A study of Nonlinear Galerkin Finite Element for time-dependent incompressible Navier-Stokes equation
aa r X i v : . [ m a t h . NA ] J un A study of Nonlinear Galerkin Finite Element for time-dependentincompressible Navier-Stokes equations
Deepjyoti Goswami ∗ Abstract
In this article, we discuss a couple of nonlinear Galerkin methods (NLGM) in finite elementset up for time dependent incompressible Navier-Sotkes equations. We show the crucial roleplayed by the non-linear term in determining the rate of convergence of the methods. We haveobtained improved error estimate in L norm, which is optimal in nature, for linear finite elementapproximation, in view of the error estimate available in literature, in H norm. Key Words . nonlinear Galerkin method, Navier-Stokes equations, optimal error estimates.
In the study of dynamical systems, generated by evolution partial differential equations, we lookinto long time behavior of the solutions. In certain cases, solutions converge asymptotically toa compact set called global attractor. Owing to complicated structures of such sets, notion ofinertial manifold (IM) was introduced, which is a smooth finite dimensional manifold, attractingall the solution trajectories exponentially. Once we can prove the existence of an IM for a evolutionequation, it is far easier to study it than to study the global attractor. Unfortunately, the existenceof an IM for Navier-Stokes equations could not be established, even for a simpler case like 2 D spatially periodic flow. Hence came the concept of artificial inertial manifold (AIM), a sequenceof smooth and finite dimensional manifolds of increasing dimensions and that the global attractorlies in a small neighborhood of each such manifold with the distance vanishing exponentially asthe dimension increases. AIM has been shown to exist for the Navier-Stokes equations in 2 D . Anumerical technique based on AIM was introduced by Marion and Temam in [12] ans was coinednon-linear Galerkin method (NLGM). The method involves splitting the solution of a dissipativesystem into large and small scales, simplifying the small scale equation, thereby obtaining smallscale in terms of large scale, relatively easily. In other words, small scales are made slaves to largescales. This technique and its modifications are studied in depth in early nineties. For details anda history of developments, we refer to [2, 15].NLGM was originally developed in the context of spectral Galerkin approximations. Themethod in [12] was based on the eigenvectors of the underlying linear elliptic operator. Laterin [13], it was expanded to more general bases and more specifically to finite elements. But fornon-spectral Galerkin discretizations, very few results are available. The problem was that theextension to such cases is not natural, since the splitting of discrete solution space into large andsmall scales (or low and high frequency modes) is not obvious any longer. Marion and Temam [13]developed various NLGMs based on finite element and later on, Marion and Xu [14] and, Ammi ∗ Department of Mathematical Sciences, School of Sciences, Tezpur University, Napaam, Sonitpur, Assam, India-784028. E-mail: [email protected] H -norm for a NLGM type two-level finite element method. There,the strong orthogonality properties of spectral Galerkin approximation was substituted by weaker L -orthogonality of finite element approximation. In [14], Marion and Xu considered a semi-linearevolution equation and showed that the H -error estimate to be of O ( H ) (here H is a space dis-cretizing parameter). They expected L -error estimate to be of O ( H ) and remarked that this isan open problem . We prove this higher order estimate in a couple of cases and believe that it cannot be extended to other possible cases.The advent of NLGMs created a lot of interest, since it outperformed the standard Galerkinmethod, which is restricted to the large scales only. Also, it was thought of as well-suited forturbulence modeling. But Heywood and Rannacher [10] argued that it is not turbulence modelingthat is responsible for the better performance of NLGM over Galerkin method. On the other hand,it is the ability of NLGM to handle better, the Gibb phenomenon induced by higher order boundaryincompatibilities induced by no-slip boundary condition. They substantiated it by showing that,in periodic domain, both perform identically. Later, Guermond and Prudhomme, in [2], revisitedNLGM and showed that, in case of arbitrary smoothness, NLGM will always outperform Galerkinmethod. And it has nothing to do with turbulence modeling, but due to the well-known fact ofsuperconvergence of elliptic projection in H -norm. We have also exploited this simple fact toimprove the existing L -error estimate.The incompressible time dependent Navier-Stokes equations are given by ∂ u ∂t + u · ∇ u − ν ∆ u + ∇ p = f ( x, t ) , x ∈ Ω , t > ∇ · u = 0 , x ∈ Ω , t > , (1.2)and initial and boundary conditions u ( x,
0) = u in Ω , u = 0 , on ∂ Ω , t ≥ . (1.3)Here, Ω is a bounded domain in R with boundary ∂ Ω and ν is the inverse of the Reynolds number. u and p stand for the velocity and the pressure of a fluid occupying Ω, respectively. Initial velocity u is a solenoidal vector field and f is the forcing term.Error estimations of NLGM in mixed finite element set up for Navier-Stokes’ equations arecarried out in [1]. The small scales equations carry both nonlinearity and time dependence and thefollowing estimates were established. k ( u h − u h )( t ) k H (Ω) ≤ c ( t ) H , k ( p h − p h )( t ) k L (Ω) ≤ c ( t ) H , where ( u h , p h ) is the Galerkin approximation and ( u h , p h ) is the nonlinear Galerkin approximation.These results were improved to O ( H ) and similar result was obtained in L -norm for velocityapproximation, but for semilinear parabolic problems, by Marion and Xu, in [14]. We feel that itis not straight forward to carry forward these results to Navier-Stokes equations. The proof of [14]depends on one important estimate involving the nonlinear term f ( u ), namely | ( f ( y h + z h ) − f ( y h ) , χ ) | ≤ k z h k L (Ω) k χ k L (Ω) , where y h and z h are the large and small scales of u h , and χ is any element from the appropriatespace containing small scales, see [14, pp. 1176]. But it may not be possible to establish similarestimate for the nonlinear term ( u · ∇ ) u . On the other hand, it is observed that the improvement2n the order of convergence is due to the fact that the splitting in space is done on the basis of L projections, unlike previous approaches, where the splitting is based on hierarchical basis. Similarapproach is