a residual-based stabilization tecnique since it controls te inter-element jumps of te non-symmetric terms in (3) and (4). Anoter approac consists in projection-based stabilization tecniques. e first step was done in [16] were weigted global ortogonal projections of te non-symmetric terms in (3) and (4) are added to te Galerkin sceme. A related local projection tecnique as been applied to te Oseen problem in [3] wit low-order equal-order interpolation. Anoter projection-based stabilization was introduced in [3,9]. e projection-based metods are closely related to te framework of variational multiscale metods introduced in [5]. e latter metod provides a new approac to large eddy simulation (LES) of incompressible flows wic does not possess important drawbacks of te classical LES like commutation errors. e goal of te present paper is a unique presentation of residual-based and projection-based stabilization tecniques to te numerical solution of te Oseen problem (3) and (4), togeter wit a critical comparison. For brevity, we consider only conforming FEM. An extension to a non-conforming approac like DG-metods in an element- or patc-wise version can be found, e.g., in [14,1]. e latter metods are not robust wit respect to te viscosity m. An overview of appropriate stabilization mecanisms in te DG framework was given in [4]. e paper is organized as follows: In Section, we describe te basic Galerkin discretization of te Oseen problem. en, we consider residual-based stabilization metods including te classical SUPG/PSPG/grad div stabilization following [36], see Section 3, and te edge/facestabilization metod following [1,13], see Section 4. Next, we present projection-based stabilization tecniques. Here, we review te local projection approac proposed in [3], see Section 5, and anoter projection-based stabilized sceme due to [3,9], see Section 6. A critical comparison of te scemes can be found in Section 7.. e standard Galerkin FEM for te Oseen problem rougout tis paper, we will use standard notations for Lebesgue and Sobolev spaces. e L -inner product in a domain x is denoted by (Æ,Æ) x. Witout index, te L - inner product in is meant. is section describes te standard Galerkin FEM for te Oseen-type problem (3) and (4), for simplicity of presentation wit omogeneous Diriclet data: L Os ðb u pþ :¼ mdu þðb rþu þ cu þrp ¼ f in ð5þ ru ¼ 0 in ð6þ u ¼ 0 on o ð7þ wit b [H 1 () \ L 1 ()] d, m, c L 1 (), f [L ()] d and m > 0 ðr bþðxþ ¼0 cðxþ P c min P 0 a:e: in : ð8þ Let H 1 0 ðþ :¼ fv H 1 ðþ jvj o ¼ 0g and L 0ðÞ :¼ fq L ðþj R q dx ¼ 0g. e variational formulation reads: find U ¼fu pg V Q :¼ ½H 1 0 ðþšd L 0ðÞ s.t. Aðb U V Þ¼LðV Þ 8V ¼fv qg V Q ð9þ wit AðbUV Þ¼ðmrurvÞþððb rþu þ cuvþþbðvpþ bðuqþ ð10þ LðV Þ¼ðf vþ ð11þ bðvpþ¼ ðprvþ: ð1þ Suppose an admissible triangulation of te polyedral domain. We assume tat is sape-regular, i.e., tere exists a constant C s, independent of te messize wit =, suc tat C s d 6 measð Þ for all S. In particular, we exclude anisotropic elements trougout te paper. Moreover, we assume tat eac element is a smoot bijective image of a given reference element ^, i.e., ¼ F ð^ Þ for all. Here, ^ is te (open) unit simplex or te (open) unit ypercube in R d. For p N, we denote by P p ð^ Þ te set f^x a : 0 6 a i 0 6 P d i¼1 a i 6 pg on a simplex ^ or f^x a : 0 6 a i 6 k 1 6 i 6 pg on te unit ypercube ^ and define p ¼fv Cð Þjvj F P p ð^ Þ8 g: ð13þ We introduce conforming FE spaces on for velocity and pressure, respectively, by V r :¼½H 1 0 ðþ\ r Šd Q s :¼ L 0 ðþ\ s ð14þ wit r, s N and we set W rs :¼ V r Qs. Clearly, oter conforming discrete spaces for te velocity and te pressure can be cosen (e.g., enriced wit bubble functions). Moreover, for brevity, we will not present possible extensions to non-conforming metods. A key point in te analysis of some metods is local inverse inequalities on wit a constant l inv depending only on te sape-regularity

3 kr wk L ð Þ 6 p ffiffiffi d krwk½l ð ÞŠ dd 6 l inv r 1 kwk ½L ð ÞŠ d 8w Vr : ð15þ For simplicity, we assume tat te solution U W:¼V Q of (9) is smoot enoug suc tat fi u r u I p s pgwrs can be cosen as te global Lagrange interpolants of {u, p}. More precisely, we want to apply te local interpolation result M. Braack et al. / Comput. Metods Appl. Mec. Engrg. 196 (007) Stability of te metod Stability of te stabilized metod (19) (1) wit d ¼ d u ¼ dp is proved w.r.t. jjjv jjj rbs :¼ j½v Šj 1 rbs þ rkqk L ðþ ðþ j½v Šj rbs :¼km1 rvk L ðþ þkc1 vk L ðþ þ J rbsðv V Þ ð3þ kv I r vk H m ð Þ 6 C l m I r kvk k m H k ð Þ 0 6 m 6 l ¼ minðr þ 1 kþ ð16þ for te Lagrange interpolation I r v of v Hk () wit k > d, [4], Section 4. Here C I is a constant independent of, r, v, but dependent on m, k, C s. e standard Galerkin FEM of (9) reads as follows: find U ¼fu p gw rs, s.t. Aðb U V Þ¼LðV Þ 8V ¼fv q gw rs : ð17þ Well-known sources of instabilities stem from te case of dominating advection, kbk L 1 ðþ m, and from te violation of te discrete inf sup condition for V r Qs ðq 9b 0 > 0 : inf sup rv Þ q Q s v V r krv k ½L ðþš kq P b 0 ð18þ k dd L ðþ were b 0 can be cosen independent of. is is te case, e.g., for equal-order velocity pressure finite element spaces. Note tat te discrete inf sup constant b 0 depends in general on r and s. 3. Classical residual-based stabilization metods e classical stabilization of te Galerkin sceme is a combination of pressure stabilization (PSPG) and streamline-upwind stabilization for advection (SUPG) togeter wit a stabilization of te divergence constraint: find U ¼fu p gw rs, s.t. A rbs ðb U V Þ¼L rbs ðv Þ wit 8V ¼fv q gw rs A rbs ðb U V Þ :¼ Aðb U V Þþ c ðr u rvþ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} grad div stabilization ð19þ þ ðl Os ðb u pþ d u ðb rþv þ dp rqþ ð0þ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} SUPG=PSPG stabilization zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ L rbs ðv Þ :¼ LðV Þþ ðf d u ðb rþv þ dp rqþ ð1þ containing te tree parameter sets fd u g fdp g and {c } depending on te coice of te FE spaces, see below. e metod simultaneously stabilizes spurious oscillations coming from dominating advection and te violation of te discrete inf sup condition (18). For details and full proofs of te following presentation, we refer to [36]. J rbs ðv V Þ :¼ d kðb rþv þrqk L ð Þ þ c kr vk L ð Þ ð4þ wit parameters d, c, r > 0 to be determined. A simplified analysis is possible since [Æ] rbs is a mes-dependent norm on W rs if d > 0. Assume tat ( ) 0 < d 6 1 min l inv r4 m c kck : ð5þ L 1 ð Þ e inverse inequalities (15) and (5) imply tat te bilinear form A rbs ðb Þ defined in (0) satisfies A rbs ðb V V Þ P 1 j½v Šj rbs 8V