MIXED FINITE ELEMENT FORMULATION AND ERROR ESTIMATES BASED ON PROPER ORTHOGONAL DECOMPOSITION FOR THE NON-STATIONARY NAVIER STOKES EQUATIONS*

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1 MIXED FINITE ELEMENT FORMULATION AND ERROR ESTIMATES BASED ON PROPER ORTHOGONAL DECOMPOSITION FOR THE NON-STATIONARY NAVIER STOKES EQUATIONS* zenong luo, jing cen, an i. m. navon Abstract. In tis paper, proper ortogonal ecomposition (POD) is use for moel reuction of mixe finite element (MFE) for te non-stationary Navier Stokes equations an error estimates between a reference solution an te POD solution of reuce MFE formulation are erive. Te basic iea of tis reuction tecnique is tat ensembles of ata are first compile from transient solutions compute equation system erive wit usual MFE meto for te non-stationary Navier Stokes equations or from pysics system trajectories by rawing samples of experiments an interpolation (or ata assimilation), an ten te basis functions of usual MFE meto are substitute wit te POD basis functions reconstructe by te elements of te ensemble to erive te POD reuce MFE formulation for te non-stationary Navier Stokes equations. It is sown by consiering numerical simulation results obtaine for illustrating example of cavity flows tat te error between POD solution of reuce MFE formulation an te reference solution is consistent wit teoretical results. Moreover, it is also sown tat tis result valiates feasibility an efficiency of POD meto. Key wors. mixe finite element meto, proper ortogonal ecomposition, te nonstationary Navier Stokes equations, error estimate 2000 Matematics Subject Classifications. 65N30, 35Q0 PII.. Introuction. Mixe finite element (MFE) metos are one of te important approaces for solving systems of partial ifferential equations, for example, te non-stationary Navier Stokes equations (see [], [2], or [3]). However, te computational moel for te fully iscrete system of MFE solutions of te non-stationary Navier Stokes equations yiels very large systems tat are computationally intensive. Tus, an important problem is ow to simplify te computational loa an save time consuming calculations an resource emans in te actual computational process in a way tat guarantees a sufficiently accurate an efficient numerical solution. Proper ortogonal ecomposition (POD), also known as Karunen Loève expansions * Receive by te eitors Scool of Science, Beijing Jiaotong University, Beijing 00044, Cina (zluo@bjtu.eu.cn). Tis autor was supporte in part by te National Science Founation of Cina (NSF04700 an NSF ) an Beijing Jiaotong University Science Tecnology Founation. Corresponing Autor: College of Science, Cina Agricultural University, Beijing 00083, Cina (jing Scool of Computational Science an Department of Matematics, Floria State University, Dirac Sci. Lib. Blg., #483, Tallaassee, FL , USA (navon@scs.fsu.eu)

2 2 Z. D. LUO, J. CHEN, AND I. M. NAVON in signal analysis an pattern recognition (see [4]), or principal component analysis in statistics (see [5]), or te meto of empirical ortogonal functions in geopysical flui ynamics (see [6 7]) or meteorology (see [8]), is a tecnique offering aequate approximation for representing flui flow wit reuce number of egrees of freeom, i.e., wit lower imensional moels (see [9]) so as to alleviate te computational loa an provie CPU an memory requirements savings, an as foun wiesprea applications in problems relate to te approximation of large scale moels. Altoug te basic properties of POD meto are well establise an stuies ave been conucte to evaluate te suitability of tis tecnique for various flui flows (see [0 2]), its applicability an limitations for reuce MFE formulation for te Navier Stokes equations are not well ocumente. Te POD meto mainly provies a useful tool for efficiently approximating a large amount of ata. Te meto essentially provies an ortogonal basis for representing te given ata in a certain least squares optimal sense, tat is, it provies a way to fin optimal lower imensional approximations of te given ata. In aition to being optimal in a least squares sense, POD as te property tat it uses a moal ecomposition tat is completely ata epenent an oes not assume any prior knowlege of te process use to generate te ata. Tis property is avantageous in situations were a priori knowlege of te unerlying process is insufficient to warrant a certain coice of basis. Combine wit te Galerkin projection proceure, POD provies a powerful meto for generating lower imensional moels of ynamical systems tat ave a very large or even infinite imensional pase space. In many cases, te beavior of a ynamic system is governe by caracteristics or relate structures, even toug te ensemble is forme by a large number of ifferent instantaneous solutions. POD meto can capture tese temporal an spatial structures by applying a statistical analysis to te ensemble of ata. In flui ynamics, Lumley first employe te POD tecnique to capture te large ey coerent structures in a turbulent bounary layer (see [3]); tis tecnique was furter extene in [4], were a link between te turbulent structure an ynamics of a caotic system was investigate. In Holmes et al [9], te overall properties of POD are reviewe an extene to wien te applicability of te meto. Te meto of snapsots was introuce by Sirovic [5], an is wiely use in applications to reuce te orer of POD eigenvalue problem. Examples of tese are optimal flow control problems [6 8] an turbulence [9, 3, 4, 9, 20]. In many applications of POD, te meto is use to generate basis functions for a reuce orer moel, wic can simplify an provie quicker assessment of te major features of te flui ynamics for te purpose of flow control as emonstrate by Kurila et al [8] or esign optimization as sown by Ly et al [7]. Tis application is use in a variety of oter pysical applications, suc as in [7], wic emonstrates an effective use of POD for a cemical vapor eposition reactor. Some reuce orer finite ifference moels an MFE formulations an error estimates base on POD for te upper tropical Pacific Ocean moel (see, [2 25]) as well as a finite ifference sceme base on POD for te non-stationary Navier Stokes equations (see [26]) ave been erive. However, to te best of our knowlege, tere are no publise results aressing te use of POD to reuce te MFE formulation of te nonlinear non-stationary Navier Stokes