adopted in this article.No further improvement is observed for (piecewise) linear finite element discretization and withforcing term f ∈ L (Ω). Results, similar to the ones mentioned above, are observed in [3, 4], but for f ∈ H (Ω) and for Navier-Stokes equations. In [15], various nonlinear Galerkin finite elements arestudied in depth for one-dimensional problems and similar results are obtained, although termed asoptimal in nature. For example, the method is studied for piecewise polynomials of degree 2 n − L -norm are as follows k ( u h − u h )( t ) k H n (Ω) ≤ c ( t ) H n − , k ( u h − u h )( t ) k L (Ω) ≤ c ( t ) min { t − / H n − , H n/ − } , respectively. For n = 1, we have the piecewise linear finite element approximation and errorestimates are of O ( H ) and O ( H / ), in energy-norm and L -norm, respectively. Later on He et.al have studied NLGM and modified NLGM in both spectral and finite element set ups, [5, 6, 8, 7]to name a few. In these articles, fully discrete NLGMs were considered and were shown to exhibitbetter convergence rate than Galerkin method. But none were optimal.Recently in [11], a new projection is employed for a two-level finite element for Navier-Stokesequations. Error estimates of O ((log h ) / H ) and O ((log h ) / H ), in energy-norm and L -norm,respectively, are obtained. Logarithmic term appears due to the use of the finite dimensional caseof the Brezis-Gallouet inequality. Note that here the forcing term is taken in L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)).In this article, we have established optimal L -error estimate for a couple of NLGMs, whilethe forcing term is in L . The approach here is an heuristic one and we basically highlight theimportance of nonlinearity of the small scales equations as the key to the higher or lower ordererror estimate. Since these methods can be considered as two-level methods, this suggests ways ofconstructions or limitations of two or multi-level methods for lower-order approximations.The article is organized as follows. In section 2, we briefly recall the notion of a suitable weaksolution of the Navier-Stokes equations. Section 3 deals with the Galerkin approximation and statethe error estimates. In Section 4, we introduce the NLGMs. Error estimates for the methods arediscussed in Section 5. Finally, in Chapter 6, we summarize the results obtained in this article. For our subsequent use, we denote by bold face letters the R -valued function space such as H = [ H (Ω)] , L = [ L (Ω)] . Note that H is equipped with a norm k∇ v k = (cid:0) X i,j =1 ( ∂ j v i , ∂ j v i ) (cid:1) / = (cid:0) X i =1 ( ∇ v i , ∇ v i ) (cid:1) / . Further, we introduce divergence free function spaces: J = { φ ∈ H : ∇ · φ = 0 } J = { φ ∈ L : ∇ · φ = 0 in Ω , φ · n | ∂ Ω = 0 holds weakly } , n is the outward normal to the boundary ∂ Ω and φ · n | ∂ Ω = 0 should be understood in thesense of trace in H − / ( ∂ Ω), see [16]. For any Banach space X , let L p (0 , T ; X ) denote the space ofmeasurable X -valued functions φ on (0 , T ) such that Z T k φ ( t ) k pX dt < ∞ if 1 ≤ p < ∞ , and for p = ∞ ess sup
From now on, we denote h with 0 < h < H h and L h , 0 < h < H and L / R , respectively, approximating velocity vector and the pressure. Assume that the followingapproximation properties are satisfied for the spaces H h and L h :( B1 ) For each w ∈ H ∩ H and q ∈ H / R there exist approximations i h w ∈ H h and j h q ∈ L h such that k w − i h w k + h k∇ ( w − i h w ) k ≤ K h k w k , k q − j h q k ≤ K h k q k . Further, suppose that the following inverse hypothesis holds for w h ∈ H h : k∇ w h k ≤ K h − k w h k . (3.1) 4or defining the Galerkin approximations, set for v , w , φ ∈ H , a ( v , φ ) = ( ∇ v , ∇ φ ) , b ( v , w , φ ) = 12 ( v · ∇ w , φ ) −
12 ( v · ∇ φ , w ) . Note that the operator b ( · , · , · ) preserves the antisymmetric property of the original nonlinear term,that is, b ( v h , w h , w h ) = 0 ∀ v h , w h ∈ H h . The discrete analogue of the weak formulation (2.1) now reads as: Find u h ( t ) ∈ H h and p h ( t ) ∈ L h such that u h (0) = u h and for t > (cid:26) ( u ht , φ h ) + νa ( u h , φ h ) + b ( u h , u h , φ h ) − ( p h , ∇ · φ h ) = ( f , φ h ) , φ h ∈ H h ( ∇ · u h , χ h ) = 0 , χ h ∈ L h . Here u h ∈ H h is a suitable approximation of u ∈ J . For continuous dependence of the discretepressure p h ( t ) ∈ L h on the discrete velocity u h ( t ) ∈ J h , we assume the following discrete inf-sup(LBB) condition for the finite dimensional spaces H h and L h :( B2 ′ ) For every q h ∈ L h , there exists a non-trivial function φ h ∈ H h and a positive constant K , independent of h , such that | ( q h , ∇ · φ h ) | ≥ K k∇ φ h kk q h k . In order to consider a discrete space, analogous to J , we impose the discrete incompressibilitycondition on H h and call it as J h . Thus, we define J h , as J h = { v h ∈ H h : ( χ h , ∇ · v h ) = 0 ∀ χ h ∈ L h } . Note that J h is not a subspace of J . With J h as above, we now introduce an equivalent Galerkinformulation as: Find u h ( t ) ∈ J h such that u h (0) = u h and for φ h ∈ J h , t > u ht , φ h ) + νa ( u h , φ h ) = − b ( u h , u h , φ h ) + ( f , φ h ) . Since J h is finite dimensional, the problem (3.3) leads to a system of nonlinear differential equations.For global existence of a unique solution of (3.3) (or unique solution pair of (3.2)), we refer to [9].We further assume that the following approximation property holds true for J h .( B2 ) For every w ∈ J ∩ H , there exists an approximation r h w ∈ J h such that k w − r h w k + h k∇ ( w − r h w ) k ≤ K h k w k . The L projection P h : L J h satisfies the following properties : for φ ∈ J h ,(3.4) k φ − P h φ k + h k∇ P h φ k ≤ Ch k∇ φ k , and for φ ∈ J ∩ H , (3.5) k φ − P h φ k + h k∇ ( φ − P h φ ) k ≤ Ch k ˜∆ φ k . With the definition of the discrete operator ∆ h : H h H h through the bilinear form a ( · , · ), a ( v h , φ h ) = ( − ∆ h v h , φ ) ∀ v h , φ h ∈ H h , (3.6)we set the discrete analogue of the Stokes operator ˜∆ = P ( − ∆) as ˜∆ h = P h ( − ∆ h ). Using Sobolevimbedding and Sobolev inequality, it is easy to prove the following Lemma5 emma 3.1. Suppose conditions ( A1 ), ( B1 ) and ( B2 ) are satisfied. Then there exists a positiveconstant K such that for v , w , φ ∈ H h , the following holds: (3.7) | ( v · ∇ w , φ ) | ≤ K k v k / k∇ v k / k∇ w k / k ∆ h w k / k φ k , k v k / k ∆ h v k / k∇ w kk φ k , k v k / k∇ v k / k∇ w kk φ k / k∇ φ k / , k v kk∇ w kk φ k / k ∆ h φ k / , k v kk∇ w k / k ∆ h w k / k φ k / k∇ φ k / The following lemma and theorem present a priori estimates of the semi-discrete solution andoptimal error estimate, respectively.