W rs : ð6þ e coercivity estimate (6) yields uniqueness and existence of te discrete solution, owever it provides no control of te L -norm of te pressure. Assume now additionally ( ) 0 < l 0 r 6 d 6 1 min l inv r4 m 1 kck L 1 ð Þ 0 6 d kbk L 1 ð Þ 6 c ð7þ wit some positive constant l 0. aking advantage of Verfürt s trick, cf. [19,41], we can sow tat tere exists a constant b > 0, independent of m, and te spectral orders r and s, suc tat te bilinear form A rbs ðb Þ in (0) satisfies inf U W rs sup V W rs A rbs ðb U V Þ jjju jjj rbs jjjv jjj rbs P b ð8þ wit te weigt 0 1 pffiffiffi Bpffiffi 1 r c þ þ pffiffi m þkck 1 L l 1ðÞCF þ C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fkbk L 1ðÞ kbk L q þ max 1 ð pffiffi A 0 m þ c min C m F ð9þ of te L -norm of te pressure in (). Moreover, it denotes c ¼ max c and C F te Friedrics constant. Note tat r is only used for te analysis. Remark. e lower bound of d in (7) can be removed in case of div-stable velocity pressure interpolations. But ten one as to replace te constant b in (8) by te inf sup constant b 0 = b 0 (r,s) from (18). 3.. A priori error estimates e following continuity result is derived using standard inequalities. It reflects te effect of stabilization wit 1

4 856 M. Braack et al. / Comput. Metods Appl. Mec. Engrg. 196 (007) assumption (5): For eac U W wit Duj ½L ð ÞŠ d 8 and V W rs tere olds A rbs ðb U V ÞQ rbs ðuþjjjv jjj rbs ð30þ wit Q rbs ðuþ :¼ j½ušj rbs þ þ!1 1 kuk L d ð Þ 1 maxðm c Þ kpk L ð Þ þ!1 d k mdu þ cuk L ð Þ : ð31þ e L -terms in (31) explode for m, c! 0ifd = c =0. Consider solutions U W and U W rs of te continuous and of te discrete problem, respectively. Let fi u r u I p s pgwrs be an appropriate interpolant of U, e.g., te Lagrange interpolant. en we obtain te quasioptimal a priori estimate of sceme (19) (1): jjju U jjj rbs Q rbs fu I u r u p I p s pg: ð3þ Now we ave to fix te stabilization parameters d, c using (3) and te local interpolation inequalities (16). Let te assumptions (8) and (7) be valid. en, we obtain jjju U jjj rbs M u ðlu 1Þ!1 ðlp 1Þ r ðku 1Þ kuk H ku ð Þ þm p s ðkp 1Þ kpk H kp ð Þ ð33þ wit l p :¼ min{s +1,k p }, l u :¼ min{r +1,k u }and! M u ¼ kck L þ d 1 ð Þ r þkbk d r L 1 ð Þ þ r m þ c þ m þ kck L 1 ð Þ r r4 m M p ¼ d þ s maxðm c Þ : First, we consider te case of equal-order interpolation of velocity and pressure, i.e., r ¼ s N. Suc pairs do not fulfill te discrete inf sup condition (18). e equilibration of te d - and c -dependent terms in M u and M p togeter wit te stability conditions (5) and (7) yields d þ rkbk! 1 L 1 ð Þ þkck L 1 ð Þ c : ð34þ r d en, a sufficiently smoot solution U of (9) wit U [H k ()] d H k () for eac, obeys te error estimate (wit l = min(r +1,k)) jjju U jjj rbs M ¼ mr þ kbk L 1 ð Þ r ðl 1Þ r M ðk 1Þ kuk H k ð Þ þkpk H k ð Þ þ kck L 1 ð Þ r : ð35þ! Remark. e estimate (35) is optimal wit respect to. Unfortunately, it is suboptimal in te spectral order r in a transition region between te diffusion-dominated and te advection-dominated limits. is is caused by te term r4 m in (34) in order to fulfill te stability conditions (7). Itis possible to refine te coefficient in front of te L -term of u on te rigt-and side of (31), tus giving an optimal estimate w.r.t. r at least in te diffusion-dominated limit, see [36]. Next, we consider interpolation pairs V r Qs wit r = s + 1. (An extension to r P s + 1 is straigtforward.) is includes te div-stable aylor Hood pairs wit s ¼ r 1 N on a sape-regular mes. A balance of te c - and d -dependent terms in M u and M p yields d r ðm þ cþ c m þ c ð36þ r d wit c 1. In tis case, a sufficiently smoot solution U of te Oseen problem (9) wit U [H k+1 ()] d H k () for eac obeys te error estimate jjju U jjj rbs l ðm þ cþkuk r k H kþ1 ð Þ þ 1 m þ c kpk H k ð Þ ð37þ wit l = min(r +1,k), provided tat is sufficiently small. e estimate (37) is optimal w.r.t. bot and r. e coice (36) reflects te importance of te grad div stabilization term and a decreasing influence of te SUPG/PSPG term wit increasing spectral order r Variants of te metod Oter variants containing te SUPG-/PSPG-stabilization wit d ¼ d u ¼ dp are te Galerkin/least-squares (GLS) metod [18] and te Douglas/Wang- or algebraic subgrid-scale (ASGS) metod [16] adding ðl Os ðb UÞ f d L Os ðb V ÞÞ and ðl Os ðb UÞ f d L Os ðb V ÞÞ respectively, to te Galerkin formulation (17). L Os denotes te adjoint operator of L Os. e analysis of tese metods is similar to te SUPG/PSPG/grad div sceme using te stabilizing effect of te term J s (Æ,Æ) defined in (4). For div-stable interpolation pairs, a reduced stabilized sceme by omitting te PSPG terms P ðl Os ðb u pþ f d rqþ from sceme (19) (1) is analyzed in [0]. Practical calculations surprisingly sow tat te scemes wit and witout PSPG give almost identical results. e grad div stabilization is always necessary for 0 < m 1, wereas te SUPG stabilization is useful for problems wit

5 M. Braack et al. / Comput. Metods Appl. Mec. Engrg. 196 (007) layers. Moreover, an order reduction of 1 was observed by using instead of te parameter coice (36) te design (34) for equal-order interpolations Implementation issues e system matrices of te Galerkin FEM and te SUPG/PSPG stabilized sceme ave te form A B A s B 1 and B 0 B C respectively, wit B 1 5 B, C 5 0. e blocks A and A s as well as B, B 1 and B ave a similar sparsity pattern. us, te SUPG/PSPG metod can be easily incorporated into an existing code for solving te Galerkin FEM. One as to store one additional off-diagonal block and te additional sparse matrix C for te pressure couplings arising from te term P ðrp d p rqþ in te stabilization. Note tat for te reduced stabilized sceme from [0] it olds B = B and C =0. One drawback of te SUPG/PSPG sceme consists in needing to evaluate second order derivatives of te velocity if r P. However, tese derivatives are multiplied wit te small factor m suc tat teir omission is an option in practical computations Coupled vs. decoupled stabilization Several drawbacks of te classical stabilization metods presented so far stem from te strong velocity pressure coupling in te stabilization terms, see te discussion in Section 7. In Sections 4 6, we will consider tecniques wit decoupled stabilization terms. Let us take as a starting point te stabilization terms of Eq. (0) ðl Os ðb u pþ d u ðb rþv þ dp rqþ : e subgrid viscosity concept in te sense of Guermond [] leads to te idea tat te stabilization of te residual does not ave to act on te wole residual but only on its projection into some appropriate subspace. We introduce an abstract projection operator (I P) and te modified stabilization term ðði PÞL Os ðb u pþ ði PÞðd u ðb rþv þ dp rqþþ ð38þ (wit a similar modification of te rigt-and side of te equation). aking now v = u, q = p and (for simplicity) d ¼ d u ¼ dp, we deduce tat (38) becomes n ðði PÞð mdu þ cuþ d ði PÞððb rþu þrpþþ þkðd Þ 1 ði PÞððb rþu þrpþk L ð Þ o : Clearly, te first part of tis sum is necessary for consistency and te last part gives te positivity. If te projection operator (I P) is cosen in suc a way tat te first part vanises sufficiently fast as! 0, ten te consistency part could be dropped witout spoiling te rate of convergence. In tis case, we may also drop te Petrov Galerkin type modification of te rigt-and side. Moreover, te positive part may be split into two in order to decouple velocities and pressure. Introducing separate stabilization terms for pressure and velocity does not cange te consistency properties of te sceme since te weak consistency is given by te approximation properties of te projection and not by te residual. en, (38) is transformed to te decoupled and symmetric form ðd u ði PÞððb rþuþ ði PÞððb rþvþþ þðd p ði PÞrp ði PÞrqÞ : A similar argument can be applied to te grad div stabilization term. Coosing te subspaces and te projection operators in a specific way, we obtain te stabilization tecniques proposed in Sections 4 6 were we will use for te stabilized bilinear form te unified notation Aðb U V ÞþS ðb U V Þ: 4. Face oriented stabilization metod e face oriented stabilization metod (or edge oriented for d = ) takes its origin in te paper [17] on interior penalty procedures for elliptic and parabolic problems. e idea was to increase te robustness of te Galerkin approximation of elliptic problems (using continuous approximation spaces) by introducing additional least squares control of te gradient jump over element boundaries. is metod was revived more tan 0 years later in [8]. For te advection diffusion problem, it was sown tat te added penalty term yields a metod tat is stable independent of te local Peclet number Pe :¼ kbk L 1 ð Þ m and allows optimal a priori error bounds uniform in Pe. e metod was ten extended to te generalized Stokes problem in [9] and to te Oseen equation wit arbitrary polynomial degree in [1]. Oter work on te face oriented stabilization includes te papers [6] were a discrete maximum principle is rigorously proved for a face oriented sock capturing sceme and [11] were te metod is extended to ig order polynomial approximations in a p-framework. Since te stabilization is based on te faces of te elements, we introduce te set of all interior faces of te mes E, we denote te jump of te quantity x over some face e by [x] e (te orientation of te jump is arbitrary, but fixed). e jump is extended to vector valued functions componentwise. For eac face we set a fix (but arbitrary) orientation of te normal vector n e. We let e denote te diameter of te face e and = max e e te messize of te element. Moreover, we assume tat te mes is locally quasi

6 858 M. Braack et al. / Comput. Metods Appl. Mec. Engrg. 196 (007) uniform in te sense tat for any two elements 0 aving at least one common node tere olds 6 q 0 were q P 1 is a parameter depending on te mes regularity. e formulation takes te form, find U W rr, suc tat A fos ðb U V Þ¼LðV Þ 8V ¼fv q gw rr ð39þ were A fos ðb U V Þ :¼ Aðb U V ÞþS fos ðb U V Þ S fos ðb U V Þ :¼ Z c u e ðb eþ½ru n e Š e ½rv n e Š e ds ee e Z ð40þ þ c e ðb e Þ½r u Š e ½r v Š e ds ð41þ Ze þ c p e ðb m eþ½rp n e Š e ½rq n e Š e ds : e ð4þ e numerical analysis sows tat te tree parameter sets in te stabilization term sould be cosen as c u e ðb Þ :¼ kb n e ek L 1 ðeþ r a e c e ðb Þ :¼ kbk L 1 ðeþ r a c p e ðb m Þ :¼ minð1 Re e eþ kbk L 1 ðeþ ra wit Re e :¼ kbk L 1 ðeþ e and a :¼ 7. mr Stability for face oriented stabilization e stability of te metod is obtained by te key observation tat te operator controlling te jump of te gradient actually controls te part of te gradient of te discrete approximation tat is ortogonal to te finite element space. anks to tis observation, one may obtain te crucial control of te streamline derivative and te pressure gradient independently. o prove stability we need te following interpolation result from discontinuous to continuous spaces. ere exists an interpolation operator p disc : ½W r Šd! ½W c r Šd, were W disc r denotes te space of discontinuous functions being piecewise polynomials of order r on eac element and W c disc r ¼fv W r : v C0 g, and constants c 0, c 1 depending on te local mes geometry and te polynomial degree, but not on te local mes size, suc tat c 0 Jðv v Þk 1 ðrv p rv Þk L ðþ c 1Jðv v Þ for all v r, were Jðv v Þ¼ Z e j½rv Š e j ds: ee e en one uses te fact tat [$p ] e = [$p Æ n e ] e for continuous finite element spaces. Well-posedness of te discrete problem is assured tanks to te following discrete inf sup condition. Independently of m and tere olds A fos ðb U V Þ jjju jjj fos sup 06¼V W r jjjv jjj fos for all U W r were te triple norm is defined by jjjv jjj fos :¼j½V Šj fos þ rkp k L ðþ wit ð43þ j½v Šj fos :¼km1 rv k L ðþ þkc1 v k L ðþ þ S fosðb V V Þ ð44þ and r similar to (9). Note tat (43) defines a norm for bot velocity and pressure wereas (44) only is a seminorm on te product space. 4.. A priori error estimates for face oriented stabilization e use of coercivity in te seminorm (44), Galerkin ortogonality, continuity and finally approximation leads to te following a priori error estimate ðlu 1Þ j½u U Šj fos M u kuk H r ku ðþ ðlp 1Þ þ M p kpk H r kp ðþ ð45þ were l p :¼ min{r +1,k p }, l u :¼ min{r +1,k u } and te constants are given by ( ) M u ¼c r þmax min r 1 kbk L 1 ð Þ þ r kbk W 11 ð Þ m Re c r 1 þm and M p ¼ max min kbk 1 r 1 L 1 ð Þ : mr 3 Here, we denote by Re :¼ kbk L 1 ð Þ and assume additionally tat b [W 1,1 ()] d. e convergence of te pres- mr 1 sure in te L -norm may ten be estimated leading to kp p k L ðþ lp kpk H r kp ðþ þ r 1 rj½u U Šj fos : ð46þ e above dependencies on te polynomial order may so far be proven rigorously only on quasi-uniform tensor product meses. Note te sligt suboptimality of te pestimates in te ig Reynolds number regime. In te case < r 5, te estimate in te triple norm is suboptimal by a power of r 1 4 and for te estimate of te pressure in te L - norm we get an additional factor of r 1 due to te use of H 1 - stability of te L -projection. In te low Reynolds number regime, te triple norm estimate is optimal, but te suboptimality of te pressure remains. For details on te p-analysis for face oriented stabilization, see [11] Variants of te face oriented stabilization metod In (40), one may take te jump in te streamline derivative only, instead of building te streamline diffusion

