A unified analysis of mixed and stabilized finite element solutions of Navier-Stokes equations

A uni®ed analysis of mixed and stabilized ®nite element solutions of Navier±Stokes equations q Tom as Chac on Rebollo a,* , Antonio Dom õnguez Delgado b a Departamento de Ecuaciones Diferenciales y An alisis Num erico, Universidad de Sevilla. C/Tar®a, s/n. 41080 Sevilla, Spain b Departamento de Matem atica Aplicada I, Universidad de Sevilla. Avda. Reina Mercedes s/n. 41080 Sevilla, Spain Received 1 September 1998 Abstract This paper performs a uni®ed numerical analysis of Mixed and Stabilized Finite Element numerical solutions of 2D and 3D Steady Navier±Stokes Equations. We introduce a general internal discretization of Navier±Stokes equations of which both Mixed and Stabilized Methods are particular cases. We prove convergence of Stabilized Methods with non linear stabilization coecients, in particular for ¯ows with convection dominance. We also analyze the approximation of branches of regular solutions, in the case of convection dominance. We introduce a stabilized post-processing of Galerkin Finite Element (FE) solution of convection-dominated ¯ows. Some numerical test for nonlinear ¯ows show the good performances of this technique. 1. Introduction This paper deals with the analysis of the solution of the steady Navier±Stokes equations for incompressible ¯ows by some Finite Element (FE) discretizations. Concretely, we develop an analysis of Mixed and a large class of Stabilized Methods by means of a unique theory and apply this theory to obtain an accurate post-processing of Galerkin Finite Element solution of convection-dominated ¯ows. It is well known that the numerical solution of incompressible ¯ows by the FE Method may present spurious oscillations in pressure and/or velocity, respectively due to the incompressibility restriction and to convection dominance. Mixed Methods can stabilize the ®rst of these restrictions, provided the pair of discrete spaces for velocity and pressure satis®es the Brezzi±Babu ska condition (see [4,21]). However, for realistic Reynolds numbers, Mixed Methods are able to stabilize the discretization of convection dominance only for very small grid sizes. Stabilized Methods provide ecient stabilizations of the spurious modes of pressure and velocity, virtually without restrictions on the discrete spaces or the grid size. The ®rst of such methods was introduced in [11], being called ``Streamline-Upwind Petrov±Galerkin'' (SUPG) Method. This method considered only piecewise ane discretizations of velocity and pressure. Lately, many generalizations to high order polynomial discretizations of Stokes and Navier±Stokes equations were introduced (see [6,16±19,23,26], and references therein). The most recent development of Stabilized Methods has been focused on the optimization of the stabilizing process, in order to construct a numerical solution not only q This research was partially supported by Spanish M. E. C. Projects DGICYT PB93-1196 and MAR97-1055-C02-02. * Corresponding author. Tel.: +34-54-557989; fax: +34-54-552898. E-mail addresses: [email protected]s (T. Chao Ân), [email protected] (A. Domõ Ânguez). 0045-7825/00/$ - see front matter PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 195-4 stable, but as accurate as possible. Optimal stabilizations via condensation of optimal bubbles have been obtained for some discretizations (see [27,10]). As could be expected from heuristic considerations (dimensional analysis), this optimal stabilization has been proved to be necessarily non linear, as it essentially depends on the local Reynolds number. The numerical analysis of the solution of Navier±Stokes equations by Mixed Methods is today a virtually achieved task, being the object of a well developed Theory ([21,7]). The analysis of stabilized methods has experimented in the last years a progressive improvement, having been the object of some recent works. The original SUPG Method for the unsteady Navier±Stokes equations was analyzed in [24,22]. For the steady Navier±Stokes equations, some modi®cations of the SUPG method have been analyzed in [25,28]. This analysis was generalized in [29], leading to a well-established theory that can analyze all kinds of Stabilized Methods. However, this analysis does not include nonlinear stabilization coecients. Thus, this analysis does not include realistic computations, in which the stabilization coecients depend on the local Reynolds number (in particular, with optimal stabilization). In this paper we analyze the solution of 2D and 3D Navier±Stokes equations by both Stabilized and Mixed Methods. This technique has both theoretical and practical applications. The main contributions of the paper are the following: To give a proof of convergence of Stabilized Methods with nonlinear stabilization coecients. To develop an accurate stabilized post-process of solutions of the Galerkin Finite Element solution of nonlinear convection-dominated convection±diusion problems. The main technical tool that we use is the splitting of the velocity space into two subspaces, representing large and small scales. All stability results related to Stabilized Methods (in both velocity and pressure) turn out to be strongly dependent on the fact that this splitting be uniform in the discretization parameter h, with respect to the H1norm. We also perform the approximation of branches of regular solutions. In this case, we consider stabilizing terms independent of the local Reynolds number. We develop an analysis parallel to that developed in [29]. Essentially, we systematize the discretization and analysis techniques developed in this paper, and obtain the essential estimates in a simpli®ed way. Moreover, we are able to handle the usual antisymmetric weak formulation of the nonlinear convection operator, with optimal error estimates. This allows to prove existence of discrete solutions for any grid size. The paper is organized as follows. In Section 2 we introduce the basic Augmented Mixed (AM) discretization of Navier±Stokes equations, with ``Adjoint Stabilized'' structure, that shall be considered in Sections 3 and 4. Section 3 treats the general AM Method, analyzing existence and eventual uniqueness of solutions, so as obtention of error estimates. Section 4 applies the analysis of Section 3 to the Adjoint Stabilized method. Section 5 extends the previous analysis to AM Methods with dierent structure (similar to SUPG, Galerkin-Least Squares and some new others). In Section 6 we analyze the approximation of branches of regular solutions of Navier±Stokes equations by means of AM Methods. In Section 7 we develop our technique of stabilized post-process of Galerkin and Mixed solution of convection±diusion problems. This technique is applied in Section 8 to some numerical tests for nonlinear ¯ows. Stable ®ltered solutions are obtained, in some cases with high precision. 