Lemma 3.2.
Let < α < νλ , where λ > is the smallest eigenvalue of the Stokes’ operator. Letthe assumptions ( A1 ),( A2 ),( B1 ) and ( B2 ) hold. Then the semi-discrete Galerkin approximation u h of the velocity u satisfies, for t > , k u h ( t ) k + e − αt Z t e αs k∇ u h ( t ) k ds ≤ K, (3.8) k∇ u h ( t ) k + e − αt Z t e αs k ˜∆ h u h ( s ) k ds ≤ K, (3.9) ( τ ∗ ( t )) / k ˜∆ h u h ( t ) k ≤ K, (3.10) where τ ∗ ( t ) = min { t, } . The positive constant K depends only on the given data. In particular, K is independent of h and t . Theorem 3.1.
Let Ω be a convex polygon and let the conditions ( A1 )-( A2 ) and ( B1 )-( B2 ) besatisfied. Further, let the discrete initial velocity u h ∈ J h with u h = P h u , where u ∈ J . Then,there exists a positive constant C , that depends only on the given data and the domain Ω , such thatfor < T < ∞ with t ∈ (0 , T ] k ( u − u h )( t ) k + h k∇ ( u − u h )( t ) k ≤ Ce Ct h t − / . A detail account of the finite element spaces, estimates of the semi-discrete solutions and errorestimates can be found in [9].
Remark 3.1.
From numerical point of view, it is a standard practice to work with mixed formu-lation (3.2) rather than (3.3). But following [9], we prefer to base our analysis on the divergencefree formulation (3.3). The same analysis can easily be carried forward to mixed formulation. It isjust a matter of taste.
In this section, we introduce another space discretizing parameter H such that 0 < h << H andboth h, H tend to 0. And based on that, we split finite element space J h into two. J h = J H + J Hh , with J Hh = ( I − P H ) J h (4.1)Note that, by definition, the spaces J H and J Hh are orthogonal with respect to the L -inner product( · , · ). In practice, J H corresponds to a coarse grid and J Hh corresponds to a fine grid. In case ofmixed method, we only split the velocity space and pressure space remains the same.6he following properties are crucial for our error estimates. For a proof, we refer to [1]. k χ k ≤ cH k χ k , χ ∈ J Hh . (4.2)And there exists 0 < ρ < h and H such that(4.3) | a ( φ , χ ) | ≤ (1 − ρ ) k φ k k χ h k , φ ∈ J H , χ ∈ J Hh . From (4.3), we can easily deduce that(4.4) ρ {k φ k + k χ k } ≤ k φ + χ h k , φ ∈ J H , χ ∈ J Hh . In nonlinear Galerkin methods, we look for a solution u h ( ∈ J h ) in terms of its components y H (coarse grid) and z h (fine grid). u h = y H + z h ∈ J H + J Hh . In the first of the two methods (NLGM I), for t > t >
0, we look for a pair ( y H , z h ) satisfying(4.5) (cid:26) ( y Ht , φ ) + νa ( u h , φ ) + b ( u h , u h , φ ) = ( f , φ ) , φ ∈ J H ,νa ( u h , χ ) + b ( u h , u h , χ ) = ( f , χ ) , χ ∈ J Hh . In the second one (NLGM II), we again look for a pair ( y H , z h ) satisfying, for t > t > (cid:26) ( y Ht , φ ) + νa ( u h , φ ) + b ( u h , u h , φ ) = ( f , φ ) , φ ∈ J H ,νa ( u h , χ ) + b ( u h , y H , χ ) + b ( y H , z h , χ ) = ( f , χ ) , χ ∈ J Hh . We set y H ( t ) = P H u h ( t ) . Note that the two systems differ only by the term b ( z h , z h , χ ), whichhas been dropped from the second system, assuming z h is small. Remark 4.1.
Here and henceforth, subscript means the classical Galerkin method and superscriptmeans the nonlinear Galerkin method.
Remark 4.2.
The nonlinear Galerkin approximations are carried out away from t = 0 . We canemploy Galerkin approximation to obtain ( u h , p h ) on the interval (0 , t ] . This is done to avoidnonlocal compatibility conditions. In [9], Heywood and Rannacher showed that to assume higherregularity for the solution demands some nonlocal compatibility conditions to be satisfied by initialvelocity and initial pressure. These conditions are very difficult to verify and do not arise in physicalcontext. In order to avoid them, we must admit singularity of velocity field in a higher order normat t = 0 , like (1.6) from [9]. Since the error analysis of NLGM demands higher regularity of thevelocity and this means higher singularity at t = 0 , the idea is to avoid these kinds of singularityby staying away from t = 0 . Remark 4.3.
Both the NLGs can be heuristically derived from (3.3) as follows. We split theGalerkin approximation u h with the help of the L projection P H . (4.7) u h = P H u h + ( I − P H ) u h = y H + z h , And we project the system (3.3) on J H , J Hh to obtain the coupled system: (4.8) (cid:26) ( y ht , φ )+ νa ( u h , φ ) = − b ( u h , u h , φ ) + ( f , φ ) , ( z ht , χ )+ νa ( u h , χ ) = − b ( u h , u h , χ ) + ( f , χ ) , for φ ∈ J H , χ ∈ J Hh . Assuming the time derivative and higher space derivatives of z h are small,various (modified) NLG methods are defined. In case, time derivative of z h is retained in theequation, different time-steps can be employed for the two equations; (much) smaller time-step forthe equation involving y H , thereby keeping z h remains steady. et. al. [1, 13, 14]. For the sake of completeness, we present below, the a priori estimates for theapproximate solution pair { y H , z h } . And for the sake of brevity, we only sketch a proof. Lemma 4.1.
Under the assumptions of Lemma 3.2, the solution pair ( y H , z h ) of (4.8) or (4.6)satisfy the following estimates, for t > t (4.9) k y H k + e − αt Z tt e αs k∇ ( y H + z h ) k ds ≤ K. And if H is small enough to satisfy (4.10) ν − cL H k y H k > , where L H ∼ | log H | / , then the following estimate holds (4.11) k z h k ≤ K. The constant
K > depends only on the given data. In particular, K is independent of h, H and t .Proof. Choose φ = y H , χ = z h in (4.8) or (4.6) and add the resulting equations. Note thatthe nonlinear terms sum up to 0. Multiply by e αt . Use Poincar´e inequality and then kickbackargument. Finally, integrate from t to t and multiply by e − αt to complete the first estimate. Forthe second estimate, we put χ = z h in the second equation of (4.8) or (4.6). ν k∇ z h k ≤ k f kk z h k + ν k∇ y H kk∇ z h k + | b ( y H + z h , y H , z h ) | . (4.12)Using (3.1) and (4.2), we find b ( y H , y H , z h ) ≤ / k y H k / k∇ y H k / k z h k / k∇ z h k / ≤ c k y H kk∇ y H kk∇ z h k b ( z h , y H , z h ) ≤ k z h kk∇ z h kk y H k ∞ ≤ cL H k z h kk∇ z h kk∇ y H k We have used the finite dimensional case of Brezis-Gallouet inequality (see [11, (3.12)]) k u h k ∞ ≤ cL h k∇ u h k ; L h ∼ | log h | / . Incorporate these estimates in (4.12). With the re-use of (4.2), we have ν k∇ z h k ≤ cH k f k + ν k∇ y H k + c k y H kk∇ y H k + cL H k z h kk∇ y H k . Again use (3.1), we find ν k z h k ≤ cH k∇ z h k ≤ cH k f k + c k y H k + c k y H k + cL H k z h kk y H k Apply (4.10) and (4.9), we conclude the proof.