3 MFE FORMULATION AND ERROR ESTIMATES BASED ON POD 3 equations an provie estimates of te error between reference solution an te POD reuce MFE solution. In tis paper, POD is use to reuce te MFE formulation for te non-stationary Navier- Stokes equations an to erive error estimates between reference solution an te POD reuce MFE solution. It is sown by consiering te results obtaine for numerical simulations of cavity flows tat te error between POD solution of reuce MFE formulation an reference solution is consistent wit teoretically erive results. Moreover, it is also sown tat tis valiates te feasibility an efficiency of POD meto. Toug Kunisc an Volkwein ave presente some Galerkin proper ortogonal ecomposition metos for parabolic problems an a general equation in flui ynamics in [27] an [28], our meto is ifferent from teir approaces, wose metos consist of Galerkin projection were te original variables are substitute for linear combination of POD basis an te error estimates of te velocity fiel terein are only erive, teir POD basis being generate wit te solution of te pysical system at all time instances. Especially, te velocity fiel is only approximate in [28], wile bot velocity an pressure fiels are simultaneously approximate in our present meto. Wile te singular value ecomposition approac combine wit POD metoology is use to treat te Burgers equation in [29] an te cavity flow problem in [2], te error estimates ave not completely been erive, in particular, a reuce formulation of MFE for te non-stationary Navier-Stokes as not yet been erive up to now. Terefore, our meto improves upon existing metos since our POD basis is generate wit te solution of te pysical system only at time instances wic are bot useful an of interest for us. 2. MFE approximation for te non-stationary Navier-Stokes equations an Snapsots Generation. Let Ω R 2 be a boune, connecte an polygonal omain. Consier te following non-stationary Navier-Stokes equations. Problem (I) Fin u = (u, u 2 ), p suc tat for T > 0, (2.) u t ν u + (u )u + p = f in Ω (0, T ), ivu = 0 in Ω (0, T ), u(x, y, t) = ϕ(x, y, t) on Ω (0, T ), u(x, y, 0) = ϕ(x, y, 0) in Ω, were u represents te velocity vector, p te pressure, ν te constant inverse Reynols number, f = (f, f 2 ) te given boy force, ϕ(x, y, t) te given vector function. For te sake of convenience, witout lost generality, we may as well suppose tat ϕ(x, y, t) is a zero vector in te following teoretical analysis. Te Sobolev spaces use in tis context are stanar (see [30]). For example, for a boune omain Ω, we enote by H m (Ω) (m 0) an L 2 (Ω) = H 0 (Ω) te usual Sobolev spaces equippe

4 4 Z. D. LUO, J. CHEN, AND I. M. NAVON wit te semi norm an te norm, respectively, v m,ω = α =m Ω D α v 2 xy /2 { m /2 an v m,ω = v i,ω} 2 v H m (Ω), were α = (α, α 2 ), α an α 2 are two non-negative integers, an α = α + α 2. Especially, te subspace H 0 (Ω) of H (Ω) is enote by H0 (Ω) = {v H (Ω); u Ω = 0}. Note tat is equivalent to in H0 (Ω). Let L2 0 {q (Ω) = L 2 (Ω); i=0 Ω } qxy = 0, wic is a subspace of L 2 (Ω). It is necessary to introuce te Sobolev spaces epenent on time t in orer to iscuss te generalize solution for Problem (I). Let Φ be a Hilbert space. For all T > 0 an integer n 0, for t [0, T ], efine { H n (0, T ; Φ) = wic is enowe wit te norm [ n v H n (Φ) = i=0 v(t) Φ; T 0 T 0 i t i v(t) n i t i v(t) i=0 were Φ is te norm of space Φ. Especially, if n = 0, An efine L (0, T ; Φ) = wic is enowe wit te norm 2 Φ t ] 2 ( T v L 2 (Φ) = v(t) 2 Φt 0 2 Φ t < } for v H n (Φ), ) 2 { } v(t) Φ; esssup 0 t T v(t) Φ <, v L (Φ) = esssup v(t) Φ. 0 t T Te variational formulation for te problem (I) is written as: Problem (II) Fin (u, p) H (0, T ; X) L 2 (0, T ; M) suc tat for all t (0, T ), (u t, v) + a(u, v) + a (u, u, v) b(p, v) = (f, v) v X, (2.2) b(q, u) = 0 q M, u(x, 0) = 0 in Ω, were X = H0 (Ω)2, M = L 0 (Ω), a(u, v) = ν u vxy, a (u, v, w) = 2 v j [u i w j Ω 2 Ω x i, i w j u i v j ]xy, u, v, w X, b(q, v) = q ivvxy. x i Ω.,

