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ETNA Electronic Transactions on Numerical Analysis. Volume 32, pp. 106-122, 2008. Copyright  2008, Kent State University. ISSN 1068-9613. Kent State University etna@mcs.kent.edu LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOWS: EQUAL-ORDER VS. INF-SUP STABLE INTERPOLATION G. LUBE , G. RAPIN , AND J. LÖWE Abstract. A standard approach to the non-stationary, incompressible Navier-Stokes model is to split the problem into linearized auxiliary problems of Oseen type. In this paper, a unified numerical analysis for finite element discretizations using the local projection stabilization method with either equal-order or inf-sup stable velocitypressure pairs in the case of continuous pressure approximation is presented. Moreover, a careful comparison of both variants is given. Key words. incompressible flows, Oseen model, stabilized FEM, local projection stabilization AMS subject classifications. 65M60, 65N15, 76M10 1. Introduction. Consider the non-stationary incompressible Navier-Stokes model )*+" in ! in # "%$'&($ #"%$'&($ 435$76 for velocity and pressure in a bounded domain -,/.10 , 2 . A usual approach is to semi-discretize in time first with an 8 -stable implicit scheme [17]. In each time step, the resulting problems can be solved via a fixed-point or Newton-type scheme [17, 22]. This leads to auxiliary problems of Oseen type (1.1) (1.2) 9;:=< >?$#@BACD? #EFGHIJKLM NO" in $ in $ E P)R UJV 1W 9BX T QP)R 9;S T with 0 , and 0 , with coefficients KMYZ" a new right-hand side (stemming from time discretization). The Galerkin approximation of (1.1), (1.2) may suffer from two problems: violation of the discrete inf-sup ![ \E]\_^ `@ac bedfhgji (or Babuška-Brezzi) stability condition and dominating advection, i.e., . The traditional way to cope with both problems in a common framework is the combination of the streamline-upwind/Petrov-Galerkin method (SUPG) [7] and the pressure-stabilization/Petrov-Galerkin method (PSPG) [18]. An overview about residual stabilized methods can be found in [5, 24]. More recent results for k - finite elements are proven in [20]. This class of residual based stabilization techniques is still quite popular, since they are robust and easy to implement. Nevertheless, they have severe drawbacks which mainly stem from the strong coupling between velocity and pressure in the stabilization terms [5]. Therefore, other stabilization techniques have appeared recently, in particular the edge-stabilization method [5, 8] and variational multiscale (VMS) methods [9, 13, 15, 16]. We emphasize that almost all stabilization methods can be interpreted as special VMS methods. The key idea of VMS methods is a separation of scales: large scales, small scales, and unresolved scales. The influence of the unresolved scales on the other scales has to be modeled. Mostly, it is assumed that the unresolved scales do not influence the large scales. l Received November 29, 2007. Accepted for publication July 4, 2008. Published online on February 9, 2009. Recommended by A. Rösch. Department of Mathematics, NAM, University of Göttingen, Lotzestrasse 16-18, D-37083 Göttingen, Germany (lube, grapin, loewe@math.uni-goettingen.de). 106 ETNA Kent State University etna@mcs.kent.edu LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW 107 Currently, there are two variants to apply VMS methods to the full Navier-Stokes model. In most papers, an equal-order interpolation of velocity-pressure is applied; see, e.g., [2, 9]. Besides the rather simple implementation into existing codes, a formal reason is appearantly mn " that in the Euler limit no second-order derivatives occur. Other authors prefer discrete inf-sup stable velocity-pressure pairs [16] as this is the “natural” choice from regularity point poq" of view for fixed . A comparison of both approaches is still missing. Local projection stabilization (LPS) as special VMS-type methods are of current interest [4, 21]. Here the influence of the unresolved scales on the small scales is modeled by additional artificial diffusion terms for the small scales. In particular, the sub-grid viscosity model [14] can be interpreted as a special LPS method. In Section 3 of this paper, we present a unified theory of LPS methods for equal-order and inf-sup stable pairs in the case of continuous pressure approximation. In Section 4, a comparison of both variants is given with respect to theory and simple numerical experiments. \Br\tsGu v s wxy/z sGu this for the norm in thew Sobolev spaces w|{ the standard notation U >Throughout 9;S S >wx*$'paper, are used. The -inner product Ie$O} v \?\ ` a b v f w~in a domain is denoted by 9BX #wx , the index is eventually . The norm in is denoted by . For omitted. 2. Variational formulation and stabilization. Here, the basic Galerkin finite element formulation and its stabilized variants via local projection (LPS) are introduced. Moreover, different technical tools are given. 2.1. Basic Galerkin approximation. The basic variational formulation Q>?$#@ for AC P the Oseen (1.2) with homogeneous Dirichlet data reads: Find problem (1.1), R UJ V T 91S 0 , such that %y?