Remark 4.4.
Under the smallness assumption on H , that is, (4.10), we could similarly provehigher order estimates of y H and z h . For details, we refer to [1, 13, 14]. Error Estimate
In this section, we work out the error between classical Galerkin approximation and nonlinearGalerkin approximation of velocity.Before actually working out the error estimates, we present below the Lemma involving the estimateof z h . For a proof, we refer to [1]. Lemma 5.1.
Under the assumptions Lemma 3.2 and for the solution u h of (3.3), the followingestimates are satisfied for z h = ( I − P H ) u h and for t > t (5.1) k z h k + H k z h k ≤ K ( t ) H , k z ht k + H k z ht k ≤ K ( t ) H , k z htt k + H k z htt k ≤ K ( t ) H . In order to separate the effect of the nonlinearity in the error, we introduce¯ u ( ∈ J h ) = P H ¯ u + ( I − P H )¯ u = ¯ y + ¯ z ∈ J H + J Hh satisfying the following linearized system ( t > t )(¯ y t , φ )+ νa (¯ u , φ ) + Z tt β ( t − s ) a (¯ u ( s ) , φ ) ds = − b ( u h , u h , φ ) + ( f , φ ) φ ∈ J H νa (¯ u , χ ) + Z tt β ( t − s ) a (¯ u ( s ) , χ ) ds = − b ( u h , u h , χ ) + ( f , χ ) χ ∈ J Hh , (5.2)and ¯ y ( t ) = y H ( t ) . Being linear it is easy to establish the well-posedness of the above system andthe following estimates.
Lemma 5.2.
Under the assumptions of Lemma 3.2, we have k∇ ¯ u h k + e − αt Z tt e αt k ˜∆ h ¯ u h k ds ≤ K, k ˜∆ h ¯ u h k ≤ K, where the constant depends on u and f . We define e := u h − u h = ( y H − y H ) + ( z h − z h ) =: e + e . We further split the errors as follows: (cid:26) e = y H − y H = ( y H − ¯ y ) − ( y H − ¯ y ) = ξ − η ∈ J H e = z h − z h = ( z h − ¯ z ) − ( z h − ¯ z ) = ξ − η ∈ J Hh . (5.3)For the sake of simplicity, we write ξ = ξ + ξ , η = η + η . And the equations in ξ and η : subtract (5.2) from (4.8) and subtract (5.2) from (4.5) to obtain (cid:26) ( ξ ,t , φ )+ νa ( ξ , φ ) = 0 νa ( ξ , χ ) = − ( z ht , χ ) , (5.4) (cid:26) ( η ,t , φ )+ νa ( η , φ ) = b ( u h , u h , φ ) − b ( u h , u h , φ ) νa ( η , χ ) = b ( u h , u h , χ ) − b ( u h , u h , χ ) , (5.5)for φ ∈ J H and χ ∈ J Hh . emma 5.3. Under the assumptions of Lemma 3.2, the following holds. (5.6) e − αt Z tt e ατ k ξ ( τ ) k dτ ≤ K ( t ) H . Proof.
Choose φ = e αt ξ , χ = e αt ξ in (5.4), add the two resulting equations and with thenotation ˆ ξ = e αt ξ , we get(5.7) 12 ddt k ˆ ξ k − α k ˆ ξ k + ν k ˆ ξ k ≤ e αt k z ht kk ˆ ξ k . Using (4.2) and (5.1), we can bound the right-hand side as: ≤ e αt .K ( t ) H .cH k ˆ ξ k ≤ νρ k ˆ ξ k + K ( t ) H .e αt . And using (4.4), we have − α k ˆ ξ k + ν k ˆ ξ k ≥ ( νρ − αλ ) k ˆ ξ k + νρ k ˆ ξ k . As a result, we obtain from (5.7)(5.8) ddt k ˆ ξ k + 2( νρ − αλ ) k ˆ ξ k + νρ k ˆ ξ k ≤ K ( t ) H .e αt . Integrate from t to t and multiply the resulting inequality by e − αt .(5.9) k ξ k + e − αt Z tt ( k ˆ ξ k + k ˆ ξ k ) ds ≤ K ( t ) H . To obtain L ( L )-norm estimate, we consider the following discrete backward problem: for fixed t ,let w ( τ ) ∈ J h , w = w + w such that w ∈ J H , w ∈ J Hh be the unique solution of ( t ≤ τ < t )(5.10) ( φ , w ,τ ) − νa ( φ , w ) = e ατ ( φ , ξ ) − νa ( χ , w ) = e ατ ( χ , ξ ) w ( t ) = 0 . With change of variable, we can make it a forward problem and it turns out to be a linearizedversion of (4.5) and hence is well-posed. The following regularity result holds.(5.11) Z tt e − ατ k w k dτ ≤ C Z tt k ˆ ξ k dτ. Now, choose φ = ξ , χ = ξ in (5.10) and use (5.4) with φ = w , χ = w to find that k ˆ ξ ( τ ) k = ( ξ , w ,τ ) − νa ( ξ , w ) ≤ ddt ( ξ , w ) + ( z ht , w ) . Integrate from t to t .(5.12) Z tt k ˆ ξ ( τ ) k dτ = (( ξ ( t ) , w ( t )) − ( ξ ( t ) , w ( t )) + Z tt ( z ht , w ) dτ. w ( t ) = 0 and ξ ( t ) = y H ( t ) − ¯ y ( t ) = 0. Next, we observe that w ∈ J H = ⇒ P H w = w , i.e., w − P H w = 0 w ∈ J Hh = ⇒ P H w = 0 , i.e., w − P H w = w (cid:27) = ⇒ w − P H w = w Therefore, ( z ht , w ) = ( z ht , w − P H w ) ≤ k z ht kk w − Φ H w k ≤ K ( t ) H .cH k w k . (5.13)From (5.12), we get Z tt k ˆ ξ ( τ ) k dτ ≤ K ( t ) e αt H (cid:16) Z tt e − ατ k w k dτ (cid:17) / . Use (5.11) to conclude.In order to obtain optimal L ∞ ( L ) estimate, we would like to introduce Stokes-type projections( S H , S Hh ) for t > t defined as below: S H : J h → J H , S Hh : J h → J Hh , and with the notations ζ := y H − S H u h ∈ J H , ζ := z h − S Hh u h ∈ J Hh the following system is satisfied. (cid:26) νa ( ζ , φ ) = 0 , φ ∈ J H ,νa ( ζ , χ ) = − ( z ht , χ ) , χ ∈ J Hh . (5.14)For the sake of convenience, we have written ζ = ζ + ζ . Note that, given a semi-discrete Galerkinapproximation u h of NSE with a priori estimates, the system (5.14) is a Stokes system, with Stokesproblem in J h projected to subspaces J H and J Hh , and hence is well-posed. Lemma 5.4.