5 MFE FORMULATION AND ERROR ESTIMATES BASED ON POD 5 Trougout te paper, C inicates a positive constant wic is possibly ifferent at ifferent occurrences, being inepenent of te spatial an temporal mes sizes, but may epen on Ω, te Reynols number, an oter parameters introuce in tis paper. Te following property for trilinear form a (,, ) is often use (see [], [2], or [3]). (2.3) a (u, v, w) = a (u, w, v), a (u, v, v) = 0 u, v, w X. Te bilinear forms a(, ) an b(, ) ave te following properties (2.4) a(v, v) ν v 2 v H 0 (Ω)2, (2.5) a(u, v) ν u v u, v H 0 (Ω)2, an (2.6) sup v H 0 (Ω)2 b(q, v) v β q 0 q L 2 0(Ω), were β is a positive constant. Define a (u, v, w) (2.7) N = sup ; u,v,w X u v w (f, v) f = sup. v X v Te following result is classical (see [], [2], or [3]). Teorem 2.. If f L 2 (0, T ; H (Ω) 2 ), ten te problem (II) as at least a solution wic, in aition, is unique provie tat ν 2 N f L 2 (H ) <, an tere is te following prior estimate: u L 2 (L 2 ) ν f L 2 (H ) R, u 0 ν /2 f L 2 (H ) = Rν /2. Let {I } be a uniformly regular family of triangulation of Ω (see [3], [32], or [33]), inexe by a parameter = max K I { K ; K =iam(k)}, i.e., tere exists a constant C, inepenent of, suc tat C K K I. We introuce te following finite element spaces X an M of X an M, respectively. Let X X (wic is at least te piecewise polynomial vector space of mt egree, were m > 0 is integer) an M M (wic is te piecewise polynomial space of (m )t egree). Write ˆX = X M. We assume tat (X, M ) satisfies te following approximate properties: v H m+ (Ω) 2 X an q M H m (Ω), (2.8) inf v X (v v ) 0 C m v m+, inf q q 0 C m q m, q M togeter te so calle iscrete LBB (Layzenskaya-Brezzi-Babuška) conition, i.e., (2.9) sup v X b(q, v ) v 0 β q 0 q M,

6 6 Z. D. LUO, J. CHEN, AND I. M. NAVON were β is a positive constant inepenent of. Tere are many spaces X an M satisfying te iscrete LBB conitions (see [33]). Here, we provie some examples as follows. Example 2.. Te first orer finite element space X M can be taken as Bernari Fortin Raugel s element (see [33]), i.e., X = {v X C 0 ( Ω) 2 ; v K P K K I }, (2.0) M = {ϕ M; ϕ K P 0 (K) K I }, were P K = P (K) 2 span{ n i 3,j i λ Kj, i =, 2, 3}, n i are te unit normal vector to sie F i opposite te vertex A i of triangle K, λ Ki s are te barycenter coorinates corresponing to te vertex A i (i =, 2, 3) on K (see [3-32]), an P m (K) is te space of piecewise polynomials of egree m on K. Example 2.2. element, i.e., (2.) Te first orer finite element space X M can also be taken as Mini s were P K = P (K) 2 span{λ K λ K2 λ K3 } 2. (2.2) Example 2.3. X = {v X C 0 (Ω) 2 ; v K P K K I }, M = {q M C 0 (Ω); q K P (K) K I }, Te secon orer finite element space X M can be taken as X = {v X C 0 (Ω) 2 ; v K P K K I }, were P K = P 2 (K) 2 span{λ K λ K2 λ K3 } 2. (2.3) Example 2.4. M = {q M C 0 (Ω); q K P (K) K I }, Te tir orer finite element space X M can be taken as X = {v X C 0 (Ω) 2 ; v K P K K I }, M = {q M C 0 (Ω); q K P 2 (K) K I }, were P K = P 3 (K) 2 span{λ K λ K2 λ K3 λ Ki, i =, 2, 3} 2. It as been prove (see [33]) tat, for te finite element space X M in Example , tere exists a restriction operator r : X X suc tat, for any v X, (2.4) b(q, v r v) = 0 q M, r v 0 C v 0, (v r v) 0 C k v k+ if v H k+ (Ω) 2, k =, 2, 3. Te spaces X M use trougout next part in tis paper mean tose in Example , wic satisfy te iscrete LBB conition (2.9) (see [33] for a more etaile proof). In orer to fin a numerical solution for Problem (II), it is necessary to iscretize Problem (II). We introuce a MFE approximation for te spatial variable an FDS (finite ifference