$7y d #EKL$ d qC=$7NO d >r$N; d ) $ d (2.1) (bh u f (`L b f = = N>$' PJ . for all $ Pq 35$765 A SSUMPTION 2.1. 0 oq 2 " beR a bounded, polyhedral ),. P 9X Let +, P R 9BX W domain. UMV T More9;S T E P over, assume that with in , , 0 with 0 NGE" KMYZ" a.e. in and constant . E R EMARK equation. In particular, #2.2. E$'GUsually, ¡¢x" is a finite element solution G¡ of the OseenQ ¢E there holds for certain test functions . Therefore is small but does not vanish inG general. within is to V '£for V >£tH '£M I¤;$¥= d A remedy tIiterative ¤;$¥= d methods I¥$'¤¦a d Navier-Stokes V )O£¦¤1simulation $'¥= d;§ replace by S S S R EMARK 2.3. It is possible to extend the! analysis rE in this paper to a situation resulting from Newton K¨ iteration including the term . A small time step, resulting in a Y©3\OE]\_bª`]a?bªd]ff>i*«_i sufficiently large , ensures coercivity of the Oseen operator since >KL$B d 'tH'E]$; d Y!#K¬q\HE]\tbª`@a?bªdff#i*«_itO\(\ S Y V K\*\ S . ¡ S We consider a shape-regular, admissible decomposition ® of into 2 -dimensional simk°¯ plices, quadrilaterals in the& two-dimensional case or hexahedra for three& dimensions. Let & P ¡ P ¡ ® and k ¡ be the maximum of all k@¯ , ® . Let ± be a be the diameter of a cell reference element of the decomposition ® . We set P¬¹ s &1$& P ¡ $ ² stu ³´AC µ ¡HP 9 S B¶ µ ¡ ¶ ¯·B¸=¯ ± ® & ¹ sr &? ± is the space of all polynomials of degree º defined on ± , and where » stu ³´ AC Gµ ¡ P 9 S ;¶ µ ¡@¶ P¬¼ s &$'& P ¡$ ¯ ·B¸ ¯ ± ® ETNA Kent State University etna@mcs.kent.edu 108 G. LUBE, G. RAPIN, AND J. LÖWE & ¼ s &? ± the space of all polynomials on ± with maximal degree with s¾u ³ ´ coordinate T W ¡ u s¾º inR » each ½ 0 direction. The finite element space of the velocity is given by or ¡ u s ¾ R ² s ¾ u ³G´ T W ¡ us ¾ ¡ us ¾ ½ 0 with scalar components ¿ of ½ . ¡ u s'À simplicity, we restrict the analysis to continuous discrete pressure spaces » s'À_u For ³G´ WÁ ¡ u s'À ² s'À_u ³G´ WÁ or . For an extension to discontinuous pressure spaces, we refer to [23]. The subsequent numerical analysis takes advantage of the inverse inequality V \*\ u $ÌË@& P ¡]$°ËL¡ PM ¡ u s ¾]$ S ¶Ç¶ 1¶ u V ¯È ÃÄªÅÆº É k°¯ Ê ¯ (2.2) Â5ÃÄeÅÆ ® ¡ u s¾ ½ V .W For and of the interpolation properties of the finite element space o V µ Î P J U U the Scott-Zhang ¡É u s ¾ Í ,Ï [26, 1], one obtains for quasi-interpolant operator S with Y &?Ü Ô AC×Ö µ ¶ ÐrÑ PUÓÒ Ô ¯ ,Õ Ï , on the patches ¯ , Û ¯@ØªÙ ¯Ú Þ ÁÝoZ"H¶Ç\ µ ¡ É u s ¾ µ \OÞu Á kßà¯ Ê Þ \ µ \ u Ð¢Ñ=$á" ÎäHåeæ= Îç$ § Ò Ò ÈZâ¨È+ã Í ¯ È º É Õ (2.3) Â É º Ê É¡ u s ¾ A m ¡ This property can be extended to the vector-valued case with è . Similarly, an ¡ u s'À interpolation operator Í é satisfying (2.3) can be defined for the pressure. 2.2. Local projection stabilization (LPS). The idea of LPS-methods is to split the discrete function spaces into small and large scales and to add stabilization terms of diffusiontype acting only on the small scales. Consider two obvious choices of the large scale space: 6, 21]të is to (i) The first variant [4, ¡ determine the large scales with the help of a coarse ¡ mesh. The coarse mesh ê Ä ë P Ä>ì í ¡ is constructed by coarsening the basic mesh ® , such that each macro with diameter kLî ¡ is the union of one or more ê & P element ¡ neighboring cells of is non-overlapping and ® . Assume that the decomposition ê shape-regular. Moreover, the interior cells are supposed to be of the same size as the macro cell: Á)oq"y¶ Á $ÌËL& P ¡$ ëïP ¡5$ & ë § (2.4) Â k@î È k ¯ ® ê with , ¡É Following the approach in [21] we define the discrete space ð for ¡ the velocity as a disê continuous finite element space defined on the macro partition . The restriction on a ëïP ¡ ¡É ë BA} µ ¡ ¶ ¶ µ ¡ñP ¡É § ð ê macro-element is denoted by ð î É A 91S ë Hm ¡É ë ð The next ingredient is a local projection , which ¡É µ ò ¶ î A} ëó P ¡ defines the É µ ¶ ¡É A 91S cm ¡É ð ò î global projection9;S ò by ò î î for É allA 91S ê pm . Denoting 9;S ¡ the identity on AC by ô2 , the associated fluctuation operator õ is É¡ ¡É defined by õ ô2 ò . These operators are applied to vector-valued A functions R 91S ÷T inm a É¡ component-wise manner. This is indicated by boldface notation, e.g., ö 0 R ¡É T É¡ A R 9;S T m R 91S ÷T ð 0 and ø 0 0 . ¡É (ii) The second choice [10, 21]¡ consists of choosing a finite element of ð ¡ u s'À ¡ u s ¾ discretization lower order on the original mesh ® or by enriching the spaces and . The same ¡y ¡ abstract framework as in the first approach can be used by setting ê ® . ¡ ¡ Analogously a discrete space ðé and a¡ fluctuation N> ¡ $> ¡ operator PM¡ u s¾ õ%é can ¡ u s À be defined. The stabilized discrete formulation reads: find , such that ¡]$ ¡¢+ù¡ ¡$ ¡ 9 ¡¢($ Ë ¡) µ ¡$'G¡ PM ¡ u s ¾ ¡ u s'À $ (2.5) 8 ¬ where the additional stabilization term is given by ù ¡ ¡ $ ¡ BAC ú É¡ 7#Ex ¡ L$ É¡ '#EFt ¡ = ø î ´xýhþ î ø îûì ü ¡ #)* ¡ L$ ¡ ÷ ¡ ' ÿ É¡ ¡ ($ É¡ H ¡ Ã î õré õ5é ø (2.6) î î ø î § ETNA Kent State University etna@mcs.kent.edu 109 LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW R EMARK 2.4. Another variant is to replace the first right-hand side term of ú by É¡ #y ¡ L $ É¡ y ¡ 7 ø ø î î ´ îpì ü ù ¡ Ie$O} see the corresponding result in ÿ Remark 3.9. The constants þ î , Ã î , î , and ù ¡ Ie$Oî } will be determined later, based on an a priori estimate. Notice that the stabilization acts solely ù ¡ on the small scales. Of course, there are some more degrees of freedom in the choice of ; see also [4, 21]. É¡ In order to control the consistency error of the õ -dependent stabilization terms, the É¡ space ð has to be large enough; more precisely we assume the following. " É¡ A SSUMPTION 2.5. The fluctuation operator õ satisfies, for È!