Under the assumptions of Lemma 3.2, we have (5.15) k ζ k + k ζ t k ≤ K ( t ) H . Proof.
Choose φ = e αt ζ , χ = e αt ζ in (5.14) to obtain(5.16) ν k ˆ ζ k ≤ e αt k z ht kk ˆ ζ k . As in (5.7), we establish(5.17) k ζ k ≤ k ζ k + k ζ k ≤ K ( t ) H . In order to obtain optimal L ∞ ( L )-norm estimate, we would use Aubin-Nitsche duality argument.For that purpose, we consider the following Galerkin approximation of steady Stoke problem: let w h ∈ J h be the solution of νa ( v , w h ) = ( v , ˆ ζ + ˆ ζ ) , v ∈ J h . Writing w h = P H w h , w h = ( I − P H ) w h , we split the above equation as(5.18) (cid:26) νa ( φ , w h ) = ( φ , ˆ ζ ) , φ ∈ J H ,νa ( χ , w h ) = ( χ , ˆ ζ ) , χ ∈ J Hh . k w h = w h + w h k ≤ c k ˆ ζ + ˆ ζ k . Now, put φ = ˆ ζ , χ = ˆ ζ in (5.18) and use (5.14) with φ = w h , χ = w h to find that k ˆ ζ k = νa (ˆ ζ , w h ) = − e αt ( z ht , w h ) . As in (5.13) along with (5.19), we find k ζ k ≤ K ( t ) H . For the remaining part, we differentiate (5.14) and proceed as above to complete the rest of theproof.Now we are in a position to estimate L ∞ ( L )-norm of ξ , that is, of ξ and ξ . Using thedefinitions of ξ i , ζ i , i = 1 ,
2, we write (cid:26) ξ = y H − ¯ y = ( y H − S H u h ) − (¯ y − S H u h ) =: ζ − θ , ξ = z h − ¯ z = ( z h − S Hh u h ) − (¯ z − S Hh u h ) =: ζ − θ . From (5.4) and (5.14), we have(5.20) (cid:26) ( θ ,t , φ ) + νa ( θ , φ ) = ( ζ ,t , φ ) , φ ∈ J H ,νa ( θ , χ ) = 0 , χ ∈ J Hh . Lemma 5.5.
Under the assumptions Lemma 3.2, we have k ξ k ≤ K ( t ) H . Proof.
Put φ = e αt θ , χ = e αt θ in (5.20) to find(5.21) 12 ddt k ˆ θ k − α k ˆ θ k + ν k ˆ θ k ≤ e αt k ζ ,t kk ˆ θ k . We recall that the spaces J H and J Hh are orthogonal in L -inner product. That is,for φ ∈ J H , χ ∈ J Hh , ( φ , χ ) = 0 . Hence k ˆ θ k ≤ k ˆ θ k + k ˆ θ k = k ˆ θ k ≤ k ˆ ξ k + k ˆ ζ k , k ζ ,t k ≤ k ζ t k . And − α k ˆ θ k + ν k ˆ θ k = ( ν − αλ ) k ˆ θ k + ν k ˆ θ k . As a result, after integrating (5.21) with respect to time from t to t , we obtain(5.22) k ˆ θ k + Z tt (cid:0) k ˆ θ k + k ˆ θ k (cid:1) ds ≤ (cid:16) Z tt e αt k ζ t k ds (cid:17) / (cid:16) Z tt (cid:0) k ˆ ξ k + k ˆ ζ k (cid:1) ds (cid:17) / . We now use Lemmas 5.3 and 5.4 to conclude from (5.22) that(5.23) k θ k + e − αt Z tt e αs (cid:0) k θ k + k θ k (cid:1) ds ≤ K ( t ) H .
12e again choose χ = e αt θ in (5.20) to find ν k ˆ θ k = − νa (ˆ θ , ˆ θ ) = ⇒ k θ k ≤ k θ k . Since θ ∈ J H , we use inverse inequality (3.1) and (5.23) to note that k θ k ≤ cH − k θ k ≤ K ( t ) H . Hence, we conclude that k θ k ≤ K ( t ) H . Now use (4.2) to see that(5.24) k θ k ≤ K ( t ) H . Combining (5.23)-(5.24), we establish k θ k ≤ K ( t ) H . Use triangle inequality and the estimates of ζ and θ to complete the proof.We are now left with the estimate of η , the error due to the nonlinearity. Lemma 5.6.
Under the assumptions of Lemma 3.2 and that H is small enough to satisfy (4.10)and µρ − H k ¯ u k ≥ , µ − H ( k ¯ u k + k y H k ) ≥ , we have k ( u h − u h )( t ) k ≤ K ( t ) H . Proof.