2. Description of discretization method In this section we describe the abstract discretization of steady Navier±Stokes equations that we shall consider to develop our analysis. Let Xbe a bounded domain of Rd(d2 or 3), with boundary C. We shall consider the steady Navier± Stokes equations in Xwith homogeneous Dirichlet boundary conditions, kinematic viscosity m>0 and given source term f:X!Rd: Find a velocity ®eld u:X!Rd, and a pressure function p:X!Rsuch that uruÿmDurpf;ru0inX; u0onC:1 302 T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 Denote by Vthe space H1 0X  d, and by Ythe space Vd. Denote also by Mthe space L2 0Xfq2L2Xj RXqdx0g. Denote respectively by V0,Y0and M0the topological dual spaces of V,Y and M. We assume Mand M0to be identi®ed by means of the L2inner product. Thus, we have the usual inclusions VMM0V0;with compact injections. We shall formulate problem (2) in variational form as follows: Given f2Y0;obtain u;p2YMsuch that BNSu;u;p;v;q hf;vi8v;q2YM;2 where BNSu;y;r;v;qbu;y;vmry;rvÿr;rv ÿ r  y;q: Here, b;; is a trilinear form on YYY, de®ned by bu;y;v1 2ury;vÿurv;y;u;y;v2Y:3 The notation ; stands for the LPXÿLP0Xduality, either for scalar, vector or tensor functions. The form bis continuous and antisymmetric with respect to its second and third arguments. We shall use the notation kk kfor the standard norm in either HkXor HkX d(k0 corresponds to L2norms). We shall also use the notation jj kfor the Hkseminorm. Finally, kk k;pshall represent the Wk;pXnorm, either for scalar or vector functions. Following [21], Problem (2) has at least one solution if f2Hÿ1X d. Moreover, this solution is unique if N m2kfkÿ1<1;where Nsup bu;v;w juj1jvj1jwj1 for u;v;w2Y  :4 In order to describe our discretization of Navier±Stokes equations (2) in a general framework, we shall consider two families fYhgh>0and fZhgh>0of subspaces of ®nite dimension of Y, indexed by the discretization parameter h, assumed positive. As usual, the parameter hmay describe the values of any given sequence of positive real numbers convergent to zero. This is what we mean by the symbolic expression ``h>0''. We assume the family of pairs of spaces fYh;Zhgh>0to satisfy the following hypothesis: Hypothesis 1. (i) For any v2Y, there exists a sequence fvhgh>0with vh2Yhsuch that lim h!0kvÿvhk10: (ii) There exists a constant C0>0 independent of h such that jyhj1jzhj16C0jyhzhj18yh2Yh;zh2Zh8h>0:5 (iii) There exists a constant C0 0>0 independent of h such that kyhk0kzhk06C0 0kyhzhk08yh2Yh;zh2Zh8h>0:6 Remark 1. In [13] it is proved that Hypothesis (ii) and (iii) are verified when Yhand Zhare Finite Element spaces that are affine-equivalent to reference elements with intersection reduced to the zero function. Remark 2. Hypothesis (iii) is equivalent to the inequality jyh;zhj 61ÿd0 0kyhk0kzhk08yh2Yh;zh2Zh;where d0 02 C0 02:7 A similar inequality is equivalent to Hypothesis (ii). To understand the meaning of the Hypothesis 1, let us state the following result. T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 303 Lemma 1. Let fzhgh>0be a bounded family of elements of Y such that zh2Zh,for all h>0. Then,it converges weakly to zero in Y. Proof. If the sequence zh fg h>0is bounded, then it contains a subsequence, that we still denote in the same way, weakly convergent in Y, and strongly in L2X d, to a function v. Due to Hypothesis 1 (i), there exists a sequence yh fg h>0with yh2Yh, strongly convergent in Yto v. Thus, using Hypothesis 1 (iii), kvk2 0lim h!0yh;zh61ÿd0 0lim h!0kyhk0kzhk061ÿd0 0kvk2 0: Consequently, v0. As the limit of all weakly convergent subsequences is unique, the whole sequence converges to it.  Remark 3. The Hypothesis 1 (i) means that the family fYhgh>0is an internal approximation of Y in the usual sense.If we consider the space XhYhZh,we may interpret Hypothesis 1 (ii) and (iii) in the sense that that space Yhis uniformly (in h) complemented with space Zhto form Xh.Then,we may view the spaces Yhand Zhas ``low'' and ``high'' frequency components of space Xh. Next, we shall need a family fMhgh>0of subspaces of M. We also assume that this is an internal approximation of M. Hypothesis 2. For any q2M, there exists a sequence fqhgh>0with qh2Mhsuch that lim h!0kqÿqhk00: Also, to each w2Y, we shall associate a family of inner products on Y;fShw;;gh>0. We denote by Rhw; the ``static condensation'' operator on space Zh, associated to the inner product Shw;;, de®ned as follows. Given /2Y0,Rhw;/is the only element of Zhthat satis®es Shw;Rhw;/;zhh/;zhi8zh2Zh;8 where h;i denotes the Y0±Yduality. We require the inner forms fShgh>0to satisfy the following. Hypothesis 3. (i) Shis symmetric. (ii) There exists a constant ms>0 such that Shu;v;vPmsjvj2 1;u;v2Y: (iii) There exists a nondecreasing function Ms:R!Rsuch that jShu;v;wj 6Msjuj1jvj1jwj1;u;v;w2Y: (iv) The function u2Y!Shu;v;w2Ris continuous, for each ®xed v;w2Y. The symmetry of the forms Shstated in (i) is not essential in our derivation, but it avoids many technical complexities. Hypothesis (ii) states the uniform coerciveness of the forms Sh. Hypothesis (iii) implies in particular the continuity of the forms Shu;; for each actual u, while Hypothesis (iv) states its continuity with respect to u. Finally, for technical reasons, we shall consider a family of approximations ffhgh>0in Hÿ1X dto f.We need a technical assumption on fh. Hypothesis 4. If f2HkX dwith kÿ1ork0, then there exists a constant Cf>0 independent of h such that kfhkÿ16Cfkfkk: Here, H0Xmeans L2X. 304 T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 With these ingredients, our discretization of Navier±Stokes equations (2) is the following: Find uh;ph2YhMhsuch that Bhuh;uh;ph;vh;qhFhuh;vh;qh8vh;qh2YhMh;9 where Bhu;y;r;v;qBNSu;y;r;v;qÿShu;Rhu;ÿurvÿmDvrq;Rhu;uryÿmDyrr; Fhu;v;qhf;viÿShu;Rhu;ÿurvÿmDvrq;Rhu;fh: We shall call the above method ``Augmented Mixed'' (AM) discretization of Navier±Stokes equations. In the case of Stokes equations, this method may be viewed as an inner discretization of an ``Augmented'' alternative Galerkin formulation of the continuous equations (see [13]). We shall prove in Section 3 that method (9) bears the same essential properties as a Mixed Method. The additional terms that appear in the de®nition of forms Bhand Fhintend to bound the range of high frequencies of the residual uhruhÿmDuhrpÿfhthat are representable in Zh. Thus, possibly some spurious oscillations of velocity and pressure are going to be stabilized in this way. The structure of the additional term introduced is similar to that of the usual Adjoint Stabilized (AdS) Method. We shall prove in Section 4 that the AdS Method is a stable particular AM Method, for a speci®c choice of the spaces Zhand the forms Sh. Let us emphasize at this point that the dependence on uof the forms Shin (9) allows the stabilizing eect to be dependent on the local Reynolds number. Also, the introduction of the approximate source term fhis needed to represent the AdS method as an AM Method. 3. Analysis of AM Method In this section we shall analyze the AM discretization (9) of Navier±Stokes equations. We shall prove existence of solutions, and uniqueness, so as obtain optimal error estimates, for diusion dominated ¯ows. We shall need some properties of the static condensation operator Rh, reported in [13], that we adapt here to our context. Lemma 2. Assume Hypothesis 1 and 2 to hold.Consider an element u2Y.Denote by Rhthe static condensation operator acting on Zh,with respect to the inner product Shu;;.The following statements hold: (i) The operator Rhis linear and bounded from Y0on to Y.Moreover, jRh/j161 ms k/kÿ18/2Y0: (ii) For any q2M, msjRhrqj16sup v2Zh r  v;q jvj1 6Msjuhj1jRhrqj1:10 Lemma 3. Assume Hypothesis 1 and 3 to hold.Consider an element u2Y.Denote XhYhZh.Denote respectively by RXh,RYhand Rhthe static condensation operators acting on Xh,Yhand Zh,with respect to the inner product Shu;;. Assume that the family fXh;Mhgh>0satisfies the discrete BB condition.Then,there exists a constant g>0 independent of h such that kqhk06gC0Msjuj12jRYhrqhj1 jRhrqhj18qh2Mh;11 where C0is the constant appearing in Hypothesis 1 (ii). Inequality (11) is a re-formulation of the discrete BB condition introduced in [1] and [4]. The existence of solutions of the AM Method (9) is stated next. T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 305 Theorem 1. Assume that Hypothesis 1±4 hold.Assume f2HkX dwith kÿ1ork0. Assume that the following statements hold: (i) The family fXh;Mhgh>0,XhYhZh,satisfies the discrete BB condition. (ii) Either ms>m,or Yhand Zhare orthogonal with respect to the H1 0X  dinner product. Then,the AM Method (9) admits at least one solution that satisfies,for some constant C>0independent of h, juhj1kphk0jchj16Ckfkk; where chRhuh;uhruhÿmDuhrph: Proof. Our proof is based upon Brouwer's Fixed Point Theorem, applied to a linearised form of problem (9). Given uh2Yh, consider the linear problem Find yh;rh2YhMhsuch that Bhuh;yh;rh;vh;qhFhuh;vh;qh8vh;qh2YhMh:12 This is a generalized Stokes problem that ®ts into the framework of [13, Section 3]. Problem (12) is equivalent to a square linear system of dimension dim Yhdim Mh. Thus, to prove the existence of solution, it is enough to prove its uniqueness. Assume then that problem (12) admits a solution yh;rh. Then, Bhuh;yh;rh;yh;ÿrhmjyhj2 1Shch;yh2mShRhDyh;ch:13 Along this proof we drop the dependency of Shand Rhon uhfor the sake of simplicity of notation. This will not be source of confusion. Let us now assume msPm. Due to Hypothesis 1 (ii), ShRhDyh;chhDyh;chi  ÿryh;rchPÿ1ÿd0jyhj1jchj1;14 for some constant d0verifying 0 <d061 (cf. Remark 2). Let us take ein the open interval 1ÿd0;1ÿd0ÿ1, and de®ne ~ mm1ÿe1ÿd0;~ msmsÿm1ÿd0eÿ1:15 Then, both ~ mand ~ msare positive because msPm. Also, if Yhand Zhare orthogonal with respect to the H1 0X  dinner product, d01 so that ~ mm>0;~ msms>0:16 Using Young's inequality in (14), from (13) we deduce ~ mjyhj2 1~ msjchj2 16Fsuh;yh;ÿrh6hf;yhiShch2mRhDyh;Rhfh:17 Due to the symmetry of the forms Sh, property (i) of Lemma 2 and Hypothesis 4, ~ mjyhj2 1~ msjchj2 16hf;yhi2mhDyh;Rhfhihfh;chi6C1jyhj1jchj1kfkk; for some constant C1independent of h. Notice that the above bound is independent of uh. We now use Young's inequality to obtain jyhj16C2kfkk;jchj16C3kfkk;with C2C1 1 ~ m 1 ~ m1 ~ ms  s;C3C1 1 ~ ms 1 ~ m1 ~ ms  s:18 To estimate the pressure, we ®rst bound the ``low'' frequency of rphthat are representable in Yh, by means of the momentum equation in (12). Take qh0. Then, rh;rvhbuh;yh;vhmryh;rvhÿhf;vhiShRhÿuhrvhÿmDvh;Rhfh 6Njuhj1jyhj1mjyhj1kfkÿ1jvhj1 jhÿuhrvhÿmDvh;Rhfhij 6C2Njuhj1kfkk ÿC2m1kfkÿ1jvhj1juhrvh;Rhfhj  mjrvh;rRhfhj 6C41juhj1kfkkjvhj1; 306 T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 where C4is a constant depending only on m,msand N. By Lemma 2 (ii), we deduce jRYhrrhj 6C51juhj1kfkk: Next, we bound the ``high'' frequencies of rrhthat are representable in Zh, by means of the de®nition of Rh. Consider an element zh2Zh. Using the de®nition of ch, and property (i) of Lemma 2, we obtain jRhrrhj 6jchj1jRhuhryhÿmDyhj 6jchj1mÿ1 skuhryhÿmDyhkÿ1 6jchj1C61juhj1kyhk1 6C71juhj1kfkk: If we now apply Lemma 3, we obtain krhk06C81juhj1Msjuhj12kfkk;19 for some constant C8independent of h. Thus, when f0, problem (12) has the zero as unique solution. When f6 0, problem (12) admits a unique solution satisfying estimates (18) and (19). Let us now denote by F:Yh!Yhthe transformation that maps uhinto yh. Notice that F is continuous. This is proved by a standard ®nite-dimensional compactness argument that uses the continuity of forms Shwith respect to uh, and the uniqueness of solutions of problem (12). De®ne the set Afvh2Yhsuch that jvhj16C2kfkkg. Then, Ais a nonempty, convex and compact subset of Yh. Moreover, FAA. By Brouwer's Fixed Point Theorem, Fmust have at least a ®xed point uh2A. This ®xed point is a solution of problem (9).  