7 M. Braack et al. / Comput. Metods Appl. Mec. Engrg. 196 (007) caracter into te stabilization parameter c u e in te form of te factor kb n e k L 1 ðeþ. Following [7], note tat [b Æ $u ] e =(b Æ n e ) [$u n e ] e. Anoter possibility is not to empasize te streamline direction in te stabilization. In fact, if c e is used as stabilizing parameter in (40), ten te divergence stabilization (41) may be omitted, at te expense of a possibly larger constant in te a priori error estimate. It is noteworty tat suc a modification, introducing crosswind diffusion, will ave no influence on te order. Note also tat ere only te pressure stabilization is depending on te viscosity m. is can be understood as using te local Reynolds number as non-dimensional weigt for a iger order viscosity term in te velocity stabilization. An equivalent term, controlling bot instabilities due te convective terms and te divergence free constraint (ence replacing (40) and (41)), is Z e Re e r ½mru n 3 e Š e ½rv n e Š e ds: ð47þ ee e Remark. o simplify te analysis te boundary conditions are imposed weakly. Weak imposition of te boundary conditions was considered in [1] using an approac due to [37]. Details ave been omitted above for brevity. Remark. For velocity/pressure pairs of aylor Hood type satisfying te discrete inf sup condition, te coice c p e ¼ 0 is allowed. However, in tis case it is recommended to replace te jump stabilization (41) of te divergence by a term c ($ Æ u,$ Æ v ) wit c = 1, compare Section 3.. It is unclear weter te use of te term (47) is sufficient to stabilize bot advection and incompressibility in suc a case also. For details on finite element metods wit velocity pressure pairs satisfying te discrete inf sup condition, see [9,13] Implementation issues e implementation of face oriented stabilization tecniques requires an additional nearest neigbor data structure, or a table giving te two elements associated to eac face. Suc structures are necessary also for a posteriori error estimation and ence for adaptive finite element codes. e task of implementing te gradient jumps only requires te addition of te stabilizing matrix. An efficient implementation will use te symmetry of te matrix and moreover consider eac face only once. On te oter and in case te streamline diffusion caracter is abandoned bot velocities and pressures are stabilized using isotropic gradient jumps. In tis case te stabilization matrix for bot velocities and pressures may be set up (on a fixed mes) as a preprocessing step. At eac time step te stabilization matrix for velocities or pressures are constructed from tis precomputed matrix simply by multiplying te indices wit te appropriate weigts accounting for varying b. is may allow to diminis te computational cost compared to te residual based stabilization were for consistency reasons te wole matrix as to be recomputed at eac time step. 5. Local projection-based stabilization metod e local projection-based stabilization (LPS) is designed for equal-order interpolation of pressure and velocities, i.e. r = s, and te stabilization of convective terms. For te formulation of te local projection, we restrict ourselves to a certain class of meses. We assume tat te mes results from a coarser mes by one global refinement. Hence, te mes consists of patces of elements for instance in two dimensions, tree triangles can be grouped togeter in order to form one triangle of. is restriction can be omitted for a certain variant of te local projection discussed in Section 5.3. As furter notation, we introduce te space of patc-wise discontinuous finite elements of degree r 1: r 1 :¼ fv L ðþjvj F P r 1 ð^ Þ8 g: ð48þ We introduce te L -projection p r 1 : r! r 1, caracterized by ð/ p r 1 / wþ ¼0 8/ r 8w r 1 and te fluctuation operator wit respect to p r 1 by, :¼ I p r 1 were I stands for te identity mapping. For te Stokes system, it was proposed in [1] to account for te violation of te inf sup condition (18) by adding te stabilization term ð, rp d p, rqþ to te Galerkin formulation. Similar to te PSPG metod, te parameter d p depends on te local mes size: d p for te Stokes problem. For te Oseen system, te same term is added but te parameter d p sould be cosen differently. is will be specified later. Remark. Due to te ortogonality property ð, rp p r 1 rqþ ¼0 8p q r te local projection can be applied only onto te test function or onto te ansatz function. Hence, it olds ð, rp d p, rqþ ¼ð, rp d p rqþ ¼ðrp d p, rqþ: ð49þ en, te convective term is stabilized by introducing ð, ððb rþuþ d u ðb rþvþ: In order to avoid furter notations, we extend te definition of, onto vector-valued functions. Additional control over te divergence is obtained by te term ð, ðr uþ crvþ: Note tat tese terms are also symmetric by te same argument as before (49). Summarizing all tese terms leads to te stabilization

8 860 M. Braack et al. / Comput. Metods Appl. Mec. Engrg. 196 (007) S lps ðb U V Þ :¼ ð, rp d p rqþþð, ru crvþ þð, ððb rþuþ d u ðb rþvþ ð50þ and te discrete bilinear form of te Oseen problem: A lps ðb U V Þ¼Aðb U V ÞþS lps ðb U V Þ: In contrast to te residual-based stabilization tecniques, te rigt-and side keeps uncanged, suc tat te discrete system reads: find U ¼fu p gw rr, s.t. A lps ðb U V Þ¼LðV Þ 8V ¼fv qg W rr : ð51þ By te same argument as in te remark above, te stabilization term can be written as S lps ðb U V Þ¼ð, rp d p, rqþþð, ðr uþ c, ðr vþþ þð, ððb rþuþ d u, ððb rþvþþ: Similar to PSPG/SUPG, tis stabilization contains tree sets of parameters {d u }, {d p } and {c} Stability for local projection stabilization For te stabilization (50), we define te mes-dependent semi-norm jjjvjjj lps by jjjv jjj lps :¼km1= rvk L ðþ þkc1= vk L ðþ þkdp1=, rqk L ðþ þkc 1=, rvk L ðþ þkdu1=, ððb rþvþk L ðþ wic contains te fluctuations wit respect to,. e main parts in tis semi-norm including te energy-norm and te L -norm of v are te same as for te residual-based metods, see (). e difference is in te -dependent parts because te pressure and velocity in jjj Æ jjj lps are separated, but tese parts include only te fluctuations. Stability is acieved directly by diagonal testing: A lps ðb V V Þ ¼ jjjv jjj lps 8V W: Control over te L -norm of te pressure is obtained by te upper bound for te discrete solution U in te seminorm jjj Æ jjj lps and te data f, cf. [3]: kp k L ðþ jjju jjj lps þkfk L ðþ : is result induces uniqueness of te pressure. 