Postprocess Galerkin method for steady Navier-Stokes equations

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Postprocess Galerkin method for steady Navier-Stokes equations Citation for published version (APA): Hou, Y., & Li, K. (1999). Postprocess Galerkin method for steady Navier-Stokes equations.

1 Postprocess Galerkin method for steady Navier-Stokes equations Citation for published version (APA): Hou, Y., & Li, K. (1999). Postprocess Galerkin method for steady Navier-Stokes equations. (RANA : reports on applied and numerical analysis; Vol. 9913). Eindhoven: Technische Universiteit Eindhoven. Document status and date: Published: 01/01/1999 Document Version: Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 5fa of the Dutch Copyright Act, indicated by the Taverne license above, please follow below link for the End User Agreement: Take down policy If you believe that this document breaches copyright please contact us at: openaccess@tue.nl providing details and we will investigate your claim. Download date: 1. Apr. 019

2 Postprocess Galerkin Method for Steady Navier-Stokes Equations Yanren Hou y & Kaitai Li Department of Scientic Computing, Xi'an Jiaotong University, Xi'an , CHINA Abstract A kind of postprocess Galerkin method for steady Navier-Stokes equations with nonsingular solution, which is based on the virtue of Inertial Manifold and Approximate Inertial Manifold, is presented in this paper. Dierent from other construction of approximate inertial manifolds, we use a kind of so-called G; decomposition to get the large and small eddy components of the true solution. Its attractive advantage is that one could get the large eddy components exactly. Then the error of the scheme only comes from the approximation of the small eddy components. We prove that the proposed postprocess Galerkin scheme can greatly improve the convergence rate of the approximate solution with lower computing eort. And we also give annumerical example to verify our result. Key words. Navier-Stokes Equations, Galerkin method, convergence, nonsingular solution 1 Introduction To improve the convergence rate of the standard Galerkin approximate solutions, many authors derived new techniques and methods in last decade. For example, Lin Qun[1], W. Layton[], J. Xu[3] used extrapolation and two level meshes respectively. Especially, since 19 when the concepts of Inertial Manifold (IM)[4] and Approximate Inertial manifold (AIM)[5] for dissipative evolutionary partial dierential equation were given by C. Foias, G. R. Sell, R. Temam and C. Foias, O. Manley, R. Temam respectively, a kind of new numerical method called nonlinear Galerkin method is studied and developed by many authors, see [7], [] and [9], etc. This method is designed for numerically simulating the long time behavior of the solution and its main advantage is that the related numerical scheme has better stability and convergence rate than that of standard Galerkin method. The main ideal of this kind of inertial scheme is to nd some kind of interactive or approximate interactive rulesbetween large eddy and small eddy components. Based upon the virtue of IM and AIM, we apply this ideal to simulating the nonsingular solution of steady Navier-Stokes equations and give a kind of postprocess Galerkin procedure in this paper. We will show that it can greatly improve the convergence rate of standard Galerkin solution by using this procedure. Suppose that u H be a solution of Navier-Stokes equations, where H is a Hilbert space, H m H is a nite dimensional subspace, u m is the standard Galerkin solution, then Supported by NSF of CHINA ku ; u m kc ; 1 m+1 y Post-Doc of Eindhoven University oftechnology, Department of Mathematics and Computing Science, P.O.Box 513, 5600 MB Eindhoven, The Netherlands 1

3 and for general nonlinear Galerkin solution ~u m ku ; ~u m kc ;1 m+1 : The basic technique of our postprocess Galerkin method is to decompose the true solution u with respect to u m, that is u =u m +^u where ^u = u ; u m u m = u m : We identify u m and ^u with the large and small eddy components respectively. Then wecan exactly derive u m, the large eddy components of u, by standard Galerkin method. For convenience, we will use u m to denote u m directly. Under this presupposition, we construct an interactive rule between u m and ^u, thatisamappingfrom H m to H, suchthat(u m ) can generate a suitable approximation of ^u. Then we use the postprocess Galerkin solution u m = u m +(u m ) to approximate u and get Preliminary ku ; u mkc ; 3;" m+1 : Consider the Navier-Stokes equations in bounded domain R d, d = 3 < ;u +(ur)u + rp = F in divu =0 in : boundary conditions (1) where u :!R d is the ow eld, p :!Rthe pressure, F the exterior force which drives the ow and = Re 1, Re > 0 is the Reynolds number. and r denote the Laplace and gradient operator respectively. The boundary conditions can be either xed boundary of periodic boundary conditions ( is a cube under this circumstances). Now weintroduce a Hilbert space H. For xed boundary case, we take H = fu L () d u =0 divu = 0 under week sense g: And for periodic boundary case, we take P P H = fu = u k e ik(x) c k = c ;k ju k j < +1 kz k6=0 kz divu =0underweek sense g where (x) is a linear function of x related to the shape of. Denote by P : L () d! H the Leray orthogonal projection, we can obtain the abstract Navier-Stokes equation by projecting (1) onto H Au + B(u u) =f () where A = ;P, B(u u) =P [(u r)u], f = PF.Aswell known, A is a linear unbounded selfadjoint and positive denite operator which possesses bounded inverse A ;1, then its eigenvalues and related eigenfunctions are 0 < 1 m!+1 m N and 1 m :