We choose φ = e αt η , χ = e αt η in (5.5).12 ddt k ˆ η k + ν k ˆ η k = e αt Λ h ( η , η ) , (5.25)where Λ h ( η , η ) = Λ h, ( η ) + Λ h, ( η ) , and Λ h, ( η ) = b ( u h , u h , η ) − b ( u h , u h , η )= b ( ξ − η , u h , η ) + b ( u h , ξ − η , η ) − b ( ξ − η , ξ − η , η )Λ h, ( η ) = b ( u h , u h , η ) − b ( u h , u h , η )= b ( ξ − η , u h , η ) + b ( u h , ξ − η , η ) − b ( ξ − η , ξ − η , η ) . Therefore Λ h ( η , η ) = b ( ξ − η , ¯ u , η ) + b ( u h , ξ , η ) . (5.26)We estimate the nonlinear terms as follows: b ( u h , ξ , η ) + b ( ξ − η , ¯ u , η ) ≤ (cid:8) k ξ kk u h k + ( k ξ k + k η k ) k ¯ u k (cid:9) k η k ,b ( η , ¯ u , η ) ≤ k η k k ¯ u k ( k η k + k η k ) ≤ k η k k ¯ u k k η k + H k ¯ u k k η k . ǫ, ǫ > h ( η , η ) ≤ ǫ k η k + ǫ k η k + c ( ǫ )( k u h k + k ¯ u k ) k ξ k + c ( ǫ, ǫ ) k ¯ u k k η k + H k ¯ u k k η k . Now, from (5.25), we find that ddt k ˆ η k + 2 νρ ( k ˆ η k + k ˆ η k ) ≤ ǫ k ˆ η k + 2 ǫ k ˆ η k + c ( ǫ )( k u h k + k ¯ u k ) k ˆ ξ k + c ( ǫ, ǫ ) k ¯ u k k ˆ η k + 2 H k ¯ u k k ˆ η k . (5.27)We choose ǫ = ǫ = νρ and assume that H small enough such that νρ − H k ¯ u k ≥ k η k + e − αt Z tt ( k ˆ η k + k ˆ η k ) ds ≤ K ( t ) H + K Z tt k η ( s ) k ds. Apply Gronwall’s lemma to establish L ∞ ( L )-norm estimate of η . We note that k η k ≤ cH − k η k ≤ K ( t ) H . For η , we again put χ = e αt η in (5.5). ν k ˆ η k = + e αt Λ h, ( η ) − νa (ˆ η , ˆ η ) . (5.28)Recall that Λ h, ( η ) = b ( ξ − η , ¯ u , η ) + b ( u h , ξ − η , η ) + b ( ξ − η , η , η ) . And b ( ξ − η , ¯ u , η ) + b ( u h , ξ − η , η ) ≤ ( k ξ k + k η k )( k ¯ u k + k u h k ) k η k b ( ξ − η , η , η ) ≤ ( k ξ k + k η k ) k η k k η k b ( − η , ¯ u , η ) + b ( − η , η , η ) ≤ H ( k ¯ u k + k η k ) k η k . Note that k η k ≤ k y H k + k ¯ y k . And under the assumption ν − H ( k ¯ u k + k y H k ) ≥ k η k ≤ K ( t ) H and hence k η k ≤ cH k η k ≤ K ( t ) H . Now, triangle inequality completes the proof. 14 .2 NLGM II
In this subsection, we deal with the error estimate for NLGM II. As earlier, we split the error intwo, that is, e = u h − u h = ξ − η . The equations and hence the estimates regarding ξ remain sameand are optimal in nature. The equation in η reads as follows: (cid:26) ( η ,t , φ )+ νa ( η , φ ) = b ( u h , u h , φ ) − b ( u h , u h , φ ) , φ ∈ J H νa ( η , χ ) = b ( u h , u h , χ ) − b ( u h , u h , χ ) + b ( z h , z h , χ ) , χ ∈ J Hh . (5.29) Lemma 5.7.
Under the assumptions of Lemma 5.6, we have k ( u h − u h )( t ) k ≤ K ( t ) H . Proof.
We choose φ = e αt η , χ = e αt η in (5.29).12 ddt k ˆ η k + ν k ˆ η k = e αt (cid:8) Λ h ( η , η ) + b ( z h , z h , η ) (cid:9) , (5.30)Since z h = z h + η − ξ , we have b ( z h , z h , η ) = b ( z h + η − ξ , z h − ξ , η ) . Now b ( ξ , z h − ξ , η ) ≤ k ξ k ( k z h k + k ξ k ) k η k b ( η , z h − ξ , η ) = b ( η , ¯ z , η ) ≤ cH k ¯ z k k η k b ( z h , z h − ξ , η ) ≤ k z h k / k z h k / ( k z h k + k ξ k ) k η k / k η k / + k z h k / k z h k / k η k ( k z h k / k z h k / + k ξ k ) ≤ cH k z h k ( k z h k + k ξ k ) k η k + cH / k z h k k η k .cH / ( k z h k + k ξ k ) ≤ cH k z h k k η k + cH k z h k k ξ k k η k . Incorporate these in (5.30). Integrate and as earlier, for small H , we obtain k η k + e − αt Z tt ( k ˆ η k + k ˆ η k ) ds ≤ K ( t ) H + K ( t ) H k z h k + K Z tt k η ( s ) k ds, which results in k η k + e − αt Z tt ( k ˆ η k + k ˆ η k ) ds ≤ K ( t ) H . That is k η k ≤ K ( t ) H , k η k ≤ K ( t ) H . As in the previous section, using only the second equation of (5.29) we can easily conclude that k η k ≤ K ( t ) H , k η k ≤ K ( t ) H . Remark 5.1.
The analysis reveals that the decrease in the order of convergence is due to thepresence of b ( z h , z h , χ ) in the error equation. So, whereas in NLG I, we keep the nonlinearityin both the equations, in NLG II, the second equation is made linear in z h by dropping the term b ( z h , z h , χ ) and which in turn appears in the error equation and is responsible for bringing downthe rate of convergence in the above analysis. .3 Improved Error Estimate In this section, we try to improve the rate of convergence, using the technique of Marion & Xu [14].But this is not straightforward, as the estimate of the function f ( u ) in their semi-linear problemdoes not hold for our f ( u ) and we have to be careful in order to obtain similar results.First, we note that the second equation of (4.6) can be written as(5.31) z h = Φ( y H ) , where Φ : J H → J Hh . Using this, we can write the equation in Φ( y H ), for χ ∈ J Hh νa ( y H + Φ( y H ) , χ ) + b ( y H + Φ( y H ) , y H , χ ) + b ( y H , Φ( y H ) , χ ) = ( f , χ )(5.32) Lemma 5.8.
Under the assumptions of Lemma 3.2 and that H is small enough to satisfy ν − cH k u h k ≥ , we have (5.33) k z h − Φ( y H ) k + H k z h − Φ( y H ) k ≤ K ( t ) H . Proof.
With the notation Φ e := z h − Φ( y H ) ∈ J Hh , we have, by deducting (5.32) from the secondequation of (4.8) νa (Φ e , χ ) = − ( z h,t , χ ) − b ( u h , u h , χ ) + b ( y H , Φ( y H ) , χ ) + b ( y H + Φ( y H ) , y H , χ ) . (5.34)Put χ = Φ e in (5.34) to obtain ν k Φ e k = − ( z h,t , Φ e ) − b (Φ e , u h , Φ e ) − b (Φ( y H ) , Φ( y H ) , Φ e ) . (5.35)Note that − ( z h,t , Φ e ) ≤ k z h,t kk Φ e k ≤ K ( t ) H k Φ e k − b (Φ e , u h , Φ e ) ≤ c k u h k k Φ e kk Φ e k ≤ cH k u h k k Φ e k − b (Φ( y H ) , Φ( y H ) , Φ e ) = − b ( z h − Φ e , z h , Φ e ) ≤ ( k z h k / k z h k / + k Φ e k / k Φ e k / ) k z h k k Φ e k / k Φ e k / ≤ K ( t ) H k Φ e k + cH k z h k k Φ e k . Put these estimates in (5.35) to find ν k Φ e k ≤ K ( t ) H + cH k u h k k Φ e k . (5.36)We have used the fact that k z h k ≤ k u h k + k y H k ≤ c k u h k . And assuming H to be small enough tosatisfy ν − cH k u h k ≥ k Φ e k ≤ K ( t ) H . And hence k Φ e k ≤ cH k Φ e k ≤ K ( t ) H . This completes the proof. 16 emma 5.9.