We next prove the convergence of the solution of AM Method to a solution of Navier±Stokes equations. Theorem 2. Under the conditions of Theorem 1,the sequence fuh;phgh>0provided by the AM Method (9) contains a subsequence weakly convergent in YMto a weak solution of Navier±Stokes equations (2). Moreover,if this solution is unique,the whole sequence converges to it. Proof. By Theorem 9 and Hypothesis 4, the sequence fuh;phgh>0is bounded in YM. Then, it contains a subsequence, still denoted in the same way, weakly convergent in YMto a pair u;p. Let us prove that this pair is a solution of Problem (2). Denote chRhuh;uhruhÿmDuhrph,dhchÿRhuh;fh. By Theorem 1, the sequence fdhgh>0is bounded in Y. Then, by Lemma 1, it is weakly convergent to zero in Y, and, up to a subsequence, strongly in L4X d. We may also assume that the sequence fuhgh>0is strongly convergent to uin L4X d. Given v;q2YM, there exists a sequence fvh;qhgh>0, with vh;qh2YhMh, strongly convergent to v;qin YM. Then, lim h!0Shuh;Rhuh;ÿuhrvhÿmDvhrqh;dh lim h!0fÿuhrvh;dhmrvh;rdhÿqh;rdhg0: The remaining terms in Problem (9) pass to the limit in a standard way. This proves that u;pis a solution of Navier±Stokes equations.  We next analyze the obtention of error estimates for the AM Method, in the case of diusion dominated ¯ows. We shall ask the high frequencies space Zhto verify the following: Hypothesis 5. There exists a constant CZ>0 such that kzhk06CZhjzhj18zh2Zh: Remark 4. Hypothesis 5 is verified,for instance,by Finite Element spaces whose reference element does not contain the constant functions (see [13],Section 6). T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 307 Remark 5. Under Hypothesis 5,we may use Sobolev's injections to prove that for any q2[2,6], there exists a constant Cq>0such that (cf. [3]) kzhk0;q6Cqhbjzhj18zh2Zh;where b 21ÿb 61 q:20 Our error estimates theorem is stated as follows. Theorem 3. Assume that Hypothesis 1±5 hold.Consider two integer numbers mP1, lP1. Assume the following: (i) Given v2Y\Hm1X dand q2M\HlX,there exist vh2Yh,qh2Mhsuch that kvÿvhkj6Cjvjm1hm1ÿj;j0;...;m1; kqÿqhki6Cjqjlhlÿi;i0;...;l;21 for some constant C>0independent of vand q. (ii) The solution of Navier±Stokes equations (2) satisfies u;p2 Hm1X dHlX. (iii) There exists a constant Cf>0such that jRhuh;fÿfhj 6Cfhmhl: (iv) The uniqueness condition (4) holds,and either msPm e0 with e01ÿN m2kfkÿ1;22 or Yhand Zhare orthogonal with respect to the H1 0X  dinner product. Then,there exists a constant h0>0such that if 0<h<h0,the following error estimate holds, juÿuhj1kpÿphk06Chmhl;23 where C is a positive constant independent of h. Proof. Due to Hypothesis (i) above, there exist Uh2Yh,Ph2Mhsuch that kuÿUhkj6Cjujm1hm1ÿj;j0;...;m1; kpÿPhki6Cjpjlhlÿi;i0;...;l; 24 for some constant C>0. We shall denote by Cthe generic constants that will appear along this proof. These constants may depend on f,uand p, but not on h. Let us de®ne the consistency errors ah, bh2Y0by hah;vibUh;Uh;vmrUh;rvÿPh;rvÿhf;vi;v2Y; bhUhrUhÿmDUhrPhÿfh: Due to (24), kahkÿ16Chmhl: Also, bhUhrUhÿmDUhrPhÿuruÿmDurpfÿfh: Along this proof, we shall denote Shuh;; by Sh; and Rhuh; by Rh to lighten the notation. This will not be source of confusion as we do not use Shv;; for vdierent from uh. By Hypothesis (iii), jRhbhj16mÿ1 skUhrUhÿmDUhrPhÿuruÿmDurpkÿ1jRhfÿfhj16Chmhl: 308 T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 Observe that for v2Y,q2M, BNSUh;Uh;Ph;v;qÿShRhÿuhrvÿmDvrq;RhUhrUhÿmDUhrPh hf;vihah;vi ÿ r  Uh;qÿShRhÿuhrvÿmDvrq;RhbhRhfh:25 Denote ehuhÿUh,khphÿPh. If we subtract (25) from (9), we obtain for vh2Yh,qh2Mh, Bhuh;eh;kh;vh;qh ÿbeh;Uh;vhÿhah;vhi  r  Uh;qh ShRhÿuhrvhÿmDvhrqh;RhbhÿRhehrUh:26 De®ne dhRhuhrehÿmDehrkh. By taking vheh,qhÿkh, similarly to the proof of Theorem 1 we obtain ~ mjehj2 1~ msjdhj2 16jbeh;Uh;ehj  kahkÿ1jehj1 jr  Uh;khj  Shdh;Rhbh jj Shdh;RhehrUh jj 2mShRhDeh;Rhbh jÿRhehrUhj;27 where ~ mand ~ msare still given by (15) or (16). Notice that ShRhdh;Rhbh jj 6Msjdhj1jRhbhj1; where Msmaxh>0Msjuhj1<1:Moreover, ShRhDeh;Rhbh jj jhDeh;Rhbhij 6jehj1jRhbhj1: Also, due to estimate (20), ShRhDeh;RhehrUh jj ShRhehrUh;RhDeh jj jehrUh;RhDehj 6Ckehk0;3kRhDehk0;6jUhj16Ch 1=2jehj2 1: Similarly, Shdh;RhehrUh jj Ch 1=2jdhj1jehj1: Also, jbeh;Uh;ehj 6NjUhj1jehj2 16NjUh ÿuj11 mkfkÿ1jehj2 1: Inserting these estimates into (27), we deduce ~ mjehj2~ msjdhj2 16Ckahkÿ1jRhbhj1jehj1jdhj1Crhjehj2 1jdhj2 1 krUhk0kkhk0N mkfkÿ1jehj2 1;28 where rhmaxfh1=2;jUhÿuj1g. Then, limh!0rh0. Our next step is to obtain an estimate for the L2norm of the pressure error. Taking qh0 in (26), and proceeding as in Theorem 1, we obtain jRYhrkhj16Ckahkÿ1jRhbhj1jehj1jdhj1: Also, as RhrkhdhÿRhuhrehÿmDeh, we deduce jRhrkhj16jdhj1mÿ1 skuhrehÿmDehkÿ16Cjehj1jdhj1: By Lemma 2, as the uhare bounded, kkhk06Ckahkÿ1jRhbhj1jehj1jdhj1:29 T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 309 Remark 10. If spaces Yhand Zhare orthogonal with respect to the H1 0X  dinner product,then the hypothesis msPm=4may be dropped in (b) and (c). Remark 11. We may analyze the uniqueness of solutions of the AM Methods considered above,deriving results similar to Theorem 4.In all cases,under the Hypothesis of Theorems 9 and 4, the uniqueness is ensured if msis large enough.In the case of AM-SPG method,we should ask in addition that h be small enough. We may also apply the analysis developed in Section 3 to obtain convergence and