5.. A priori estimate for local projection stabilization e a priori estimate for tis stabilization becomes jjju U jjj lps M u! ðlu 1Þ r ðku 1Þ kuk H ku ð Þ þ M p ðlp 1Þ r ðkp 1Þ kpk H kp ð Þ ð5þ wit M u ¼ 1 d u þ 1 d p þkck L 1 ð Þ r þmþr l inv c þd u r l inv kbk W 11 ð Þ M p ¼ r l inv dp þ r maxðmc Þ : is estimate as some similarities to te one for te residual based stabilization (33). However, te considered (semi)-norm is different. e coice of te stabilization parameters d p du c can be done in dependence of te regularity of te pressure. We consider te two most important cases: At first, we study te case of same regularity of pressure and velocity, i.e. k:¼k p = k u, and l :¼ min{r +1,k}. Equilibration of te terms involving te stabilization constants leads to te optimal coice d u :¼ d p :¼ du r m þ r l inv kbk W 11 ð Þ þkck L 1 ð Þ! 1 ð53þ c :¼ r 4 d u : ð54þ l inv Provided 0 < l 0 =r 6 d p, tis leads to M u =ðdu r Þ and M p r d p l inv, suc tat te estimate (5) becomes jjju U jjj lps ðl 1Þ M u kuk H k ð Þ þ M p kpk H k ð Þ wit r ðk 1Þ M u ¼ m þ l inv kbk W 11 ð Þ þ r kck L 1 ð Þ and M p ¼ r d p l inv : ð55þ ð56þ ð57þ Let us sortly compare tis result wit (35) for te residual-based stabilization. Besides te fact tat te considered (semi-)norms on te left-and sides differ, te rigt-and sides are qualitatively te same at least for small r. In te case tat te flow is advection-dominated, te time-step is large and iger order approximation is considered (r 1) te estimate (55) (57) is readily suboptimal due to te term l inv kbk L 1 ð Þ wic is not divided by r. However, for moderate m as well as for moderate time steps, te estimate (55) (57) is optimal. At second, in te case of less regular pressure, i.e. k p = k u 1, te coice of te stabilization parameters above would give a poor a priori estimate because M p M u kbk 1 L 1 ð Þ. In order to ave M p M u, te parameter d p sould scale as. We take du as before in (53) and d p :¼ c r 4 ml :¼ m: inv is leads to M p =ðr mþ, M u ¼ð1 þ r l inv Þm þ l inv kbk W 11 ð Þ þ r kck L 1 ð Þ

9 M. Braack et al. / Comput. Metods Appl. Mec. Engrg. 196 (007) and to te a priori estimate jjju U jjj lps l p M u r kp kuk H ku ð Þ þ 1 m kpk H kp ð Þ : 5.3. Variants of local projection stabilization In tis section, we present some variants of te local projection stabilization. First of all, te stabilization term for te convective term can be replaced by te full derivative. In tis case, te stabilization becomes S lps ðu V Þ¼ð, rp d p rqþþð, ru d u rvþ: ð58þ Furtermore, instead of te fluctuation filter,, te filter wit respect to te global Lagrange interpolant onto te coarser mes can be used:, :¼ I I r : Wen suc a filter is used, te stabilization consists of building te gradients of te fluctuations wit respect to te filter. e Oseen system can now be stabilized by adding te terms: S lps ðb U V Þ :¼ ðr, p d p r, qþþðrð, uþ crð, vþþ þððb rþ, u d u ðb rþ, vþ: ð59þ Wit tis notation, te discrete equation remains as before, see (51). e semi-norm jjj Æ jjj lps contains now te stabilization terms of (59). Stability can be sown by te same tecnique as for te variant presented in Section 5.1, see [1]. e a priori estimate (5) remains valid, but in order to apply te Lagrange interpolant onto te exact solution, we ave to assume tat p H t () andv H t () d, wit t > d. is variant can be considered as a generalization of te concept of Guermond [] proposed for advec- tion diffusion equations. As last variant we will sortly discuss is te most attractive one from te practical point of view. Instead of using two different meses,and, only te principal mes is used. e discrete space W rr sould be at least of is used to for- order r P. e additional space W r 1r 1 mulate te local projection: ~, :¼ I I r 1 : e stabilized form reads as (59) wen, is replaced by ~,. is stabilization term keeps optimal for te Stokes system. However, for te Oseen system it becomes suboptimal because te advection stabilization ensures only te convergence order of te lower order space W r 1r Implementation issues e price for tis symmetric minimal stabilization tecnique is te larger stencil of te corresponding stiffness matrix due to te projection, acting on patces. If te complete stencil is included in te sparsity structure of te matrix, te memory requirement is about a factor of two larger in comparison to te Galerkin part (in -D and in 3-D as well). However, wit a ceap preconditioner, as for instance S prec ðb U V Þ :¼ ðrp d p rqþ þ ðr u crvþ þððb rþu d u ðb rþvþ te larger stiffness matrix can be avoided. We refer to [1] for teoretical and practical results of suc a preconditioner in te case of te Stokes problem. Anoter necessity for te use of suc local projection is te availability of patces (cells of te mes ). 6. A coarse space projection based metod e local projection based metod presented in Section 5 can be cast into a more general framework. Let G H,U be a finite dimensional space of d (d + 1)-tensor-valued functions and d be a non-negative function. e index H sould indicate tat G H,U is a coarse or large scale space, eiter defined on a coarser grid or by low order finite elements on te finest grid. e abstract coarse space projection formulation seeks fu G HU gw rs G HU suc tat Aðb U V ÞþðrU G HU drv Þ ¼ LðV Þ 8V W rs ðru G HU L HU Þ ¼ 0 8L HU G HU : ð60þ e second equation in (60) is te L () projection of te pressure gradient and of te velocity gradient into G H,U. Note tat te (trivial) coice G HU :¼ðrV r Þ ðrq s Þ, were, e.g., rvr stands for te space consisting of all derivatives of functions in te space V r defined in (14), avoids any projection and te Galerkin formulation is obtained. We will sortly discuss several non-trivial coices of G H,U in order to recover several types of stabilization tecniques. 1. One of te first projection metods for equal-order interpolation of te Stokes system, wic was proposed in [15], projects te pressure gradient only. is metod can be cast into tis framework by taking G HU :¼ ðrv r Þð r Þd. e gradient of te velocity is not projected, but te pressure gradient becomes projected onto a discrete space equal to te discrete velocity space witout Diriclet conditions. In tis case, te projection acts globally due to te continuity of te functions of G H,U.. aking a discontinuous space G H,U leads to a local projection and as te benefit tat te additional degrees of freedom, G HU, can be locally condensed. In particular, wit te notation of (48), te coice G HU :¼ ð r 1 Þdd ð r 1Þd leads to G HU ¼ p r 1 ru. Due to te ortogonality property (49), te local projection terms in (58) are recovered: S lps ðu V Þ¼ð, rp d p, rqþþð, ru d u, rvþ: 3. Finally, te case of inf sup stable pairs of finite element spaces was studied in [8]. Since te pressure stabilization is not necessary in tis case, G HU ¼ G H ðrq s Þ.