Under the assumptions of Lemma 5.8, we have k e k ≤ K ( t ) H + K k e k , (5.37) k e k ≤ K ( t ) H + KH k e k . (5.38) Proof.
Recall that e = z h − z h = ( z h − Φ( y H )) − ( z h − Φ( y H )). With the notation Φ e = z h − Φ( y H ),we have e = Φ e − Φ e . The equation in Φ e can be obtained by deducting (5.32) from the secondequation of (4.6). νa (Φ e − e , χ ) = − b ( u h , y H , χ ) − b ( y H , z h , χ ) + b ( y H , Φ( y H ) , χ ) + b ( y H + Φ( y H ) , y H , χ ) . (5.39)Put χ = Φ e to obtain ν k Φ e k = νa ( e , Φ e ) + b ( u h , e , Φ e ) + b ( e − Φ e , y H , Φ e ) + b ( e , Φ( y H ) , Φ e )(5.40)Note that b ( u h , e , Φ e ) = b ( u h − e − Φ e , e , Φ e ) + b (Φ e , e , Φ e ) b ( u h − e − Φ e , e , Φ e ) ≤ k u h k k e k k Φ e k + k e k k Φ e k + k Φ e k k e k k Φ e k b (Φ e , e , Φ e ) = 12 (Φ e · ∇ e , Φ e ) −
12 (Φ e , ·∇ Φ e , e ) ≤ k Φ e kk Φ e k k e k + k e k ∞ k Φ e k k Φ e k ≤ cH (1 + L H ) k e k k Φ e k b ( e − Φ e , y H , Φ e ) = b ( e − Φ e , u h − z h , Φ e ) ≤ k e k k u h k k Φ e k + k e k k z h k k Φ e k + cH (1 + L H ) k y H k k Φ e k b ( e , Φ( y H ) , Φ e ) ≤ k e k ( k Φ e k + k z h k ) k Φ e k . Incorporate these estimates in (5.40). Whenever it suits us, we bound k e k by k y H k + k y H k ≤ K .And therefore, we have, after kickback argument ν k Φ e k ≤ K k e k + K ( t ) H + cH (1 + L H )( k y H k + k y H k ) k Φ e k . (5.41)Assuming H small enough to satisfy ν − cH (1 + L H )( k y H k + k y H k ) > , we obtain k Φ e k ≤ K ( t ) H + K k e k . And so k Φ e k ≤ K ( t ) H + KH k e k . Using triangle inequality, we complete the proof.
Remark 5.2.
Recall that e i = ξ i − η i , i = 1 , and since the linearized error ξ (that is, ξ , ξ ) isoptimal in nature, we have from (5.37)-(5.38) k η k ≤ K ( t ) H + c k η k (5.42) k η k ≤ K ( t ) H + cH k η k . (5.43) 17ollowing Marion & Xu [14], we introduce the operator R Hh : J h → J Hh satisfying(5.44) a (cid:0) v − R Hh v , χ (cid:1) = 0 , ∀ χ ∈ J Hh . With the notations || v || R = k ( I − R Hh ) v k , ( v , w ) R = a (cid:0) ( I − R Hh ) v , ( I − R Hh ) w (cid:1) , we have, from Lemma 4.1 of [14],(5.45) c k v k ≤ k v k R ≤ c k v k , where c , c are positive constants independent of h, H . And similar to Lemmas 4.5 and 4.6 of [14],we find for φ ∈ J H ( y Ht , φ ) + ν ( y H , φ ) R = ( f , ( I − R Hh ) φ ) − b ( u h , u h , ( I − R Hh ) φ ) − b ( z h , z h , R Hh φ )(5.46) ( y H,t , φ ) + ν ( y H , φ ) R = ( f , ( I − R Hh ) φ ) − b ( u h , u h , ( I − R Hh ) φ ) + ( u h,t , R Hh φ ) . (5.47)Now, for φ ∈ J H , we write the equation in e = y H − y H as( e ,t , φ ) + ν ( e , φ ) R = +( u h,t , R Hh φ ) + b ( z h , z h , R Hh φ ) − b ( e + e , u h , ( I − R Hh ) φ ) − b ( u h , e + e , ( I − R Hh ) φ ) . (5.48) Lemma 5.10.
Under the assumptions Lemma 5.8, we have k e k + Z tt k e ,t k ds ≤ K ( t ) H . Proof.
Put φ = e ,t in (5.48) and observe that( u h,t , R Hh e ,t ) = ddt ( u h,t , R Hh e ) − ( u h,tt , R Hh e )= ddt ( u h,t , R Hh e ) − (( I − P H ) u tt , R Hh e ) − (( u h − u ) tt , R Hh e ) ≤ ddt ( u h,t , R Hh e ) + K ( t ) H k e k , − b ( e + e , u h , ( I − R Hh ) e ,t ) − b ( u h , e + e , ( I − R Hh ) e ,t ) ≤ c ( k e k + k e k )( k u h k + k u h k ) k e ,t k b ( z h , z h , R Hh e ,t ) = b ( z h − e , z h − e , R Hh e ,t ) = ddt b ( z h , z h , R Hh e ) − b ( z h,t , z h , R Hh e ) − b ( z h , z h,t , R Hh e ) + b ( z h , − e , R Hh e ,t ) + b ( e , z h , R Hh e ,t ) − b ( z h,t , z h , R Hh e ) − b ( z h , z h,t , R Hh e ) ≤ cH k z h k k z h,t k k e k b ( e , z h , R Hh e ,t ) ≤ c k e k / k e k / ( k z h k k R Hh e ,t k / k R Hh e ,t k / + k z h k / k z h k / k R Hh e ,t k ) ≤ cH k e k k z h k k e ,t k ≤ c k e k k z h k k e ,t k . Here, we have used that k ( I − R Hh ) e ,t k ≤ k e ,t k + cH k e ,t k ≤ c k e ,t k . And now we find k e ,t k + ν ddt k e k R ≤ K ( t ) H k e k + ddt n ( u h,t , R Hh e ) + b ( z h , z h , R Hh e ) o + K ( k e k + k e k ) k e ,t k + cH k z h k k z h,t k k e k + c k e k ( k z h k + k z h k ) k e ,t k . (5.49) 18ntegrate (5.49), use (5.45) and the fact that e ( t ) = 0 to find k e k + Z tt k e ,t k ds ≤ K ( t ) H + c Z tt ( k e k + k e k ) ds + ( u h,t , R Hh e ) + b ( z h , z h , R Hh e ) . As earlier, we estimate the last three terms to obtain k e k + Z tt k e ,t k ds ≤ K ( t ) H + c Z tt ( k e k + k e k ) ds. (5.50)We note from (5.42) and triangle inequality that k e k ≤ K ( t ) H + k η k ≤ K ( t ) H + k e k . Therefore k e k + Z tt k e ,t k ds ≤ K ( t ) H + c Z tt k e k ds. Use Gronwall’s lemma to establish k e k + Z tt k e ,t k ds ≤ K ( t ) H . Remark 5.3.