ã;È º É , the following approximation property: Â (2.7) Á ûo+"¶ \ É¡ % \ u õ î È Á k ßî ¶ %¶ u $ÌËL PU ë î ß ß º Éß ($Ë ë P ê ¡ § ¡ Due to the consistency of the õ é -dependent stabilization term, thus involving the space ¡ ðHé , we do not need such a condition for ðñé . In Section 3.5 several choices for the discrete spaces will be presented. For the analysis, the following properties of the stabilization term (2.6) are required. $ P R W UV ÷T L EMMA 2.6. There holds for all ¡ M ¶ù ¡ ô ¬ù¡] ôô ù¡] ô÷ô÷ô (2.8) (2.9) (2.10) ¶ ù ¡ $ È ;Yq"%$ $ $ îpì ü È $ ù ¡ $ ä ´ ÿ î îpì ü ¶ ¶ S Á S V Á ¶ ¶ S $ V >?$#@($ ´ \OE\ S bª` a b ff i Ã î . î ýþ î Proof. Property (ii) is trivial and (i) follows from the Cauchy-Schwarz inequality. Inequality (iii) can be derived from with Á $ A}!Á S ä ù ¡ $ È Á S ú îpì ü using Assumption 2.5 with ã ´ ýþ î " \E]\ S bª` a b î ff i ¶ ¦¶ S u V î ¶ ¦¶ S u V î Ã î ÿ î ¶ ¶ S u V î $ for the fluctuation operator. ¢¡É construct a special interpolant A 2.3. 1m UMV Special ¡ interpolation operator. Following [21], µ rwe 91S ¡É µ ¡É ¿ for the velocity, such that the error is -orthogonal to ð for all µMP+UM V . In order to conserve the standard approximation properties, let us suppose the following. oq" A SSUMPTION 2.7. There exists a constant É (possibly depending on º É ), such that µ ¡$'G¡ åªæ ¾ Y oZ"$ î ´ É ´ b f \ \ u \O ¡ \ u µ ¡ (2.11) ´ ´ ¾ ì î î Æ ì î $ !&#"% ' )( ¡ u s ¾L ë BAC Gµ ¡]¶ ¶ µ ¡ P ¡ u s ¾$ µ ¡" ë where ¿ on . î ¿ É¡ R EMARK 2.8. The space ð must not be too rich, since the inf-sup condition (2.11) ¡É has to be satisfied. On the other hand, ð must be rich enough to fulfill the approximation ¡É property (2.7) . Later on, we will present several function spaces ð which satisfy (2.11). ETNA Kent State University etna@mcs.kent.edu 110 G. LUBE, G. RAPIN, AND J. LÖWE *,+.- ÁNoq" , if there exists a constant independent In the following, we use the notation Á of all relevant sizes, such as mesh size, polynomial degree or coefficients, such that È . L EMMA 2.9. Let Assumption 2.7 be satisfied. Then there are interpolation operators É¡ A m ¬¡ u s¾ ¡ ¢¡ É A UJ V m ¿ and , such that µ ¡ É µ $7¡¢ d "%$ ËLG¡ P ¡É $IË µûP¬U V ($ (2.12) ð ç u Ð \ µ ¡ É µ \ u k]î ¶ µ ¡ É µ ¶ u ç; \ \ $ µ k ßî V î (2.13) î S É ß º É º Éß / for all µûPU W M U V and ß * 0+ 21 / É¡ $43 ¡ d Î"%$ÌË53 ç ¡ PR ð ¡É T 0 $°ËL PM ¶ ¬ É¡ 1¶ u / V î + ç; É k º ßî Éß \*1\ ß u Ð 1 $ > (2.14) / É¡ ;\ u î \* (2.15) k î S º É - $ 76 P!R U ÷T W ë P ç Dç Ô A}/Ö Ô ¡ ¯ is a ê , and È)ãÈ º É . Here î for all ß ë 0P ¯ î ¡, ê neighborhood of . For better readability, we put the proof in the Appendix. Analogously, a corresponding result can be proved for the pressure. o" L EMMA 2.10. Suppose that there exists a constant (possibly depending on º ), é é such that µ ¡$'¡¢ § åªæ À Y î ´ ´ ´ (2.16) ´ À b f \ µ ¡ \ u \* ¡ \ u ì é î î Æ ì î 9 !&8"% ' m ¡ u s'À ¡ A Then there exists an interpolation operator é µ ¡ µ $7¡¢ d % " $ Ë ¡ L (2.17) é ç \ µ ¡ µ \ u k]î ¶ µ ¡ µ ¶ V u ç ; (2.18) é î é î S º é é 0+ , such that ¡ $ ðHé k ßî \ µ \ u Ð ß º ß é P 21 $Ë µñP W U § ß R EMARK 2.11. The estimates of Lemmas 2.9 and 2.10 are optimal with respect to kLî . ¶¶ V u The estimates in the norm î are seemingly sub-optimal regarding to º É and º . é 3. A priori analysis. The next goal is an error estimate for the scheme (2.5). Therefore, ¡ u s'À ¡ u s'À ¡ u s ¾]$ ¡ u s ¾ further assumptions on the finite element spaces , and ð are required. ;: 3.1. Stability. First, the stability of the scheme will be shown. The standard approach is to provide this in the mesh-dependent norm \¶ =<¶ R ¶e\AC TI¶ S \O%\ S ?> @ y;\ S ¶ R T ¶ S ACN\ $ A 1\ S \OK ù ¡ $ ($ n>$' PÝ o~" for . Here we prefer a separated approach for with suitable ¶ R T¶ velocity and pressure by using first the semi-norm and then a post-processing argument for the pressure. L EMMA 3.1. The following a-priori estimate is valid (3.1) B y \O A ¡ \ S !