10 86 M. Braack et al. / Comput. Metods Appl. Mec. Engrg. 196 (007) Possible coices of te velocity part G H of G H,U and of d will be discussed in more detail in tis section. Let W rs ¼ V r Qs be a pair of finite element spaces wic fulfill te discrete inf sup condition (18). Let G H be a finite dimensional space of d d-tensor-valued functions. Since te stabilization parameter d can be interpreted in te coarse space projection based metod as an additional viscosity, it is denoted by m add (V,x). en, te coarse space projection based metod is defined as follows: find fu G H gw rs G H suc tat AðbU V Þþðm add ðu Þðru G H Þrv Þ ¼ LðV Þ 8V W rs ðru G H L H Þ ¼ 0 8L H G H : ð61þ Metods of tis kind ave been studied in, e.g. [8 30,3 34]. eir complete description requires to coose two parameters: te space G H and te additional viscosity m add (U,). e first parameter in (61) is te space of tensor-valued functions G H. e second equation in (61) states tat te tensor-valued function G H is just te L ()-projection of $u into G H : G H ¼ P GH ru. Wit tis notation, one can reformulate (61) as follows: find U W rs suc tat A csp ðu u v Þ¼Aðb U V ÞþS csp ðu u v Þ ¼ LðV Þ8V W rs ð6þ wit S csp ðu u v Þ :¼ ðm add ðu ÞðI P GH Þru rv Þ: In (6), G H plays te role of a large scale space suc tat ði P GH Þru represents (resolved) small scales or fluctuations of $u. o avoid negative additional viscosity, it is required tat G H frv jv W rs g. In te extreme case tat equality olds, te second term on te left-and side of (6) vanises and te Galerkin finite element discretization of te Oseen equations (5) (7) is recovered. If G H ¼fOg, one obtains an artificial viscosity stabilization of te Oseen equations wit a possible non-linear artificial viscosity. If m add (U,) is te Smagorinsky eddy viscosity model (63), te Smagorinsky LES model is recovered (in te case of te Navier Stokes equations). Since G H represents large scales, it must be in some sense a coarse finite element space. ere are essentially two possibilities. If W rs is a iger order finite element space, G H can be defined as low order finite element space on te same grid as W rs. is approac is studied in [9]. e second possibility, in particular if W rs is a low order finite element space, consists in defining G H on a coarser grid, see [30] for a study of tis approac in te case of advection-dominated advection diffusion equations. Concerning te second parameter of (61), m add (U,), almost all studies (for te Navier Stokes equations) used an eddy viscosity of Smagorinsky type [39] m add ðu Þ ¼c Sma kru k ð63þ were c Sma is a user-cosen constant, typically c Sma [0.001,0.05], and kæk denotes te Frobenius norm of a tensor. In [30], m add (U,) =c as been used in te projection based stabilization for stabilizing advectiondominated advection diffusion equations on equi-distant meses wit mes size Stability of te metod e metod (6) introduces additional viscosity by adding m add (U,) to te resolved small scales. For te subsequent analysis, we consider for simplicity te case m add (U,) being independent of U. en, te additional viscosity m add () can be written in front of te second term on te left-and side of (6) and te second equation of (61) can be used to add a elpful zero to get te following problem: find U W rs suc tat Aðb U V ÞþS csp ðu v Þ¼LðV Þ8V W rs ð64þ wit S csp ðu v Þ :¼ ðm add ðþði P GH Þru ði P GH Þrv Þ: By properties of te L ()-projection, one obtains for kru k L ðþ > 0 S csp ðu u Þ¼m add ðþ kru k L ðþ kp G H ru k L ðþ ¼ m add ðþ 1 kp! G H ru k L ðþ kru kru k k L ðþ L ðþ ¼: m þ ð G H u Þkru k L ðþ ð65þ wit 0 6 m + (,G H,u ) 6 m add (). If kru k L ðþ ¼ 0, we set m + (,G H, u ) = 0. e viscosity m + (,G H,u ) is small only if te L ()-projection of $u into te large scale space G H is close to $u itself. is is te case if tere are (almost) no small scales in te flow. We are not interested in tis situation since a stabilization is not necessary in tis case. e effective viscosity is now given by m eff ð G H u Þ :¼ m þ m þ ð G H u Þ: e stability estimate is obtained in te usual way by using U as test function. One obtains in te first step m eff ð G H u Þkru k L ðþ þ c minku k L ðþ 6 jðf u Þj: ð66þ We consider only te case tat c min > 0. e modifications for c min = 0 are obvious. e rigt-and side of (66) can be estimated by te Caucy Scwarz inequality or by te dual estimate. Eiter estimate is followed by Young s inequality. One obtains finally m eff ð G H u Þkru k L ðþ þ c minku k L ðþ ( ) 6 min kfk L ðþ kfk H 1 ðþ : c min m eff ð G H u Þ

11 M. Braack et al. / Comput. Metods Appl. Mec. Engrg. 196 (007) A priori error analysis e a priori error analysis starts in te usual way by subtraction (64) from (9) for test functions from W rs, splitting te error into e ¼fu u p p g¼fu ~u p ~p g fu ~u p ~p g ¼fg u g p g f/ u /p g wit f~u ~p gw rs and using te test function V ¼f/ u /p g. It is discussed in [8] tat f~u ~p g can be defined by te Stokes projection of {u,p} to ensure optimal interpolation estimates for {g u,g p }. After reordering terms, one obtains m eff ð G H / u Þkr/u k L ðþ þ c mink/ u k L ðþ 6 jðmrg u r/ u Þþððb rþgu / u Þþðcgu / u Þ ðp ~p r/ u ÞþS cspðg u / u Þ S cspðu / u Þj for arbitrary ~p Q s. e first terms on te rigt-and side are estimated using tecniques like te Caucy Scwarz inequality, Hölder s inequality, Poincaré Friedrics inequality and Young s inequality. One obtains for te last terms on te rigt-and side, using te definition of m + from (65), S csp ðg u / u Þ 6 mþ ð G H g u Þkg u k L ðþ þ mþ ð G H / u Þ kr/ u 8 k L ðþ 6 m þ ð G H g u Þkg u k L ðþ þ m effð G H / u Þ kr/ u 8 k L ðþ S csp ðu / u Þ 6 m addðþkði P GH Þruk L ðþ þ m effð G H / u Þ kr/ u 8 k L ðþ : Collecting terms, using (65) and applying te triangle inequality give te final estimate m eff ð G H ðu u ÞÞkrðu u Þk L ðþ þ c minku u k L ðþ " 6 C inf m eff ð G H g u Þkrg u k ð~u ~p ÞW rs L ðþ! þ c min þ kck L 1 ðþ kg u k L c ðþ þ kp ~p k L ðþ min m eff ð G H / u Þ ( ) þkbk L 1 ðþ min kgu k L ðþ krg u k L ðþ c min m eff ð G H / u Þ # þ m add ðþkði P GH Þruk L ðþ : Except te last term on te rigt-and side of tis estimate, all terms beave asymptotically as te interpolation error. e last term tends to zero as te mes widt! 0 if m add ()! 