This tells us that k η k ≤ K ( t ) H , and as a result, from Remark 5.2, we have k η k + H k η k ≤ K ( t ) H . Another application of triangle inequality results in k e k + H k e k ≤ K ( t ) H . For the final estimate, we note down the equations in terms of e i , i = 1 , (cid:26) ( e ,t , φ ) + νa ( e , φ ) = − b ( u h , u h , φ ) + b ( u h , u h , φ ) νa ( e , χ ) = − ( z ht , χ ) − b ( u h , u h , χ ) + b ( u h , u h , χ ) − b ( z h , z h , χ ) , (5.51) Lemma 5.11.
Under the assumptions of Lemma 5.8, we have k ( u h − u h )( t ) k ≤ K ( t ) H , t > t . Proof.
With the notation ˜∆ H = P H ( − ∆ h ), we choose φ = ˜∆ − H e ,t in the first equation of (5.51)to find k e ,t k − + ν ddt k e k = b ( e + e , u h , ˜∆ − H e ,t ) + b ( u h , e + e , ˜∆ − H e ,t ) . (5.52)As earlier, we have b ( e + e , u h , ˜∆ − H e ,t ) + b ( u h , e + e , ˜∆ − H e ,t ) ≤ c ( k e k + k e k )( k u h k + k u h k ) k e ,t k − . k e k + Z tt k e ,t k − ≤ c Z tt ( k e k + k e k ) ds ≤ K ( t ) H + c Z tt k e k ds. Apply Gronwall’s lemma to conclude k e k + Z tt k e ,t k − ≤ K ( t ) H . Remark 5.4.
It is clear from our above analysis is that the linearized error between NLG ap-proximation and Galerkin approximation is of order H in L -norm. However, non-linearizedpart of the error may not always be of same order. For example, if the equation in z h containsonly b ( y H , y H , χ ) , then the non-linearized part of the error (i.e. the equation in η ) will containadditional terms like b ( y H , z h , χ ) and b ( z h , y H , χ ) apart from the non-linear terms of the secondequation of (5.29). And with one of these terms, we believe, we can only manage H order ofconvergence in L -norm. In this work, our main focus is in obtaining optimal L -error estimate , that is, O ( H ) for nonlinearGalerkin finite element approximations. For that purpose, we have discussed two NLGMs. In thefirst one, small scales are assumed stationary. In the second one, we have an additional assumptionthat interactions between small scales are negligible. And in both these cases, we have managed toshow optimal L -error estimate. But any further simplification of the small scales equations willlead to sub-optimal error estimate, say O ( H ), as has been observed in the remark 5.4. Acknowledge : The author would like to thank CAPES/INCTMat, Brazil for financial grant.
References [1] Ammi, A. and Marion, M. ,
Nonlinear Galerkin Methods and Mixed Finite Elements: Two-gridAlgorithms for the Navier-Stokes Equations , Numer. Math. 57 (1994), 189-214.[2] Guermond, J. and Prudhomme, S. ,
A fully discrete nonlinear Galerkin method for the 3DNavier-Stokes equations , Numer. Methods Partial Differential Equations 24 (2008), 759-775.[3] He, Y. and Li, K. ,
Nonlinear Galerkin Method and Two-step Method for the Navier-StokesEquations , Numer. Methods Partial Differential Equations 12 (1996), 283-305.[4] He, Y. and Li, K. ,
Convergence and Stability of Finite Element Nonlinear Galerkin Methodfor the Navier-Stokes Equations , Numer. Math. 79 (1998), 77-106.[5] He, Y. and Li, K. ,
Optimum finite element nonlinear Galerkin algorithm for the Navier-Stokesequations , Math. Numer. Sin. 21 (1999), 29-38.[6] He, Y. , Li, D. and Li, K. ,
Nonlinear Galerkin method and Crank-Nicolson method for viscousincompressible flow , J. Comput. Math. 17 (1999), 139-158.207] He, Y. , Miao, H. , Mattheij, R. , and Chen, Z. ,
Numerical analysis of a modified finite elementnonlinear Galerkin method , Numer. Math. 97 (2004), 725-756.[8] He, Y. , Wang, A. , Chen, Z. and Li, K. ,
An optimal nonlinear Galerkin method with mixedfinite elements for the steady Navier-Stokes equations , Numer. Methods Partial DifferentialEquations 19 (2003), 762-775.[9] Heywood, J. and Rannacher, R. ,
Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial dis-cretization , SIAM J. Numer. Anal. 19 (1982), no. 2, 275–311.[10] Heywood, J. and Rannacher, R. ,
On the question of turbulence modeling by approximateinertial manifolds and the nonlinear Galerkin method , SIAM J. Numer. Anal. 30 (1993), 1603-1621.[11] Liu, Q. and Hou, Y. ,
A two-level correction method in space and time based on Crank-Nicolsonscheme for Navier-Stokes equations , Int. J. Comput. Math. 87 (2010), 2520-2532.[12] Marion, M. and Temam, R. ,
Nonlinear Galerkin Methods , SIAM J Numer. Anal. 26 (1989),1137-1159.[13] Marion, M. and Temam, R. ,
Nonlinear Galerkin Methods: The Finite Element Case , Numer.Math. 57 (1990), 205-226.[14] Marion, M. and Xu, J. ,
Error Estimates on a New Nonlinear Galerkin Method Based onTwo-grid Finite Elements , SIAM J Numer. Anal. 32 (1995), 1170-1184.[15] Nabh, G. and Rannacher, R. ,
A Comparative Study of Nonlinear Galerkin Finite ElementMethods for One-dimensional Dissipative Evolution Problems , East-West J. Numer. Math. 5(1997), 113-144.[16] Temam, R. ,