\*K Hence, uniqueness and existence of ¡ \ S È ¡ P ¡ u s ¾ ¶ R ¡ T¶ S È $' ¡ d § in the scheme (2.5) follows. ETNA Kent State University etna@mcs.kent.edu 111 LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW #Eñ HI ¡ $' ¡ d Proof. Integration by parts yields therefore ¡]\ S ¡]\ S V '#/_ErI ¡ $ ¡ d S Q" , and ¡¢N¶ R _¡ TI¶ S § ¡ PJ ¡ u s ¾ This implies (3.1), hence uniqueness and existence of the discrete velocity field of the scheme (2.5). ¡P ¡ us À The corresponding result for the pressure ff i )\\ ` a bedf qåefollows æ d ° Lfrom E Lemma Ý\OE@\ bª` 3.2 a bedand Lemma 3.1. Here, we use the notation X , , X . (3.2) +ù¡¢* 8 ¡$ ¡¢Ý\O \OK ù¡] ¡%$ DC E o+" L EMMA 3.2. There exists a constant dependent on the continuous inf-sup constant and on the polynomial degree º É but independent of the mesh size k , such that Á ;E ç X \IL¡\ Á Kñ Á y+Á ¶ R ¡_TI¶; \ \ V $ X È ¯ (3.3) KÁ S Ê BJ AE FHG .I J I J .I S ä îpì ü ´ î \E\ Sbª`]a?b î ff>i Ãî and Á L 7K ¡ 1 M V äN îPsOR¾QS T 1 . ÁJ Á /Á ¯ where is ýþ the constant in the Poincaré inequality. Proof. Using range theorem, the the exisÉ¡ yields V \ ¡ \ inf-sup condition PJ eN ¡ ¶ ;¶ continuous ¡ AC R U V theT closedM 0 with tence of and V È . We set . Lemma 2.9 together with the triangle inequality imply ç ç ç ¶ ¡]¶ V ¶ 1¶ V +Á ç; ¶ 1¶ V ç1+Á ç; \L¡\ yAÁ \L¡\ § È È º É º É ý É É M K F / K F Consider now $ ¡ d N (3.4) 8 VU ù ¡ O ¡ $> ¡ $7"'xj ¡ $7 d ú & Ä V Ä WU Ý\ ¡ \ S ú & with terms Ä given below. Standard inequalities and integration by parts imply & V )#¡%$y¡ d #KLB¡$¡¢ d EFGIB¡$¡¢ d Á 1E X X #KÁ \O ¡ \ S \ K ¡ \ S ¶ ¡ ¶ V È KÁ S Á 1E X X #KÁ Á ¶ R ¡ TI¶r\ ¡ \ $ È KÁ S J I J J I J =F F where J Á J K F X JK stems from the Poincaré inequality & S !ù¡] È ¡%$G¡$'"' I Á ¬¶ R È ¡_TI¶r¶ ¡@¶ V ù¡ È K J ¶ ¡@¶ V . Lemma 2.6 gives ¡¢ ù¡] ¡%$7"($G¡$'"¢ \( ] ¡ \ Á È ¡$ I Á@ Á G ¶ R ¡ T¶r\L¡\ § Integration by parts (for continuous discrete pressure space) and Lemma 2.9 yield & )¶jN¢¡xM*$>@¡¢ d ¶ N¶ª ¡ M$@¡¢ d ¶)¶ª¡xM$ É¡ @¡¢ d ¶ õ Á S S ç Á ¶ R ¡ T¶ ä ¶ R ¡ TI¶¢\I ¡ \ § ú k î ¶ 1¶ S u k î V î ÿ È È S ÿ î É º î î º É î U ZY [ Ä V G & $ Ä ETNA Kent State University etna@mcs.kent.edu 112 G. LUBE, G. RAPIN, AND J. LÖWE $ ¡ d \ \ ¶ ¡ ¶ \ @\ RU V ÷T V V with the norm V in È Furthermore, there holds Ê 0 . Ê Ê Using all these estimates, we obtain from (3.4), Á E X \I ¡ \ Á Á K Á Á ¶ R ¡ TI¶ Á ¶ª¶ ¶e¶ $ V X ý È ¯ (3.5) KÁ S Ê ¡ Á A} V ä s¾ where ¯ . î F G 1 M N ORQS T 1 I J I J J I K 3.2. Approximate Galerkin orthogonality. In LPS methods the Galerkin orthogonality is not fulfilled and a careful analysis of the consistency has to be done. s'À ¡ P ¡ u s ¾ ¡ uerror P L EMMA 3.3. Let be the solutions of (2.1) and of ¬ and (2.5), respectively. Then, there holds +ù¡* ¡%$ ¡¢1ù¡ $ ¡¢($ Ë ¡ PJ ¡ u s ¾ ¡ u s'À § (3.6) 8 Proof. The assertion (3.6) follows by subtracting (2.5) from (2.1). Now the consistency error can be estimated with the help Lemma 2.6. #Ep5I =$#@ ofP/ P ¾ L EMMA 3.4. Let Assumption 2.5 À be fulfilled and ¡ with U @ " P|U ë P V ë V ë , for all . Then, we obtain for 0 , ê ß " ß $ È+ã É ã È º É , é À ¾ S S ¶ ù¡ $ ¡¢O¶ ¶ R ¡ TI¶ § ú k î ß ¾ ¶jEF ¶ S ¾ u ÿ k î ß À ¶ ¶ S À u V î (3.7) î S î ´ þ î º ÉS ß ß ß º ß îûì ü é ^] \ ] K &] + F Proof. Lemma 2.6 yields $ ¡r ù¡ $ ù ¡] ¡]$ ¡¢ ù¡ $ ¶ R ¡T¶ § È È )*+" Assumption 2.5 and imply À ¾ S S ß ù¡] $ ú ÿ k î ß ¾ ¶jEFGI¶ S ¾ u k î À ¶ ¶ S À u § V î î S î ´þ î º É S ß ß ß º ß îûì ü é ù¡] _] + Now the assertion follows from these estimates. 3.3. A priori error estimate. The a priori estimate can be shown by using the standard technique of combining the stability and the consistency results of the last subsections. >?$>L P T HEOREM 3.5. + be the continuous solution of (2.1) and s ¾ ¡ u s'À ¡ñQ >B¡$>L¡ P ¡ u Let À ¾ the the discrete solution ofU (2.5). n?$#@ PN Ó P P)R U that T V We assume V solution is sufficiently regular, i.e., , #EGI PR U ¾ 0 , and ß ß ÷T ß 0 . Furthermore, let the Assumptions 2.5 and 2.7 for the coarse velocity ¡É ¡ space be satisfied. ð ç ç For the space äåeæ ðñ$é we assume that (2.16) is satisfied. Then, there holds for Èqã É È º É and Èqã È º º É , é é¾ S ¶R ú ¡ T¶ S k î ß ¾ \ #E t\ S ¾ u Ð (3.8) ´ þ î º ÉS ß ß îpì ¾ ü À S S ç S ç S ß ç; ç; Á \¦\ S À u Ð $ k î ¾ ß Á É \( \ S ¾ u Ð k îÀ V V S S î S S îé É ß ß º É ß Ê ß Ê º é é ] F + K ] 1 F H] 1 F K ] 1K ETNA Kent State University etna@mcs.kent.edu 113 LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW where ACN\\ ` a b Á É î Á ACÎÿ îé î ç î f Ã î ç S k î > Kû º É S k î § º é 2` ç ÿ þ î \E\ S bª`]a?b Ãî î ff>i î $ þ î ` Proof. The error is split into two parts J ¡ $>ñ ¡ ) ¡ ) We start with the approximation error yield / É¡ $#ñ ¡é @ / É¡ J ¡ $ ¡é ñ ¡ § / É¡ $>û ¡é @ . Lemma 2.9 (i) and Lemma 2.10 À S ç ß É ¶ R ¡ $#ñ ¡ @T¶ (3.9) / é +0F ç; K F ú ´ º kS ß î À Ê S ÿ î \I \ ßS Àa] V u Ð21 K îpì ü é ¾ é S ç S ç; ú \O\ ` a b f K k î \OE@\ S bª`@a?b ff>i k î ¾ ß \*\ S ¾?] u Ð 1 § V K F ÉKF ´ ý Ãî îþ î S S î º2É` ß º É ß Ê îpì ü z¡ACN\b ¡$ ¡) É¡ B¡$ ¡ ;@¡ Now we estimate the remaining part via Lemma 3.1, Õ / é ù ¡ *' É¡ M ¡ $ ¡ ñ ¡ ($z ¡ ¶ R É¡ JB¡$ ¡ ñL¡T¶ 8 / ¶ R zM¡T¶ é / é ù¡¢*' É¡ M$ ¡ ñL($7zJ¡r ù¡OJB¡$>ñ @¡¢($zM¡¢ § 8 / ¶ R zJ¡_TI¶ é 8 ¶ R zJ¡_TI¶ ( í ( íí Applying Lemmas 3.3 and 3.4, the first term is bounded by ¾ S ù ¡ $>L($7z ¡ ú k î ß ¾ \E¬] \ S ¾u ¶ R z ¡ T¶ Í î ´ þ î º ÉS ß ß îpì ü F + À S ] K § k î ß À \ \ S À u V î î S ß º ß é Now we consider the terms of ÍÍ separately. Integration by parts and property (2.14) yield #% É¡ J;($ ¡¢ d >K É¡ J;($ ¡¢ d 'EFO É¡ M($ ¬¡ d )#% É¡ JB*$7 ¡¢ d >K É¡ J;*$ ¡¢ d q É¡ '#EFt ¡¢($ É¡ ; d ø ¾ S ç ç S ç; ú \O\ ` a b f #Kû \(\ S ¾ u Ð ¶ R z ¡ TI¶ § k î¾ ß k î V î ´ º ÉS ß Ê S É É ß î º þ îpì ü / +eF / cb 4b / cb KF / f &b / !b g< The orthogonality property (2.14) results in ¶jjû ¡ $ ¡ d ¶ Ý¶jjñ ¡ $ ¡ N é é õré ç ç1 ú > 2N` h &b )ÿ db !b / ] 1 K ¡¢ d ¶ À S S V \ \ S À u Ð ¶ R zM¡TI¶ § k î ßÀ V ´ S ß S Ã¦îÊ ß º îpì ü é é Integration by parts (due to continuous discrete pressure) and (2.14) lead to ¶ª ¡]$ É¡ J;' d ¶ ¶ª# ¡]$ É¡ J; d ¶ ¶ É¡ # ¡r($ É¡ JB d ¶ È ¾ Õ (3.10) Õ õ Õ S S ç ç ç; ú \*\ S ¾ u Ð ¶ R zJ¡_TI¶ § k î ß¾ V ´ ÿ î º ÉS ß S É ß îpì ü +eF + F / KF KF H] ] / &] ] ^] 1 K _] 1 K / ETNA Kent State University etna@mcs.kent.edu 114 G. LUBE, G. RAPIN, AND J. LÖWE The estimation of the stabilization term is straightforward ¶ ù ¡ É¡ J?