0 or if G H tends to te space {$v v V}. In bot cases, te Galerkin finite element discretization of te Oseen equations is recovered asymptotically. o obtain an optimal order of convergence, m add () and G H ave to be cosen in suc a way tat te last term beaves at least as te interpolation error. L -error estimates for te pressure can be derived in te standard way by using te discrete inf sup condition (18) Implementation issues e algebraic representation of (61) consists in a large coupled system of equations. e solution of tis system in coupled form as been studied in [30] and it as been found to be a very inefficient approac. A straigtforward idea consists in condensing te coupled system by eliminating te equations describing te L ()-projection into G H to obtain an algebraic analog of (6). In comparison to te Galerkin finite element discretization of te Oseen equations, one gets an additional matrix on te left-and side. In [30], a semi-implicit-in-time approac to te nonstationary Navier Stokes problem (1) and () was found to be quite efficient wic solves in eac time step an equation of te form: Find U W rs suc tat Aðb U V Þþðm add ðu Þru rv Þ ¼ LðV Þþðm add ðu ÞP GH ru old rv Þ ð67þ for all V W rs, were uold is te solution from te previous discrete time. Note, te left-and side of (67) is in general a stable discretization, e.g., using (63) gives te same matrices as in te linearization of te Smagorinsky LES model. e fully implicit approac after te condensation of te L ()-projection was studied in [9]. In [9,30], te efficient implementation of (61) and (67) into an existing finite element code was investigated. No matter if G H is defined on te same grid as W rs or on a coarser grid, one finds for te reason of efficiency two requirements on G H : G H sould be a discontinuous finite element space, te basis of G H sould be L ()-ortogonal. If iger order finite elements are used for velocity and pressure, te definition of G H on te same grid wit low order finite elements is appealing. In tis case, te fulfillment of te two requirements on G H prevent unnecessary fill-in in te matrices wic describe te L ()-projection into G H. In particular, te discontinuity of G H allows te computation of te L ()-projection by using only local information. Altogeter, te conditions on G H ensure tat te sparsity pattern of te additional matrix is te same as of te matrix representing Aðb U V Þ. us, adding bot matrices to obtain te left-and side of (6) causes no difficulties. o summarize, te costs of te coarse space projection space metod consist essentially in storing and assembling additional matrices wic represent te second term in te first equation of (61) and te second equation of (61).

12 864 M. Braack et al. / Comput. Metods Appl. Mec. Engrg. 196 (007) ese matrices can be used eiter to modify te rigt-and side as in te semi-implicit approac (67) or to modify te system matrix like in te fully implicit approac. 7. Critical comparison and outlook Let us first come back to te discussion at te end of Section 3 were a roug motivation of te symmetric stabilization metods of Sections 4 6 was given. In particular, coosing te subspaces and te (abstract) projection operator P in te way proposed in Sections 5 and 6 leads to te local projection and coarse-space projection based metods. Coosing te projection operator as p, cf. Section 4.1, leads to an face oriented stabilization metod of Section 4. is reasoning as given a more rigorous treatment in [3] were it was sown tat SUPG stabilization on te subgrid alone is sufficient to yield optimal a priori error estimates. e following conclusions can be drawn: We ave traded te full element residual of te SUPG/ PSPG metod for a projected residual, tus loosing te Galerkin ortogonality. e approximation properties of te projection give a weak consistency of te rigt order tat allows for te decoupling of te velocity and te pressure and ence leads to a decoupled stabilization. However, at te price of a larger stencil and/or in te case P projects onto a space of lower polynomial order or a coarser space, te approximation properties of te sceme will be given by tis coarse space. is results from te weak convergence of te stabilization operator. ypically te convergence order depends on te weak consistency of te stabilization operator. If te small scale space is cosen too small ten te projection error vanises at a rate significantly lower tan te approximation error of te fine scale mes (cf. Section 5.3 and 6). e slow convergence of te projection error will ten make convergence rates deteriorate. We now compare te metods from Sections 3 6 wit respect to teir different properties concerning some relevant issues suc as velocity pressure approximation, design of stabilization parameters, cost of stabilization, a priori error estimates Velocity pressure approximation Altoug not presented for all variants, all metods sare te property tat rater arbitrary pairs of velocity pressure approximation are allowed. In particular, equalorder pairs are still attractive from te implementation point of view. e stabilization of div-stable pairs of velocity pressure spaces is necessary to treat te advection-dominated case. In tis case, te non-symmetry of te SUPG/PSPG sceme is even more botering wit te non-symmetric velocity/ pressure coupling. On te oter and, te stabilization of te face oriented metod or te local projection metod remains symmetric since te stabilization of velocity and pressure are decoupled. 7.. Design of stabilization parameters In te case of equal-order pairs, te stabilization parameters of te SUPG/PSPG/grad div sceme depend in a sensitive way on te data at te element level. is is a consequence of te non-symmetric structure of te SUPG/PSPG terms. In particular, in te proof of te stability estimate (6) is te inverse inequality needed to control te terms coupled wit te Laplacian Du. is imposes certain upper bounds on te stabilization parameters, wereas in te case of local projection stabilization and face oriented stabilization (at least wen considering iger order polynomial approximation) te metod is very robust wit respect to overstabilization. Coosing te stabilization parameter too large gives rise to a less well-conditioned matrix, but as remarkably little effect on te approximation error (see i.e. [10,38] for numerical examples). e decoupled velocity and pressure stabilizations also allow for stabilization parameters for te velocity tat are independent of te viscosity. Suc a coice migt not always correspond to te least possible perturbation, but te order of te numerical sceme will not be altered. For te case of te pressure owever te stabilization must be canged in order to keep optimal order estimates in te regime of low local Reynolds number. Reducing te viscosity dependence of te stabilization parameters is of interest in strongly nonlinear situations suc as tose arising in combustion or viscoelastic flows. For div-stable pairs, one obtains a muc simpler parameter design for te SUPG/PSPG/grad div sceme. Moreover, it seems tat te PSPG terms can be omitted in tis case, see [0]. ey