4 And we denote V = D(A 1 ), the closure of H under the sense of H 1 endowed with the norm kvk = ja 1 1 vj where jvj =(v v) = R v vdx. For any s R, As is a power operator. Then the weak form of () reads nd u V such that a(u v)+b(u u v) =(f v) v V: (3) Here a(u v) =(ru rv) From [6], we know jb(u v w)j b(u u v) =((u r)u v): b(u w v) =;b(u v w) u v w V (4) c 1 kuk s1 kwk s+1kvk s3 u D(A s 1 ) v D(A s +1 ) w D(A s 3 ) where s 1 + s + s 3 d, s 1 s s 3 0 with (s 1 s s 3 ), 6= (0 0 d ), (0 d 0), ( d 0 0). If we denote by P m the L orthogonal projection operator from H onto H m = f 1 m g,the standard Galerkin approximate solution u m H m satises a(u m v)+b(u m u m v)=(f v) v H m : (6) Because of u m H m, the L error estimate can not exceed the L estimation of u m ; P m u.in fact ju ; u m jc ;1 m+1 ku ; u mkc ; 1 m+1 (7) where c>0 is a constant depending on f and. Moreover, we state the interpolation inequality in Sobolev spaces which will be used in the following: for v D(A 1 ), 0 " 1 3 G-Decomposition (5) kvk " c kvk " jvj 1;" : () From the point of view of L orthogonal projection P m, the approximate order of P m u ; u m is restricted by (7). Due to the restriction of truncate error, u m ; P m u has already been optimal. Alternatively, from other projection point of view, the results maybe quite dierent. As we said in introduction, we can decompose u according to u m intuitively u = u m +^u ^u = u ; u m : (9) Identifying u m H m and ^u V with large and small eddy components respectively, the standard Galerkin approximate solution u m reaches the large eddy components of the true solution exactly. According to the ideal of IM and AIM, there may exist an interactive ruleor at least some approximate interactive rules between large and small eddies. In this paper, we call (9) the Galerkin-decomposition of the true solution, simply the G- decomposition. Of course, when one considers the usual L orthogonal decomposition, the small eddy components under the sense of G-decomposition consists of small eddy as well as large eddy components. In order to understand this G-decomposition deeply, weintroduce the following formal nonlinear "orthogonal projection" Q m : V! V m,wherev m = P m V. 3

5 Restricting (3) on V m and subtracting (6) from it, we derive a(u ; u m v) +b(u ; u m u ; u m v)+b(u m u ; u m v) +b(u ; u m u m v)=0 v V m : (10) That is, ^u satises the above equation if we substitute ^u for u ; u m. It is a nonlinear equation of ^u. Enlightened by (10), we can formally dene the formal nonlinear "orthogonal projection" operator Q m from V onto V m < : Under this "projection", we have V nd Q m V m such that a( ; Q m v)+b( ; Q m ; Q m v) +b(u m ; Q m v)+b( ; Q m u m v)=0 v V m : (11) u m = Q m u ^u =(I ; Q m )u if we take = u in (11). And ^u is "orthogonal" with V m under the sense of (11). G- decomposition of u is just the result of this kind of "projection". As to whether (11) can really dene a "orthogonal projection" or not, it is unnecessary for our discussion in the rest. On the other hand, ^u should be a small quantity compared with u m if m is large enough and we could get another formal linear "projection" by omitting the higher order small item b( ; Q m ; Q m v) in (11). In fact, we ever studied this kind of projection elsewhere in detail. The disadvantage is that the related algorithm is very hard to be completed in computing. On the contrary, our algorithm developed in this paper by using the formal "projection" Q m can overcome this disadvantage and obtain suciently high convergence rate. Now based upon G-decomposition of u, the Navier-Stokes equations (3) can be rewritten as a(^u v) +b(^u ^u v)+b(^u u m v)+b(u m ^u v) +a(u m v)+b(u m u m v)=(f v) v V: (1) Obviously,we could obtain the small eddy components of u under the sense of G-decomposition by solving (1). But this procedure is as dicult as to solving (3) when we consider the complex form of (1). 4 Construction of Finite Dimensional Mapping As we said in introduction, the aim of this paper is to construct a nite dimensional mapping : V m! V such that it can form a suitable approximate interactive rulebetween the large and small eddy components of the solution. After we get the large eddy components u m by standard Galerkin method, we could use it to obtain (u m )which should be a high accuracy approximation of ^u. We expect that u m +(u m ) can generate a much better approximation of u than both standard Galerkin and nonlinear Galerkin approximation. We call this procedure the postprocess Galerkin procedure. Now, the key problem here is how to construct such that the computation of (u m ) is simple and can derive a high accuracy postprocess tool. Of course, we hope to be a linear mapping. Based on the above consideration and noticing (1), we will use the linear part of (1) to determine, that is < : V m nd () V such that a(() v)+b(() u m v)+b(u m () v) a( v)+b( v)=(f v) v V: (13) 4