$ ¡ ñL($z ¡ é ù¡] É¡ J?$ ¡ ñL($G É¡ M$ ¡ ñ @' #ù¡]#zJ¡]$7zJ¡' È é é ¾ S ç ç; ú ¶ R z ¡ T¶ \OE]\ S bª` a b f>f i \*\ S ¾ u Ð k î¾ ß V Ã î î ´ º ÉS ß Ê S ý þ î É ß îûì ü À S ç ¶ R z ¡ T¶ § ç; ÿ ú k î À ß \ \ S À u Ð V S S ´ î ß º ß Ê îpì ü é é ¶ R z ¡ TI¶ S Í Í¢Í together with the estimate of Adding up all inequalities for the estimate of (3.9) gives the assertion. C ORROLARY 3.6. Under the assumptions of Theorem 3.5 and the notation of Lemma 3.2, we obtain Á ç Á \?L¡]\ Á K päåªæ E Á ¶R ¡_TI¶C$ X X K ¯ (3.11) / +0F / KF F KF E / ] 21 K ] 1 K < J > G G I J +iE F G G K Á Á with constants , , ¯ and as in Lemma 3.2. Proof. ¡y °¡$>@The ¡¢ proof mimics ¡> thepproof ;¡%$#of @Lemma ¡¢ 3.2. @$¡¢In d equation ù¡] (3.4), ¡one $ ¡ É has $'"¢to replace by and by . ÿ $ 3.4. Parameter design. Now we will calibrate the stabilization parameters î þ î , and Ã î with respect to the local mesh size k î , the polynomial degrees º É and º of the é discrete ansatz functions and problem data. The parameters are determined by minimizing and balancing the terms of the right-hand side of the general a priori error estimation. Á É First, equilibrating the þ î -dependent terms in î yields D kj el k î \E\ be` a b þ î (3.12) ff i S º É î § Á É Á ÿ Similarly, equilibration of the terms in î and î é involving Ã î and î À ¾ ¾ À V V $ ÿ $ k ßî À Ê]ß ¾ k ßî ¾ Ê@ß À Ã î (3.13) î S S º ß Ê@ß º ß Ê]ß l l ^] l ] &] ] l yields where we used º º É Dº . For the following result, we assume that the solution é the continuous Oseen problem is sufficiently smooth. Y!ç ãÉ º É º C ORROLARY 3.7. For equal-order interpolation º , let ã é and ÿ k@î § Ã k]î $ ÿ k@î $ \E\_bª`]þ a?b ff>i S (3.14) Ã=î î î S S þ º º º î ] \ =$#@ ã é F5F + F K f K ] 1 ` ] È ] Then we obtain, under assumptions of Theorem 3.5, S V ç S S V E ¶R ¡_T¶ S ú ç; k î ß \ \ S u Ð k î ß \ \ S u Ð V \ E\_bª`]a?b ff>i S S S ß ß º ß º ß î îpì ü é ç S S S ç; \O\ ` a b f K k î \E\ be` a b ff i @ k î k î ß \(\ S u Ð V î î S S S É ß º º º ßàÊ h of ] 1 K º 1 § ETNA Kent State University etna@mcs.kent.edu 115 LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW For inf-sup stable interpolation with º É ÿ S ÿ k î $ (3.15) î Ãî º É ç º é Ã $ þ î º É U , we assume ã É ã ç k@î § \OE@\tbª` þ a b f>f i S º É î ] Then we obtain, under assumptions of Theorem 3.5, ¾ ¾ V S S ç S E S ¶e¶ R ¡T¶ S ç; \ ¦ \ \ ú ß ß k î¾ k î ¾ ¾ uÐ \ E]\ ªb ` a V S S S ß º É ß º É ß îpì ü é ç S ç S ç; \O\ ` a b f K k î !\OE@\ be` a b ff i @ k î î î S É º É º É º É º + K f F 1 ] K F5F º É and set é ] h ` m b î f 1 I\ S u Ð ß 0 ¾ S k î ¾ ß \(\ S u Ð V S S É ß Ê ß ] 1 K § _] This result requires some further discussion: ¡ us ¡ us ¡ us V 1 For equal-order pairs and for (inf-sup stable) s Taylor-Hood pairs s V ¡ us k î k î , we obtain the optimal convergence rates and , respectively, with respect to k î . É¡ $ ¡ é in the Due ¶¶ to the non-optimal convergence order of the interpolation operators V -norm, these estimates are presumably not optimal with respect to polynomial degree º É . Let us assume that in Lemma 2.9 there holds ç ç; k ßî \ µ \ u Ð k î ¶ µ ¡É µ ¶ V u (3.16) î É ß º É º ß _] n m + n o] / 1 and a similar result in Lemma 2.10 as well. A careful check of the proofs leads to (3.17) Ãî k î Ã º $áÿ for equal-order pairs with º S ÿ ÿ k î $ (3.18) î S º É m ÿ k î º î º É º Ãî $ þ î þ \E]\ bª` k a î bªd]ff i º É and é Ã l ç $ þ î þ î \OE]\tbª`]k] a?bedff>i º É ç for inf-sup stable pairs with º É . Then the a-priori estimate (3.8) in Theoº é rem 3.5 would be optimal with respect to º É and º , too, with the possible exception é of the factors depending on É and . The latter reason eventually leads to a noné optimal parameter design with respect to º É . The formulas for the stabilization parameters and the error estimates are only asymptotic statements. To the best of our knowledge, there is so far no general concept for a more precise design of the stabilization parameters (with the possible exception of one-dimensional problems with constant coefficients). $7 $'ÿ Unfortunately, this leaves the practitioner with the problem of choosing þ . Table 4.1 below might give an indication of suitable values in our experiments for an academic problem. 