error estimates for all AM Methods considered above. The convergence follows under the conditions of Theorem 9, similar to Theorem 2. Also, the error analysis yields the following result. Theorem 10. Assume that all Hypothesis of Theorem 3 hold,excepting Hypothesis (iv) which is replaced by ms>0for AM-GaLS method and msPm=4e0for AM-SUPG and AM-SPG method,with e0given by (22). Then,there exist a h0>0such that if 0<h<h0,the solution uh;phprovided by the AM Method (45) satisfies the error estimate juÿuhj1kpÿphk06Chmhl; where C is a positive constant independent of h. The preceding results may be applied to prove the existence of solutions, and convergence and to obtain error estimates for the usual GaLS and Generalized SUPG methods. Indeed, assume that space Yhis given by (36) and space Mhby either (37) or (38). Let the space Zhbe the Bubble Finite Element space Bhconstructed in Section 4, and let Shbe the inner forms de®ned by (40). Then, for f2L2X d, Lemma 4 implies that X K2Th sKuhuhryh ÿmDyhrrhÿf;e1uhrvhe2mDvhrqhK Shuh;Rhe1uhrvh e2mDvhrqh;Rhuh;uhryhÿmDyhrrhÿfh 8yh;rh;vh;qh2YhMh8e1;e22R: Thus, sucient conditions for the existence of solutions of GaLS and SUPG methods are deduced from Theorem 9. This is stated next. Theorem 11. Assume the Hypothesis of Theorem 5 to hold.Let f2L2X d.Then, (a) The GaLS method,applied to the solution of Navier±Stokes equation (2),always admits at least one solution. (b) The Generalized SUPG method,applied to the solution of Navier±Stokes equation (2),admits at least one solution if A64A0,where A0is the constant given by (42). Moreover,in both cases such solution verifies juhj1kphk06Ckfk0; for some constant C>0independent of h. Remark 12. Under condition (4),it is possible to state sufficient conditions,similar to (44),to ensure the uniqueness of solutions of GaLS and Generalized SUPG methods for diffusion-dominated flows. The error estimates result for GaLS and Generalized SUPG methods is next stated, as a consequence of Theorem 10. 316 T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 Theorem 12. Assume that the Hypothesis of Theorem 7 hold,excepting Hypothesis (v), which is replaced by A>0for GaLS method,and by 0<A64A0e0for Generalized SUPG method,where A0and e0are respectively given by (42) and (22). Then,there exists a h0>0such that if 0<h<h0,the solution uh;phof either GaLS or Generalized SUPG methods satisfies the error estimate juÿuhj1kpÿphk06Chmhl; where C is a positive constant independent of h. Our analysis technique may also be applied to Penalty Stabilized methods. For instance, we may consider the method introduced in [12], intended to independently stabilize convection and pressure gradient. Instead of form Sh, let us consider two forms S1hand S2h, both satisfying Hypothesis 3. Also, instead of space Zh, we consider two subspaces Z1hand Z2hof ®nite dimension of Y. Respectively denote by R1hand R2hthe static condensation operators associated to the pairs S1h;Z1hand S2h;Z2h. Our Penalty AM Method is Find uh;ph2YhMhsuch that BPuh;uh;ph;vh;qhhf;vhi8vh;qh2YhMh;46 where BPu;y;r;v;qBNSu;y;r;v;qS1hu;R1hu;urv;R1hu;ury  ÿS2hu;R2hu;rq;R2hu;rr  : By means of the representation Lemma 4, when Z1hand Z2hare convenient Bubble Finite Element spaces, the stabilizing terms above may be explicitated as follows: S1huh;R1huh;uhrvh;R1huh;uhryh X K2Th s1Kuhuhryh;uhrvhK8yh;vh2Yh; S2huh;R2huh;rrh;R2huh;rqh X K2Th s2Krqh;rrhK8qh;rh2Mh: Here, s1Kuhand s2Kare given stabilization coecients. The second one is assume not to depend on uh,as the term in S2hin BPis intended to stabilize just pressure gradient. When s1K0, the above method is the one introduced in [8]. For the Penalty AM Method (46), our analysis proves existence of solutions under the Hypothesis of Theorem 1, excepting that there is no restriction on the coerciveness constants of forms S1hand S2h. We also prove error estimates under the Hypothesis of Theorem 3 with ml1, and Hypothesis (iv) replaced simply by the uniqueness condition (4). However, the convergence order is necessarily limited to one, due to the penalty nature of the method. A particular proof of ®rst order convergence for piecewise ane discretizations of velocity and pressure is reported in [12]. Notice that this limited accuracy would make unecient the use of Penalty AM Methods with high-order Finite Elements. 6. Approximation of branches of regular solutions In this section we analyze the approximation of branches of smooth solutions of Navier±Stokes equations by the AM Method. We are able to analyze only methods with stabilization coecients independent of the local Reynolds number. Our analysis will follow in parallel that of [29]. We shall consider a scaled form of Navier±Stokes equations (1), kuruÿDurpf;ru0inX; u0onC;47 where k1=m, and we still denote p=mand f=mby pand f, respectively. The parameter kplays the role of the Reynolds number. We shall assume that kmay vary in a compact interval Kof real numbers. T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 317 Let us denote by Vthe product space H1 0X  dL2 0X. Then, problem (47) may be written in operator form as Fk;uk;pkuk;pkTGk;uk;pk0;48 where the operator G:KV!Vis given by hGk;u;p;v;qi  kbu;u;vÿhf;vi8v;q2V;49 and, T:VV!Vdenotes the Stokes operator, which associates to a pair u;s2Vthe unique solution y;r2Vof ry;rvÿr;rv ÿ r  y;qhu;viÿs;q8v;q2V:50 We shall analyze an AM-SUPG discretization of problem (47) somewhat dierent from the one previously introduced (corresponding to e1ÿ1 and e20 in (45)). We shall consider inner forms Shu;;  Sh; independent of u, still satisfying Hypothesis 3 (i) and (ii), and Hypothesis 3 (iii) with a constant Msindependent of u. In this case, the stabilization will be independent of the local Reynolds number. This structure of the Shwill ensure the linearity of the discrete Stokes operator approximating T. Our AM-SUPG discretization is the following: Find uh;ph2YhMhsuch that Bkuk;uk;pk;vh;qhFkuk;vh;qh8vh;qh2V;51 where Bku;y;r;v;qkbu;y;vÿry;rvÿp;rv ÿ r  y;qÿShRhÿkRu;vrq;RhkRu;y ÿDyrp;Fku;v;q hf;viÿShRhÿkRu;vrq;Rhf: Here, R:YY!Y0is the operator de®ned by hRu;v;wibu;v;w8u;v;w2Y: Also, we have denoted by Rh instead of Rh the static operator on Zhassociated to form Sh. This will somehow clarify some complex expressions that appear hereafter. We have considered the above AM discretization with SUPG structure by analogy with the paper of [29]. In this paper, the approximation of branches of regular solutions of steady Navier±Stokes equations by the standard SUPG method is analyzed. Uniqueness of solutions in a neighbourhood of the branch, so as optimal error estimates for smooth solutions are proved. However, some technical dif®culties impede to use the antisymmetric form bu;v;win their formulation. We shall reproduce the analysis of [29] for discretization (51) by using the techniques developed in the present paper. This will simplify the technicalities, as we shall always consider internal approximations of H1 0X  d. However, in some sense our result is less general than that of Tobiska±Verf urth. In fact, in principle we are not able to treat discontinuous approximations of the pressure combined with low-degree approximations of the velocity. In this case, the stability of standard stabilized methods requires the inclusion of interelement jump terms of the pressure. We do not know whether this kind of terms may also ®t in the general framework of AM Methods, by means of a representation Lemma similar to Lemma 4. On another hand, we consider the formulation (51) instead of the previously introduced (45) in order to be able to formulate it in operator form, similarly to (48). Alternatively, we could keep the convection operator in the stabilizing terms, and change bu;y;vinto ury;v, in the de®nition of Bk. However, in such a case there would exist a solution only for small h. This is the case of the discretization considered in [29]. All existence, convergence and error analysis made in Sections 3 and 5 still holds for the actual AM-SUPG method (51). This is a consequence of the following. Given zh2Zhand u,v2Y, we have 318 T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 hRu;v;zhi jj urv;zh  1 2r  u;vzh 6C1juj1jvj1kzhk0;36C2h1=2juj1jvj1jzhj1:52 Thus, operator Rbehaves in the stabilizing terms exactly as the convection operator appearing in formulation (45). We next write the discretization of method (51) in operator form, similarly to (48). Denote VhYhMh. We de®ne the discrete Stokes operator break Th:V!Vhas the mapping that transforms a pair u;s2Vonto the element yh;rh2Vhsolution of ryh;rvhÿrh;rvh ÿ r  yh;qhÿShRhrqh;RhÿDyhrrh hu;vhiÿShRhrqh;Rhuÿs;qh:53 Following the analysis of [13], this problem admits a unique solution which depends continuously on the data, if either msPm=4, or Yhand Zhare orthogonal with respect to the H1 0X  dinner product. We also de®ne the approximation of operator G, as the operator Gh:KVh!Vthat transforms a pair uh;ph2Vhonto the element Ghk;uh;ph2Vde®ned by hGhk;uh;ph;v;qi  hkRuh;uhÿf;viÿShRhruh;v ÿRhruh;Rhrq;RhkRuh;uhÿDuhrphÿf;for all v;q2V; where ru;wkRuh;w8w2Y. Now, we have the following. Lemma 5. Let the operator Fh:KVh!Vhbe defined by Fhk;uh;phuh;phThGhuh;ph.Then,a pair uh;ph2Vhis solution of problem (51) if and only if Fhk;uh;ph0:54 Proof. Consider uh;ph2V. Denote w;rGhk;uh;ph2V. By de®nition of Gh,ifv;q2V, hw;vihkRuh;uhÿf;viÿShRhkRuh;v;RhkRuh;uhÿDuhrphÿf; r;qShRhkRuh;Rhrq;RhkRuh;uhÿDuhrphÿf: Then, when u;sw;r, the r.h.s. of problem (53) is hw;vhiÿShRhrqh;Rhwÿr;qh hw;vhÿRhrqhi ÿ r;qh hkRuh;uhÿf;vhiÿShRhkRuh;vh;RhkRuh;uhÿDuhrphÿf ShRhrqh;Rhf  ÿkRuh;uh: Then, uh;phis solution of problem (51) if and only if it is a ®xed point of ÿThGh. Remark 13. We may consider Fhacting on all V,as Ghis itself defined on all KV. Remark 14. We could also write all AM Methods (45) under the operator form (54).To do this,we should replace the convection operator in the stabilizing terms by operator R,and still define Thby (53) in all cases.It is operator Gh,and more concretely the form ru;w,what must be adjusted to match each actual AM Method. For instance,the AM±AdS method (9) is recovered for ru;wkRu;wDw. We shall consider in the sequel a ``branch'' of regular solutions of Navier±Stokes equations (47). We shall assume that there exists a continuous branch k2K!uk;pk2V, which is regular in the sense that for each k2K,Du;pFk;uk;pkis a homeomorphism from Vonto V. Our analysis is directly based on the Theory developed in [29], which in its turn is closely related to the general analysis for approximation of nonsingular branches developed in [9], and its application to Finite Element discretizations of Navier±Stokes equations by Girault and Raviart [21]. T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 319 The essential estimates are given by Lemma 4.2 of [29] , which basically still holds in our case. Indeed, denote respectively by Uh;k2Yh,Ph;k2Mhsome approximation of ukand pkthat satisfy the error estimates (24) (cf. Theorem 3). Denote by LVthe space of bounded linear operators on V. Then the following holds. Lemma 6. Assume that Hypothesis 1, 2, 3 and 4hold.Assume also (i) Either msPm=4, or Yhand Zhare orthogonal with respect to the H1 0X  dinner product. (ii) The solution of Navier±Stokes equations (47) satisfies uk;pk2 Hm1X dHlX. Then,there exists a positive constant C independent of h and ksuch that: (a) kFhk;Uh;k;Ph;kkV6CK m;lhmhl; (b) kDu;pFk;uk;pkÿDu;pFk;Uh;k;Ph;kkLV6Ckh; (c) kDu;pFhk;Uh;k;Ph;kÿDu;pFk;Uh;k;Ph;kkLV6Ckh1=2; (d) If ku1;p1kV6M,ku2;p2kV6M,for some M>0, then kDu;pFhk;u1;p1ÿDu;pFhk;u2;p2kLV6Ck1 h1=2kM1ku1;pÿu2;pkV; where Km;lsupk2Kmaxfkfkkÿ1;kukkm1;kpkklg,kminfm;lg. Proof. To simplify the notations in this proof, we shall drop the sux k, and denote Du;pby D. (a) As Fhk;u;p0, we have kFhk;Uh;PhkVkFhk;Uh;PhÿFhk;u;pkV6kUh;Phÿu;pkVkTÿThGk;u;pkV kThGk;u;pÿGhk;Uh;PhkV :S1S2S3:55 By estimate (24), S16CK m;lhmhl:56 The error bounds for the Stokes equations obtained in [13], Section 6, imply S26ChmhlkGk;u;pkkÿ16CkK2 m;lhmhl:57 To bound S3, we recall that the Stokes problems (53) that de®ne Thare uniformly (in h) well