6 To investigate (13) more deeply, let us recall the Navier-Stokes equations (3) and its standard Galerkin approximation (6). Indeed, they dene the following two mappings. That is < : F : V! V V F( ) V such that < F( ) v >= a( v)+b( v) ; (f v) v V and < : Then, (3) and (6) equivalent to F m : V m! Vm V m F m () Vm such that < F m () v >= a( v)+b( v) ; (f v) v V m : F(u) =0 F m (u m )=0: (14) Denoting by DF(u) and DF m (u m ) the Frechet derivatives of F and F m at points u and u m, we get the following two linear mappings from V to V and V m to V m >< >: <DF(u)w v >= a(w v)+b(w u v)+b(u w v) w v V <DF m (u m )w v >= a(w v)+b(w u m v)+b(u m w v) w v V m : In the rest, for the sake of simplicity, we employ the following symbols (15) L(w v) =<DF(u)w v > L m (w v) =<DF m (u m )w v>: And we easily know that L m (w v) =L(w v) ; b(^u w v) ; b(w ^u v): (16) Then (13) can be rewritten as Vm nd () V such that L m (() v)+ < F() v >=0 v V: (17) Generally, for arbitrary V m, whether (17) can determine an unique () or not is uncertain. At present time,we will only discuss the case in which (17) can generate a single valued mapping from V m! V.For this purpose, we need following assumptions and properties. First of all, we assume that u is a nonsingular solution of (3). For singular case, we will treat it in the future. Then, DF(u) is an isomorphism between V and V (see [10],[11]) and there must be some constant 0 > 0 such that inf sup L(w v) wv vv kwkkvk 0 (1) inf sup vv wv L(w v) kwkkvk 0 (19) The next lemma describes under what conditions could the standard Galerkin approximate solution u m be also a nonsingular solution. Lemma 4.1 Assume V m V is a nite dimensional subspace and F m is a smooth mapping from V m to V m. Let u be a nonsingular point off and denote (u) =kdf(u) ;1 k L(V V ) 5

7 If u m is close to u so much such that (u m )=kdf(u) ; DF m (u m )k L(V V ): (u)(u m ) < 1 (0) DF m (u m ) is an isomorphism between V m and V m. Hence, u m is a nonsingular point off m. Proof See [1]. Corollary 4.1 Suppose all conditions in lemma 4.1 hold. Take m large enough such that then u m is a nonsingular point off m. Proof From (1)(19), we know m+1 4c 1 c (1) 0 On the other hand, (u) =kdf ;1 (u)k L(V V ) ;1 0 : (u m )= kdf(u) ; DF m (u m )k L(V V ) = sup w vv < (DF(u) ; DF m (u m ))w v > kwkkvk c 1 ku ; u m kcc 1 ; 1 m+1 : Thanks to lemma 4.1, we could get the results. As a result of lemma 4.1 and its corollary, it holds that inf wv m inf vv m Further more, by using (16), (1) and (19), we have L m (w v) sup vv m kwkkvk 0 () L m (w v) sup wv m kwkkvk 0 : (3) inf sup L m (w v) wv vv kwkkvk 0 (4) inf sup L m (w v) vv wv kwkkvk 0 : (5) Now let us consider (17). For any given V m,weknow F() V. At the same time, (4)(5) show that the bilinear form L m ( ) isweak coercive onv V. Then, we can obtain the following uniqueness theorem of (17) by using the generalized Lax-Milgram theorem. Theorem 4.1 Suppose that u is a nonsingular solution of (3) and m is large enough such that (1) holds. Then, (17) is well-posed. Therefore, (17) can dene a nite dimensional mapping from V m to V. Moreover, is a local Lipschitz mapping, that is, for any bounded subset B V m with B = f V m : kk <g where 0 <<+1 is any xed constant there exists a positive constant l > 0 such that k( 1 ) ; ( )kl k 1 ; k 1 V m : Proof The proof is quite simple and we omit it. 6