can be omitted also for te symmetric stabilizations in Sections 4and5. In tis case te stabilization parameter may be cosen independent of te viscosity. Some grad div stabilization is still necessary in tis case to obtain a priori error estimates tat are robust wen m! 0. Simply dropping te pressure stabilization witout modifying te stabilization terms of te velocity leads to a numerical sceme of order lp 1 for te error in te triple norm (). Since te pressure space is of lower polynomial order tan te velocity space, tis estimate looks suboptimal. Increasing te least squares control of te divergence on te oter and leads to an estimate tat is optimal in te H div norm, see [13,9] A priori analysis A striking advantage of te scemes wit a symmetric stabilization is te separate control of velocity and pressure terms in te analysis. In tis respect, te pysical meaning of te stabilization term P d kðb rþu þrpk L ð Þ of te SUPG/PSPG sceme is unclear. In te case of symmetric stabilization on te oter and, te a priori error estimate

13 M. Braack et al. / Comput. Metods Appl. Mec. Engrg. 196 (007) is given in te pysically relevant triple norm augmented wit te stabilization terms tat now give a measure on ow muc artificial dissipation as been added to te equation. Hence te artificial dissipation of energy induced by te stabilization may be monitored efficiently. e accuracy of te metods presented in Sections 3 6 is comparable, see, e.g., te discussion in Section 5.. e analysis of te SUPG/PSPG/grad div sceme is te most complete so far, including local estimates for scalar advection problems, and a priori error estimates in a p-framework. Considering te oter metods, it seems tat some additional work remains to obtain a more complete analysis. In te case of te projection stabilizations, it is te approximation property of te coarse space tat gives te precision of te a priori error estimates. It would terefore seem appealing to coose te coarse space as big as possible from te point of view of precision, for stability on te oter and it sould be cosen small enoug Expense of stabilization: computational costs Only te SUPG/PSPG/grad div sceme as a non-symmetric structure of te stabilization terms. e bulk of additional terms may lead to a time-consuming assembling of te linear system. But tese terms can be easily implemented into existing codes. On te oter and, te sceme is (muc) more compact tan tat of te edge/face-oriented stabilization metod. e latter metod can be easily implemented into codes wit a data structure tat allows te application of standard a posteriori estimators. On te one and, integrations ave to performed on te faces of te elements as in a DG metod and tere are many more faces tan elements. But on te oter and, tese integrals are of lower dimension and can ence be evaluated at lower cost tan te integrals over te elements. e computational cost of te projection-based scemes depends in an essential way on te efficient implementation of te projectors, and on te coice of te coarse spaces (see Section 6 for more details). e construction of efficient algebraic solvers and preconditioners is simpler for te scemes wit symmetric stabilization. e strong velocity pressure couplings in te matrix of te SUPG/PSPG/grad div sceme makes tis task more complicated, see, e.g. [35]. For te face/edge oriented stabilization, te system matrix does not ave te same structure as te standard Galerkin metod. e same is true for projection-based scemes were te coarse space does not posses specific properties as discussed in Section 6.3, see also [16]. An appealing alternative in tis case seems to be te use of quasi Newton algoritms for te solution of te non-linear problem, using only te part of te matrix tat fits in te standard Galerkin stencil cf. Section Unsymmetric vs. symmetric stabilization Numerical flow simulation are very often used for optimization issues were beside of te Navier Stokes equations (primal problem) an associated adjoint (or dual) problem arises. is adjoint problem as probably also to be stabilized. Usually, tere are two possibilities to andle adjoint problems numerically: (i) Building te adjoint out of te discretized stabilized primal problem. (ii) Building te adjoint out of te continuous primal problem and ten stabilize it. ese two possibilities are not equal in general. Possibility (ii) as te drawback, tat te adjoint problem is in general not consistent wit te optimization problem, because te gradient is perturbed. In contrast, possibility (i) is consistent, but not necessarily discretized properly. is is exactly te situation for te classical residual based stabilization tecniques in finite elements. A symmetric stabilization cures tis problem, because te possibilities (i) and (ii) become equal. Due to te symmetry of face/edge stabilization, local projection and coarse space projection tese scemes are advantageous for optimization problems. e local projection metod as already been used in optimization, see [] Outlook e accurate numerical solution of te advection-dominated Oseen equations by finite element metods requires te addition of viscosity in a sopisticated way. Moreover, if equal order finite element spaces for te velocity and te pressure are applied, te use of pressure stabilization terms becomes necessary. We presented an overview on stabilization scemes (element and face/edge residual based, projection metods) wic differ above all in teir basic ideas for stabilizing dominating advection. Compared to te Galerkin discretization, te numerical overead increases for all approaces, in particular wit respect to te memory requirements. Eiter te sparsity pattern of matrix blocks becomes more dense (edge-stabilization and local projection metod (velocity velocity block)) or additional matrices ave to be assembled (SUPG, coarse space projection sceme). Neverteless, we discussed several aspects sowing tat te not fully consistent scemes of Sections 4 6 wit symmetric stabilization ave some potential advantages over te classical SUPG/PSPG sceme. An a priori error analysis is available for all presented scemes, leading to information about te principal coice of te parameters involved in tese scemes on sape-regular triangulations. More researc is necessary to extend te analysis to te case of anisotropic meses. We did not include a numerical comparison of te scemes, altoug all statements of te paper are supported by our numerical experience. In tis respect, we ope to invite oter groups to contribute to suc a comparison. As discussed in Section 1, te metods under consideration ave to applied as a kernel witin a full code for

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