posed (see [13, Section 7]). Then, S36CkGk;u;pÿGhk;Uh;PhkV 6CkGk;u;pÿGk;Uh;PhkV kGk;Uh;PhÿGhk;Uh;PhkV :CS31 S32:58 From the error estimates (24), S31 6Ckkuk2 mhm:59 To estimate the term S32, consider v;q2V. Denote ahkRUh;UhÿDUhrPhÿf. Then, hGk;Uh;Ph jÿGhk;Uh;Ph;v;qji  kShRhRUh;v jÿRUh;Rhrq;Rhahj khUh;v ÿRUh;Rhrq;Rhahi 6CkK1;0jRhahj1kv;qkV: As ah2L2X d, then Hypothesis 5 implies jRhahj16Chkahk06CK2 m;lhmhl: Consequently, S32 6CkK3 m;lhmhl 60 320 T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 Inserting estimates (56) through (60) into (55), we obtain (a). (b) This estimate was already proved in [29]. (c) Recall that Thk;; may be considered as an operator de®ned on all V, instead of Vh. Consider v;q2V. Then, khDFhk;Uh;PhÿDF k;Uh;Ph;v;qikV6kThhDGhk;Uh;PhÿDGk;Uh;Ph;v;qikV kThÿThDGk;Uh;Ph;v;qikV :S1S2:61 Due to the uniform well-possedness of the Stokes problems (53), S16CkhDGhk;Uh;PhÿDGk;Uh;Ph;v;qikV: Let us de®ne dhkRUh;vkRv;UhÿDvrqhDah;v;qi. Take w;r2V. Using estimate (52), hDGhk;Uh;Ph h ÿDGk;Uh;Ph;v;qi;w;riVÿV 6kShRv;w jÿRv;Rhrr;RhahjkShRUh;w jÿRUh;Rhrr;Rhdhj 6Ckh1=2kw;rkVjvj1kahkÿ1 jUhj1kdhkÿ16CkK2 1;0h1=2kw;rkVkv;qkV: Consequently, S16CkK2 1;0h1=2kv;qkV:62 To estimate S2, observe that, as mP1, hDGk;Uh;Ph;v;qi  kRUh;vkRv;Uh2 L2X  d: We may then apply the error estimates of [13, Section 6], and deduce S26ChkhDGk;Uh;Ph;v;qik06CkhkUhk2jvj1:63 Combining estimates (61)±(63), we deduce (c). (d) Take a pair v;q2V. Then, khDFhk;u1;p1ÿDFhk;u2;p2;v;qikV kThhDGhk;u1;p1ÿDGhk;u2;p2;v;qikV 6CkhDGhk;u1;p1ÿDGhk;u2;p2;v;qikV 6Ck1 h1=2kju1j1kju2j11ku1;p1ÿu2;p2kVkv;qkV: Here, the last inequality is deduced using estimate (52) after some lengthy, but elementary algebra. The term in h1=2is due to the appearance of terms of the form Ru;uin the expression of DGhk;u;p. Remark 15. Similarly to [29],we may obtain an estimate in Ohinstead of Oh1=2in (c).To do this,we use the regularity of all terms appearing in the Navier±Stokes operator,and the fact that if g2L2X d,then jRhgj16Chkgk0.Then,if g12Hÿ1X dand g22L2X d, jShRhg1;Rhg2j6ChjRhg1j1kg2k0: We may apply this with g2Rhahand g2Rhdhto obtain estimate (62) in Oh. However,we cannot extend Lemma 6 to the ``natural'' regularity u;p2 H1 0X  dL2 0X,as we need at least H2dH1regularity to obtain estimate (63).In fact,this regularity is needed to obtain a Finite Element interpolation error estimate of order Ohafor some positive a,and is not directly related to the obtention of error estimates for the discrete Stokes problem (53). T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 321 Following [29], from this Lemma we may conclude the main result of this section. Theorem 13. Assume that Navier±Stokes equations (47) admits a regular branch of solutions k2K! uk;pk2V;and that uk;pk2 Hm1X dHlXfor some integer numbers m;lP1. Assume that Hypothesis 1, 2, 3 and 4hold.Assume also that either msPm=4, or Yhand Zhare orthogonal with respect to the H1 0X  dinner product. Then,there exists a h0K>0such that for all h such that 0<h<h0K,a unique branch k2K!uh;k;ph;k2Vof solutions of problem (51) exists in a neighbourhood of uk;pk.Moreover,the following error estimate holds. kuk;pkÿuh;k;ph;kkV6CKhlhm: Excepting for some minor details, the proof of Theorem 13 is that of Theorem 4 of [29], and for that reason we shall omit it here. Remark 16. A similar analysis applies to approximate branches of regular solutions of all AM Methods considered in Sections 3 and 5: AM±AdS,AM±GaLS,Penalty AM Methods. Also,these results may be particularized to the standard Generalized SUPG,AdS and GaLS methods. 7. Stabilized post-processing of Mixed Method In this Section we shall develop a post-processing of the solution furnished by the Mixed-Method, to obtain a stable and more accurate solution. This will be based on a splitting of the velocity space Xhinto XhYhZh, where Zhwill not be, in general, a bubble Finite Element space. Our main technical tool is the uniform inverse triangular inequality (5). Let us recall at ®rst the formulation of the standard Mixed Method to solve the Navier±Stokes equations (2). Given two subspaces of ®nite dimension, Xhof Yand Mhof M, Obtain uh;ph2XhMhsuch that BNSuh;uh;ph;vh;qhhf;vhi8vh;qh2XhMh:64 This problem admits at least one solution which depends continuously on the data fif the family Xh;Mh fg h>0satis®es the discrete BB condition. This is a classical result (see [21]), which may be considered as a particular case of the Theory developed in Section 3. Let us recall the abstract process of ``static condensation'' of high frequencies (or small scales) originally developed in [5]. Consider a splitting of space Xhinto XhYhZh, where Yhand Zhare arbitrary subspaces of Xh. A solution uhof (64) is correspondingly decomposed into uhyhzhwhere yh2Yh,zh2Zh.Ifwe take vhth2Zh;qh0 in (64), we recover the following problem for zh: Su;zh;thhf;thiÿSuh;yh;thph;rthhfÿRuh;yhÿmDyhrph;thi; where Su;y;tbu;y;tÿmry;rt:Then, by De®nition 8, zhRhuh;fÿRuh;yhÿmDyhrph;65 where the static condensation operator Rhis calculated with respect to the bilineal form S. If now the expression (65) is re-injected in the Mixed Method (64), the pair yh;phturns out to be a solution of the linear stabilized problem Obtain yh;ph2YhMhsuch that Bhuh;yh;ph;vh;qhFhuh;vh;qh8vh;qh2YhMh;66 322 T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 where Bhu;y;p;v;qBNSu;y;p;v;qÿSu;Rhu;ÿurvÿmDvrq;Rhu;uryÿmDyrp; Fhu;v;qhf;viÿSu;Rhu;ÿurvÿmDvrq;Rhf:67 Problem (66) is the linearized version of the Augmented Mixed Method (9), when all inner products Shcoincide with S. From (66), yhis an approximation to the solution of the linear Navier±Stokes equations uryÿmDyrpf;ry0inX; y0onC;68 rather than an approximation to the Navier±Stokes equations (2). The essential observation to develop our post-processing is that, owing to the uniform inverse triangular inequality (5), we obtain the estimate jyhj1jzhj16C0juhj16C0 mkfkÿ1:69 From the characterization of zhgiven by (65), we may interpret this inequality in the sense that the high frequencies of the residual Ruh;yhÿÿmDyhrphÿfthat