8 5 Postprocess Galerkin Procedure In this section, we will use the nite dimensional mapping derived in section 4 to construct the postprocess Galerkin procedure for numerically solving the Navier-Stokes equations. As the result of this kind of so called G-decomposition of the true solution, u can be decomposed as the sum of large eddy and small eddy components, that is u = u m +^u where u m is the large eddy components of u. The main ideal of our postprocess is to obtain a suitable approximation of ^u by using u m andwhich is considered as an approximate interactive rule between above two parts. The theoretical basis of this procedure is the concepts of IM and AIM which suppose that there at least exist some approximate interactive rules between the large and small eddy components of the solution of some dissipative systems, in our case, they are u m and ^u. After we getu m,wewant to use (u m )toapproximate ^u and then use u m = u m +(u m ) to approximate u. We will show thatu m can approximate u much better than either u m or ~u m.we call it postprocess Galerkin procedure in this paper and we write it in following three steps. (Step 1) nd um V m such that a(u m v)+b(u m u m v)=(f v) v V m (Step ) for um V m nd (u m ) V such that L m ((u m ) v)+ < F(u m ) v >=0 v V (Step 3) u m = u m +(u m ): Following theorem describes the convergence rate of this procedure. Theorem 5.1 Assume u is the nonsingular solution of (3), m is large enough such that (1) is valid. Then the postprocess Galerkin solution u m has the following convergence rate where " = 0 for d =," = 1 for d = 3 and c 3 = c c 1 c. ku m ; uk c 3 ; 3;" m+1 (6) 0 Proof Because m is large enough such that (1) is valid, u m is also a nonsingular solution of (6) because of lemma 4.1 and its corollary. As we know, Therefore u = u m +^u u m = u m +(u m ): u m ; u =(u m ) ; ^u 4 = e: (7) Our main task is to estimate kek. For convenience, the equation of (Step ) can be rewritten as a((u m ) v)+ b((u m ) u m v)+b(u m (u m ) v) () + a(u m v)+b(u m u m v) ; (f v) =0 v V: Now, subtract (1) from (), it yields a(e v)+b(e u m v)+b(u m e v) =b(^u ^u v): 7

9 Alternatively, L m (e v) =b(^u ^u v): (9) Since u m is the nonsingular solution of (6) and (4)(5) are valid, we have sup vv L m (e v) kvk 0 kek: (30) On the other hand, by using (4)(5) and (), we know ( c1 k^uk 1 kvk c 1 c j^ujk^ukkvk d = b(^u ^u v) c 1 k^uk 3 kvk c 1 c j^uj 1 k^uk 3 kvk d =3 4 Combining (30), (31) and (7), we have 0 3;" kek c 1c j^uj 1;" k^uk 1+" c 1 c c ; m+1 c 3 ; 3;" m+1 where " = 0 for d =," = 1 for d = 3 and c 3 = c c 1 c. In practical manipulating, (Step ) of the postprocess procedure must be completed in nite dimensional subspace of V. Generally, wechoose another constant M N such thatm m and restrict (Step ) on V M. That is, we alternate (Step ) with (Step ') and alternate (Step 3) with for um V m nd M (u m ) V M such that L m ( M (u m ) v)+ < F(u m ) v >=0 v V M : (Step 3') u mm = u m + M (u m ): Corollary 5.1 Under the assumption of theorem 5.1, we take M N large enough, then (31) ku mm ; uk c 3 ; 3;" + 1 m+1 c; M+1 : (3) 0 Proof To prove this corollary, we only need to notice M is large enough and where u M is the standard Galerkin solution of u mm ; u = u mm ; u M + u M ; u a(u M v)+b(u M u M v)=(f v) v V M : (33) Then the remainder of the proof is very simple and we omit it. Remark 1. Noticing (3), to balance the two items on the right hand side, we should take 1 1 3;" M>msuch that m+1 M+1, that is, m M 3;". Thus, the computing scale of (6) is much smaller than that of (33). Remark. Observing the postprocess procedure (Step 1), (Step' ) and (Step' 3), the computational complexity mainly comes from (Step 1) which tends to solve a nonlinear system by some kind of iterative methods. Comparing with (Step 1), the computing of (Step' ) is quite simple though the scale of the equation is larger than that of (Step 1), because it is a linear system and only need solving one time during the whole procedure. On the other hand, to get a standard Galerkin approximation u M which has the same accuracy as u mm,wemust solve (33) by some iterative method for M m. As well known, (33) is harder and need much more iterative steps to convergence than (Step 1) since M m. At the same time, the computing eort in (Step' ) is almost the same as that of one iterative step of (33). Obviously, the postprocess method presented in this paper is much cheaper to complete than standard Galerkin method.