1 1wvw w s q ^ r d q v t u t p l F x y 1 < 1 > F z|{ x K 1 y K R EMARK 3.8. The SUPG parameter stabilization methods has ¡ ¡ þ î in residual-based äHåeæ a i s¾ V s ¾ the typical design þ î ; see [20]. This can¡ be rewritten as ¡ a i V V äåªæ s ¾ äåeæ ç s¾ with the local Reynolds number î . þ î This means that the design of the SUPG-like parameter þ î in (3.17)-(3.18) is strongly ù ¡ simplified for the LPS method due to the symmetric stabilization term . This choice 0l }~ 1 kq r^q4 u t 1Bw w y ETNA Kent State University etna@mcs.kent.edu 116 G. LUBE, G. RAPIN, AND J. LÖWE £ will not cause problems for locally vanishing as the corresponding stabilization term is É¡ >£MtH¤ ¡ $ É¡ >£M_¥ ¡ õ î . Clearly, a proper implementation is required. In î þ î õ £ the Stokes limit , the SUPG-type stabilization term given above does not occur, hence " îþ . Regarding the remaining stabilization terms, the PSPG-type term can be omitted for inf-sup stable elements. As a consequence, no suboptimality occurs in the analysis for the Stokes limit. R EMARK 3.9. The corresponding result for the LPS scheme with local projection of the S full velocity gradient, see Remark 2.4, leads to the design î Dk]î º É (or î !k@î º É ). Please note that the error estimates deteriorate if the local projection divergence terms ¶jjû of the ¡ $ ¡ d ¶ is omitted. Then, the critical term in the proof of Theorem 3.5 is . é l Zl ?b 3.5. Choice of the discrete paper [21] presents different variants for the À ¾ ¡ u sspaces. ¡ u s The ¡É ¡ M ðé using simplicial and hexahedral choice of the discrete spaces and ð elements. There are basically two variants: ¡ ¡ A two-level (formally denoted by ¡ ¡ variant with a suitable refinement ® of ê ê ® S ). ¡ ¡ ® , hence k î k , with a proper enrichment of A variant with ê ¾ u ³´ ² s one-level by using bubble functions. In the numerical results below, we restrict ourselves to the two-level approach, but the theory also covers the one-level approach. Note that the present analysis covers only the case of continuous pressure approximation. For san extension ² b s f to discontinuous discrete pressure V -elements, we refer to [23]. approximation, in particular to the case of ¡ Ê Ê The discontinuous coarse spaces are defined on the coarser mesh ê with polynomials of one degree less. Thus, for hexahedral elements the coarse spaces are given by s¾ u ´ $ sÀ u ´ § ¡É ¡ V ü V ü ð ðHé Ê Ê m m # For simplicial elements we obtain ð ¡É ² s¾ Ê u ´ $ V ü ðHé ¡ ² sÀ Ê u ´ § V ü 9BS É A Obviously, 2.5 is valid for our discrete spaces if the local -projection ò î 91S ë Zm Assumption ¡É ë ð for the velocity and similarly for the pressure is applied; see [21]. Moreover, for these choices the constants É in Assumption 2.7 and in (2.16) scale like Iç é çG for simplicial elements and like for quadrilateral elements in the affine º É é linear case; see [23]. n I n 4. Numerical results. 4.1. The Oseen problem. A proper calibration of the stabilization parameters requires careful numerical experiments going beyond the scope of this paper. Some papers validate the parameter design and the theoretical convergence rates for the Oseen problem 9;:=< >?$>LBA}N? #EFGHIJKLM )*+" (4.1) where #"%$çG S \C !C ? V $ S , using the smooth solution < åeæ ò C V ($ ò C |S & ò C V > $ DC 4C = V $ 1 S in in $ $ åªæ= ò C V & ò C S ($ E and . The source term and the Dirichlet data follow from . We refer to a study in [25] of the one-level variant for equal-order pairs with enrichment of the velocity space by ETNA Kent State University etna@mcs.kent.edu 117 LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW using bubble functions. The two-level variant with inf-sup stable pairs can be found in [19] for equal-order and for inf-sup stable pairs; for the latter case, see also [23]. In all these cases, É¡ 9;S the local -projection is used for the definition of the fluctuation operators õ é . As an example, some typical results are presented for the two-level variant with S V and the S S pairs on unstructured, quasi-uniform meshes. More precisely, the coarse ¡É ¡ "5 ð é spaces ð is constructed as discussed in Subsection 2.2. Moreover, we set , which ¡ õ é id. The results in Table 4.1 with the errors results in full grad-div-stab stabilization, i.e., ¬ ¡ FH ¡ ñNç" É and for the advection-dominated case Ê show comparable é results for the best variants of S V and S S , although the results for S S are slightly better due to the better pressure approximation. ~ TABLE 4.1 Two-level LPS scheme: Comparison of different variants of stabilization for problem (4.1) with pairs and . !^2Hok d ÿ XaoX þ 0.000 0.056 0.056 0.000 3.162 0.000 0.000 0.056 0.056 0.000 Pair S V S S Ã 0.000 0.562 0.562 0.562 0.000 0.000 0.000 1.000 0.000 5.623 0.000 0.010 0.000 0.000 0.000 0.000 0.018 0.018 0.018 0.018 ¶ É ¶V 2.56E-1 1.91E-3 1.91E-3 2.61E-3 1.87E-2 2.38E+1 1.65E-2 9.30E-4 1.77E-3 3.26E-3 É \ \* ¡ \ 5.42E-4 6.21E-6 6.20E-6 7.42E-6 7.50E-5 5.35E-2 3.48E-5 2.85E-6 4.18E-6 7.20E-6 2.02E-1 1.82E-4 1.66E-4 1.72E-4 1.56E-2 1.45E+1 9.37E-3 2.14E-4 1.46E-3 2.00E-4 \ \ Xao and ~é\ 2.31E-4 9.08E-5 8.06E-5 8.05E-5 1.08E-4 1.66E+3 6.96E-6 4.31E-6 3.25E-6 7.56E-6 Nevertheless, the importance of the stabilization terms is different. The small-scale SUPG- and PSPG-type terms are necessary for the equal-order case but not for the inf-sup stable pair. At least the PSPG-type term can be omitted for the