are representable in Zh± by means of static condensation by Rh± are uniformly bounded in h. This suggests a technique for post-processing the solution uhof Mixed Method (64) in order to recover a more stable solution: splitting Xhinto XhYhZh, where the family of spaces Yh;Zh fg h>0 satis®es the uniform inverse triangular inequality (UITI) (5), and retaining only the component yhon Yh of uh. This procedure is also likely to produce a still accurate solution, as yhis an approximation to problem (68). In [15] some techniques of splitting of Finite Element Spaces satisfying the UITI are developed. Basically, Yhmay be obtained from Xhby one of the following techniques: ·keeping the same kind of interpolation on a coarser grid, or ·interpolating with polynomials of lower degree on the same grid. Space Zhneed not to be constructed explicitely. Instead, one may construct a projection operator Phfrom Xhon to Yh. Indeed, if YhPhXh, then ZhIÿPhXh: This technique may be made local by eliminating only degrees of freedom located only on a preset region of the ¯ow. 8. Numerical tests We develop in this section two applications of the stabilized post-processing developed in Section 7. We show the good performances of this technique to stabilize highly oscillating numerical solutions of nonlinear ¯ow problems. 8.1. Burgers equations When applied to Burgers equations, an optimal version of the post-processing described in Section 7 provides not only stable, but very accurate numerical solutions. Let us consider the steady Burgers equation, uu;xÿmu;xx fin 0;1; u00;u11:70 In order to obtain the steady solution, we solve the unsteady Burgers equation, u;tuu;xÿmu;xx fin 0;1; T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 323 by a semi-implicit Euler scheme in time: 1 Dtun1unun1 ;xÿmun1 ;xx fn11 Dtun:71 We solve this equation by standard Galerkin Method with piecewise ane Finite Elements, on a grid of size h1=N;for a given integer number NP1. Let us denote by Xhsuch Finite Element space and X0hXh\H1 00;1. We thus solve at each time step the problem Obtain un1 h2Xhsuch that 8vh2X0h 1 Dtun1 h;vhun hun1 h;x;vhmun1 h;x;vh;xhf;vhi 1 Dtun h;vh; un1 h00;un1 h11: 72 As it is well known, at each time step this may provide a solution contamined by spurious oscillations, due to convection dominance, as soon as the grid P eclet number, Pehjun hjh m is large enough. We stabilize the above Galerkin solution of Burgers equation (72) by replacing at each time step the convection velocity un hby a stabilized projection ^ un hon a suitable subspace Yh. This space is that of piecewise ane Finite Elements on a grid of size 2h,YhX2h. Obviously, for this to be possible we need N2M, for some integer MP1. In its turn, space Zhis built up in order to optimize the post-processing for linear convection±diusion equations with constant data. The construction of this optimal post-processing follows in parallel that of SUPG Method. Concretely, consider the convection±diusion equation kwaw;xÿmw;xx 0in0;1; w00;w11;73 with constant kand a. This equation is solved by the Galerkin method on Xh. Then, one may prove that there exists a space Zhsuch that XhYhZhand such that if the Galerkin solution whis decomposed into whyhzh, then yx2iwx2i;06i6Mxjjh:74 In other words, there exists a stabilized component of the Galerkin solution which coincides with the exact solution on the nodes of the grid of size 2h. The basis functions of such a space Zhhave a minimal support of length 3h. Thus, Zhis not a bubble Finite Element space. Concretely, to each odd node x2iÿ1;16i6M, there is an associated basis function u2iÿ12Xh, given by u2iÿ1xj 1ifj2iÿ1; 1ÿha2iÿ1if j2iÿ2; 0 otherwise: 8 < : 75 The slopes a2iÿ1are calculated from the equalities (74). The slope a2iÿ1only depends on i;kh=m;and the grid P eclet number ah=m:This allows to extend this procedure to nonconstant coecients. To apply it to Eq. (71) it is enough to replace, for instance, aby an average value of unon x2iÿ2;x2i. It is this procedure that we have applied to stabilize the Galerkin solution of Burgers equation (72). The projection operator Ph:Xh!Yhis now calculated by hand. It can be computed with ONarithmetic operations. We have tested the above procedure for constant fx1=2and linear fxxright hand side in (70). We have taken N36;m10ÿ3:This provides an out¯ow grid P eclet number of 26, thus provoking a strong convection dominance eect. For both cases a steady state is readily attempted with a time step Dt10ÿ3: As ``exact'' solution, in order to compare the obtained results, we have used the numerical solution of Galerkin method with N20 36:Figs. 1±3 show the results obtained. Both cases present a sharp boundary layer close to x1. In both cases the post-processed solution yhcoincides with the ``exact'' one 324 T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 up to a precision of 10ÿ5at the even grid nodes. Thus, our post-processing not only provides a more stable, but also a very accurate solution. 8.2. Channel with obstacle This test was introduced in [11] to show the eect of convection dominance in the Galerkin solution of Navier±Stokes equations. The geometry of the ¯ow is a channel with an obstacle in one of the walls. For moderately high Reynolds numbers, the Galerkin solutions oscillates in an oblique strip between the in¯ow boundary and the obstacle (see Fig. 5). We shall consider the computational domain XR2;given by  X0;1:20;1[1:2;1:6 0:4;1[1:6;80;1. We decompose the boundary Cof Xas CC1[C2[C3with C1f0g 0;1;C20;1:2f0g[f1:2g0;0:4[1:2;1:6f0:4g[f1:6g0;0:4[1:6;8f0g[ 0;8f1g and C3f8g0;1. We shall consider the following problem: @u @turuÿmDurpf;ru0inX0;T; u1;0on C10;T; u0;0on C20;T; m@u @npn0on C30;T; ux;01;0on C1; ux;00;0in  X=C1: 76 Fig. 1. Stabilized post-processing of Burgers equation. Exact solution of (70) with m10ÿ3and f1=2 (solid line ); Galerkin-P1FE approximation (line marked by symbols) and ®ltered solution (line marked by * symbols). T. Chaco Ân, A. Domõ Ânguez / Comput. Methods Appl. Mech. Engrg. 182 (2000) 301±331 325