10 Re SGM (ku m ; uk) PGM (ku mm ; uk) (m =9) (M =m) E E E E E E E-07.95E E-0.917E E E E-0.770E E E E E E E E E E E E Numerical Test To shed light on the better convergence rate of the postprocess Galerkin method proposed in section 5, we will give some numerical tests by taking example of two-dimensional Kolmogorov ows on rectangle : = [; ] [; ], where V = f = Find u V such that a(u v)+b(u u v) =(f v) P P kz k6=0 kz k6=0 v V k e i(k1x+ky) k = ;k j k j (k 1 + k ) < +1 div =0g (34) f = 1 Re (sin y 0)T, Re > 0 is the Reynolds number dened in [13]. From the literature of Kolmogorov ows (e.g. [13], [14]), the above equations has a trivial solution u =(siny 0) T for any Reynolds number, and when we take =0:7, the bifurcation occurs at Re =3: From (3), we know ku mm ; uk c 3 ; m+1 c; M+1 : (35) 0 As we said in remark 1, we should take M m 3. On the other hand, we notice that the exterior force of the Kolmogorov ows has only two modesaswell as the trivial solution has only two modes, and for any w (I ; P m )V, b(u m u m w) = 0. That is, for M>m and some suitable iterative initial value, the solutions of standard Galerkin method (SGM) and the postprocess Galerkin method (PGM) will keep unchanged with the growth of M. Thus, for this concrete test model, we take M =m. Following table gives the comparison of the errors of SGM solutions and PGM solutions for dierent Re which is not equal to Re with =0:7. Just as we discussed in previous sections, our postprocess Galerkin method in this paper is designed for treating the nonsingular case, that is Re 6= Re. For singular case, we will deal with it later. 9

11 When Re is very close to Re,we suppose PGM will lose its higher convergence rate because 0 in (1)(19) will tend to zero. From the numerical results corresponding to Re =3:01 and Re =3:0 which are quite close to Re, PGM has almost the same convergence rate as SGM. That is, PGM will lose its higher convergence rate near singular point. When Re is far from Re, the results show PGM can improve the convergence rate of SGM at about ten times. References [1] Lin Qun, Higher Order Accuracy Finite Element Method, Hebei University Press, [] W. Layton, W. Lenferink, Two-Level Picad, Defect Correction for the Navier-Stokes Equations, Appl. Math. and Computing, 0(1995), 1-1. [3] J. Xu, A Novel Two-grad Method for Semilinear Elliptic Equations, SIAM, J. Sci. Comput., 15(1994), 31{37. [4] C. Foias, G. R. Sell & R. Temam, Inertial Manifolds for Nonlinear Evolutionary Equations, Journal of Dierential Equations, 73(19), 309{353. [5] C. Foias, O. Manley & R. Temam, On the Interaction of Small Eddies in Two-Dimensional Turbulence Flows, Math. Modeling and Numerical Analysis, M AN, (19), 93{114. [6] R. Temam, Navier-Stokes Equation and Nonlinear Functional Analysis, CBNS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 193. [7] M. Marion & R. Temam, Nonlinear Galerkin Methods, SIAM J. Numer. Anal. 6(199), 1139{1157. [] A. Debussche& R. Temam, Convergent Family of Approximate Inertial Manifolds, J. Math. Pures Appl., 73(1994), 49{5. [9] Li Kaitai & Hou Yanren, Fourier Nonlinear Galerkin Method for Navier-Stokes Equations, Discrete and Continuous Dynamical Systems, ()1996, 497{54. [10] M. Gunzburger Finite Element Methods for Viscous Incompressible Flow, A Guide to Theory, Practice and Algorithms, Academic Press, Boston (199). [11] F. Bezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Verlag, New York (1991). [1] V. Girault, P-A. Raviart, Finite Element Methods for the Navier-Stokes Equations, Theory and Algorithms, Springer Verlag, Berlin (196). [13] H. Okamoto, Nearly Singular Two-Dimensional Kolmogorov Flows for Large Reynolds Numbers, J. Dynamics and Dierential Equations, (1996), 03{0. [14] H. Okamoto, M. Shoji, Bifurcation Diagrams in the Problem of Incompressible Viscous Fluid Flows in D Tori, Japan J. Indus. Appl. Math., 10(1993), 191{1. 10

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