inf-sup stable case. On the other hand, the divergence-stabilization gives clear improvement for the inf-sup stable case and some improvement for the equal-order case. Let us remark that the divergence stabilization without local projection has better algebraic properties than its LPS variant. In Figure 4.1, convergence plots are shown for the two-level LPS) scheme ç with)optimized ç" and parameters in the diffusion- and advection-dominated cases with Ê , respectively. The numerical convergence rates confirm the theoretical results. Interestingly, no gain of the better pressure approximation for the S S pair can be observed in the diffusion-dominated case. Let us finally check the effect polynomial degree for inf-sup Taylor6$ increasing $ r ûDçGstable " $7K ç PM 3%$'of Hood pairs Ò Ò V with Õ . This is shown in Figure 4.2 for , Ê Ê and different values of k . Similar results are obtained (but not shown) for equal-order approxPM ç $35$76%$ ]$ 5 imation with Ò Ò -elements with Õ . R !¡ a¢ !¡ ;¢ 4.2. Navier-Stokes problem. Finally, we apply the LPS stabilization to theK lid-driven EJ/ ~" cavity flow as a standard Navier-Stokes benchmark problem (4.1) with , and . Homogeneous Dirichlet data are prescribed with the exception of the upper part of the Î ç $7" ¯ boundary where is given. An unstructured quasi-uniform mesh is used together with the Taylor-Hood pair S V and the equal-order S S pair using the two-level variant and Ã according to our theory and ÿ Î" . In of LPS stabilization with scaling parameter þ " particular, the cases þ correspond to no stabilization. Ã . ETNA Kent State University etna@mcs.kent.edu 118 G. LUBE, G. RAPIN, AND J. LÖWE −6 Q2Q1, ν = 1, c = 1, LPS −1 10 Q2Q1, ν = 10 , c = 1, LPS −1 10 −2 10 −2 −3 10 −4 10 10 −3 10 3/2 error (log) error (log) h 10 4/2 −5 h 10 −4 −5 h4/2 10 5/2 −6 −6 10 10 |u−u | h |u−u | h1 −7 10 h1 ||u−u || h6/2 h 0 ||u−u || h 0 −7 10 ||∇ ⋅ uh||0 ||∇ ⋅ uh||0 ||p−p || h 0 −8 10 ||p−p || −7 −6.5 −6 −5.5 −5 −4.5 gridsize h (log ) −4 −3.5 h 0 −8 10 −3 −7 −6.5 −6 −5.5 2 −2 10 −3 10 −2 −3 −4 error (log) 10 4/2 h 10 −6 −4 10 −5 h4/2 10 −6 10 10 |u−u | |u−u | h1 −7 10 h1 ||u−u || 6/2 h 0 h −7 10 ||∇ ⋅ u||0 ||u−u || 6/2 h h 0 ||∇ ⋅ u||0 ||p−p || −7 ||p−p || h 0 −8 10 −3 −6 10 error (log) −3.5 Q2Q2, ν = 10 , c = 1, LPS −1 10 10 −5 −4 2 Q2Q2, ν = 1, c = 1, LPS −1 10 −5 −4.5 gridsize h (log ) −6.5 −6 −5.5 −5 −4.5 gridsize h (log ) −4 −3.5 h 0 −8 10 −3 −7 −6.5 −6 −5.5 2 F IG . 4.1. −5 −4.5 gridsize h (log ) −4 −3.5 −3 2 -convergence for the Oseen problem with L (left) and ! £ (right) with . In Figure 4.3, we present typical solution profiles of the velocity for the case of }~ ¤¢%$'" "" . The results for k V ` for both variants are in excellent agreement with the reference data in [11], where the stream function-vorticity formulation of the Navier-Stokes "ç problem on very fine meshes with up to ¥ ¥ "ç nodes is used. In particular, the boundary layers are well even on this quasi-uniform mesh. Moreover, the solution for a coarse V resolved V is in good agreement with the data in [11] away from the boundary layers. grid with k The results confirm the previous remarks for the linearized problem of Oseen type. For the S V element, only the divergence stabilization is necessary whereas for the S S pair all stabilization terms are important. Then, we compare in Table 4.2 the positions and values of extrema of the velocity profiles for different values of ç . The results for the two-level LPS scheme with the S V pair on 3 the fine mesh with k are in good agreement ç 63 with the results in [3, 27]. Moreover, the LPS results on the coarser grid with are in good agreement with the case k ç 3 . This is verified in k -convergence studies in [19]. k G$ 7 Finally, we compareÞ in Table 4.3 the position Ô ofG$ the7 main vortex and the values of the streamfunction . for two values of ç . The ÄªÅ and of the vorticity in the 3 results for the two-level LPS scheme with the S V pair on the fine mesh with k are in very good agreement with the results in [3, 11], but slightly different from the results in [27]. Studies of k -convergence can be found in [19]. ¦ ¢ ¥ }£~ §¦ ¢ ¥ ª { g¦ DC¨ 4© ¨ «¨ DC¨ !© ¨ }~ ¬¦ ¢ ¥ 5. Summary. A unified a-priori analysis of stabilized methods via local projection (LPS) is given for equal-order and inf-sup stable velocity-pressure pairs on isotropic meshes. The error estimates are comparable to classical residual-based stabilized (RBS) methods. This shows that only stabilization of the fine scales is necessary. Compared to the RBS methods, ETNA Kent State University etna@mcs.kent.edu 119 LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW −1 −2 10 10 h ≈ 1/8 h ≈ 1/16 h ≈ 1/32 −2 10 h ≈ 1/8 h ≈ 1/16 h ≈ 1/32 −4 10 −3 −4 ||∇ ⋅u || error (log) 10 −5 10 −6 10 h 0 |u−uh|1 error (log) 10 −6 10 −8 10 −7 10 −10 10 −8 10 −9 10 qd 2 3 b ´ f −12 qs® 4 element order Q /Q É Ê É k 10 5 2 3 k−1 qd¯ É ´ qk® element order Q /Q k 4 5 k−1 −3 10 h ≈ 1/8 h ≈ 1/16 h ≈ 1/32 −4 10 −5 h 0 ||p−p || error (log) 10 −6 10 −7 10 −8 10 −9 10 −10 10 −11 10 2 3 qé ´ q® 4 element order Q /Q Ê é k 5 k−1 F IG . 4.2. Polynomial convergence for the Oseen problem with !H£ 1 Q2Q1, h=1/16 Q2Q1, h=1/64 Q2Q2, h=1/64 ECG04 0.4 v component of velocity u component of velocity 0.6 0.5 0.5 0.4 0.9 0.95 1 0 −0.5 0 0.2 0.4 0.6 y−coordinate, (x = 0.5) ´4µX¶µX¬ · µXH¸ ²o¹oº 0.8 0.2 0 −0.3 −0.2 −0.4 −0.4 Q2Q1, h=1/16 Q2Q1, h=1/64 Q2Q2, h=1/64 ECG04 −0.5 0.9 −0.6 1 0 0.95 0.2 1 0.4 0.6 x−coordinate, (y = 0.5) X°ao±X³ ²Hkoo ´!µ«H¸ o²o¹Hk¶µ«H¸ H!»Hk· µ« F IG . 4.3. Lid driven-cavity problem with pair with and and for fixed . 0.8 : Cross-sections of the discrete solutions for pair with . 1 o the design of the stabilization parameters is much simpler for LPS schemes as the strong coupling of velocity and pressure in the stabilization terms does not occur. Numerical results from the literature and our own experiments confirm both, the design of the stabilization terms and the theoretical convergence rates. A major difference between equal-order pairs and inf-sup stable pairs is that LPS-stabilization is always necessary for equal-order pairs. For inf-sup stable pairs, the necessity of stabilization is seemingly much ETNA Kent State University etna@mcs.kent.edu 120 G. LUBE, G. RAPIN, AND J. LÖWE Xao7¿¾ÀÄÃ4aÁoº 4?!¹o @;;Â°7¿± ¾³4aºo²o¹ ¿À4aºo²o¹ C]ÞLÅoÆ µ Þ ÄªÅ Þ ÄªÅ L © Þ µ ÞLÅoÆ ÄeÅ ¿ ¼Ã4aHºo¸ ²o² ¹ TABLE 4.2 : Maxima and minima on cross-sections Lid driven-cavity problem for different values of ; a) LPS , b) LPS , c) (in [12]), d) [27]), e) spectral method with (in [3]). ½ §H¸ ² }~ 100 1000 7500 a) b) d) e) a) b) d) e) a) b) c) d) 0.21399 0.21404 0.21411 0.21404 0.38512 0.38857 0.39009 0.38857 0.43940 0.45478 0.43590 0.46413 0.45703 0.45703 0.45898 0.4581 0.17578 0.17188 0.16992 0.1717 0.07031 0.06250 0.0625 0.06445 0.17951 0.17957 0.17946 0.17957 0.37404 0.37692 0.37847 0.37695 0.43749 0.45836 0.44030 0.47129 0.23828 0.23828 0.23633 0.2370 0.16016 0.15625 0.15820 0.1578 0.07813 0.06641 0.0703 0.06836 Xaa¿°¾ÃÄÏ± 4aÁo4º ?!¹o ª Þ ÄªÅ ÔÉ¨ C]Þ 0.25376 0.25378 0.25391 0.25380 0.52295 0.52701 0.52839 0.52708 0.56560 0.58043 0.55216 0.58878 and (in ÄªÅ 0.80859 0.80859 0.81055 0.8104 0.90625 0.91016 0.90820 0.9092 0.96484 0.96484 0.9609 0.96289 TABLE 4.3 Lid driven-cavity problem for different values of : Position of center of main vortex and values of stream function and of vorticity ; a) LPS , b) LPS , c) (in (in [27]), e) spectral method with (in [3]). [11]), d) ÇÉÈÉÊÌË 4aºo²o¹ Í8Î }~ 1000 7500 a) b) c) d) e) a) b) c) d) 0.118310 0.118936 0.118942 0.1193 0.118937 0.119315 0.122386 0.122386 0.1253 Xaa¿¾Ã4aºo²o¹ ¿Ã4aºo²o¹ CR¨ ©¨ 2.06527 2.06772 2.06721 – 2.06775 1.86210 1.92691 1.92697 – 0.5314 0.5308 0.5300 0.5313 0.5308 0.5192 0.5122 0.5133 0.5177 0.5676 0.5651 0.5650 0.5664 0.5652 0.5293 0.5321 0.5317 0.5313 less pronounced as for equal-order pairs. In particular, the grad-div stabilization is much more important than the fine-scale SUPG stabilization. Moreover, the fine-scale PSPG part seems to be superfluous. Appendix. L EMMA A.1. É¡ A m 2.7 ¡ Let Assumption ¡ u s ¾ be satisfied. Then there are interpolation operators ¢¡ É A UJ V m , such that ¿ and µ ¡ É µ $'G¡ d "ËLG¡ P ¡É $Ë µûP¬U V ($ (A.1) ð ç \ µ ¡ É µ \ u k@î ¶ µ ¡ É µ ¶ V u ç; k ßî \ µ \ u Ð $ (A.2) î î S É ß º É º Éß µûPU W UM V , and for all ß ¬ É¡ $ ¡ d "Ë ¡ PJR ¡É T $Ë@ PJ $ 0 (A.3) ð ç \( É¡ 1\ u k@î ¶ É¡ 1¶ u ç; k ßî \(1\ u Ð $ V î (A.4) î S É ß º É º Éß / / / ;3 / e+ + 53 1 1 ETNA Kent State University etna@mcs.kent.edu 121 LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW 76 P!R U ÷T W ë P ç Dç Ô AC/Ö Ô ¡ ê ¯ is a , and È)ãÈ º É . Here î for all ß ëá0 P ¯ î ¡, ê neighborhood of . Proof. We follow the lines of the proof of [21, Theorem 2.2], but we take into account the dependency of the constants on the polynomial constant É . ¡ñA ¡ order ë inf-sup Ü ë m and ¡É the Define the linear continuous operator ¿ ð by Ë µ ¡P ¡ ë ($7 ¡P ¡ µ ¡ $7 ¡ A}) µ ¡ $' ¡ $ ¡É ë § ¿ ð î ¡ The Theorem yields is¡]an ¾ b f thatµ ¡ P zM u Assumption zM¡] ë Closed-Range \ µ ¡]\ via \ ¡ µ ¡@\ ´2.7 ë isomorphism zJ¡]from ë ¡É ë Ü onto ð with É z ¡ ë î BÈ AC , , where Ø î ¡ ª$ GÕ is the orthogonal complement ofV with respect to î . ¡ µ $ë P ë P ¡ µÎP!UM be arbitrary. Then, there exists a unique ê Let and V \ µ ¡ u s¾ µ \ u z ¡ ë \ ¡ µ $ ë O\ È Í with î , such that É ÑÐ 4Ó Ò 4Ó Õ Ñ Ð ¡ Õ ¡% µ $ ë (A.5) Ð ($7¡ Ò .Ô Ð M Õ ¡] µ ($7¡r î A J U V Ð ~ Ð N µ Õ Í ¡ u s ¾ µ $7¡¢ The approximation property in the Õ O¶ S u V î ¶ ¡ µ $ ë È ú 2Ñ2×_Ö Ø 1 Ñ + ¡] ë $ÌËLG¡ P î ð ¡É ë § ¡ ë P ¢¡ É u µ AC , ê , by î , we can define a global operator ¢¡ É (2.3), the operator satisfies for ç; Á ç S É ÑÑ×oÖ Ø 1 ú S k ¯ ß \ µ \ S u Ð¢Ñ § S ß ´ º Éß UV -seminorm follows from inequality (2.2), S V \ ¡ µ $ ë O\ S u S \ µ ¡É u s ¾ µ \ S u $ S Ã ÄeÅÆ ú ¯ Ã ÄªÅÆ º É k ¯ Ê º É k ¯Ê Í ¯ È Ñ É Ñ ´ ´ 2×_Ö Ø 1 ` Õ and using the approximation property (2.3), ¶ µ ¡ É µ ¶ S u )¶ µ ¡ É u s¾ µ ¡% µ $ ë O¶ S u V î V î È Í V ç ÃÄeÅÆ k ßàî Ê ¶ µ ¶ u Ð § S S Éº ß É º ÉßàÊ !Ó Õ m 5¡ É u ¿ Now, ¡ É u s¾ µ we O¶ define ¡]local µ $ ë operators ¡î Í î . Since ê is a partition of ¢¡ É µ O¶ AC ¢¡ É u µ ¡ ¢¡ É A UM V m ¿ by . 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