7 MFE FORMULATION AND ERROR ESTIMATES BASED ON POD 7 sceme) for te time erivative. Let L be te positive integer, enote te time step increment by k = T/L (T being te total time), t (n) = nk, 0 n L; (u n, pn ) X M te MFE approximation corresponing to (u(t (n) ), p(t (n) )) (u n, p n ). Ten, applying a semi-implicit Euler sceme for te time integration, te fully iscrete MFE solution for te problem (I) may be written as: (2.5) Problem (III) Fin (u n, pn ) X M suc tat were n L. (u n, v) + ka(un, v) + ka (u n, u n, v) kb(pn, v) = k(f n, v) + (u n, v) v X, b(q, u n ) = 0 q M, u 0 = 0 in Ω, Put A(u n, v ) = (u n, v ) + ka(u n, v ) + ka (u n, u n, v ). Since A(u n, un ) = (un, un ) + ka(u n, un ) + ka (u n, u n, un ) = un 0 + kν u n 0, A(, ) is coercive in X X. An kb(, ) also satisfies te iscrete LBB conition in X M, terefore, by MFE teory (see [], [32], or [33]), we obtain te following result. Teorem 2.2. Uner te assumptions (2.8)-(2.9), if f H (Ω) 2 satisfies N n < ν 2, Problem (III) as a unique solution (u n, pn ) X M an satisfies, (2.6) u n kν if k = O( 2 ), n u i 2 0 kν (2.7) u n u n n 0 + k /2 (u i u i ) 0 + k /2 n n f i 2, p i p i 0 C( m + k), f i were (u, p) [H 0 (Ω) Hm+ (Ω)] 2 [H m (Ω) M] is te exact solution for te problem (I), C is a constant epenent on u n m+ an p n m, an n L. If R = ν, triangulation parameter, finite elements X an M, te time step increment k, an f are given, by solving Problem (III), we can obtain a solution ensemble {u n, un 2, pn }L n= for Problem (III). An ten we coose l (for example, l = 20, or 30, in general, l L) instantaneous solutions U i (x, y) = (u ni, uni 2, pni )T ( n < n 2 < < n l L) (wic are useful an of interest for us) from te L group of solutions (u n, un 2, pn )T ( n L) for Problem (III), wic are referre to as snapsots. 3. An reuce MFE formulation base POD tecnique for te nonstationary Navier Stokes equations. In tis section, we use POD tecnique to eal wit te snapsots in Section 2 an prouce an optimal representation in an average sense. Recall ˆX = X M. For U i (x, y) = (u ni, uni 2, pni )T (i =, 2,, l) in Section 2, we set (3.) V = span{u, U 2,, U l },

8 8 Z. D. LUO, J. CHEN, AND I. M. NAVON an refer to V as te ensemble consisting of te snapsots {U i } l at least one of wic is assume to be non-zero. Let {ψ j } l enote an ortonormal basis of V wit l = im V. Ten eac member of te ensemble can expresse as (3.2) U i = (U i, ψ j ) ˆXψ j for i =, 2,, l, were (U i, ψ j ) ˆXψ j = (( u ni, ψ uj) 0 ψ uj, (p ni, ψ pj) 0 ψ pj ), (, ) 0 is L 2 -inner prouct, an ψ uj an ψ pj are ortonormal bases corresponing to u an p, respectively. Since V = span{u, U 2,, U l } = span{ψ, ψ 2,, ψ l }, b(p ni imply b(ψ pi, ψ uj ) = 0 ( i, j l)., unj ) = 0 ( i, j l) Definition 3.. Te meto of POD consists in fining te ortonormal basis suc tat for every ( l) te mean square error between te elements U i corresponing t partial sum of (3.2) is minimize on average: (3.3) min {ψ j } l suc tat U i (U i, ψ j ) ˆXψ j 2ˆX (3.4) (ψ i, ψ j ) ˆX = δ ij for i, j i, ( i l) an were U i ˆX = [ u ni u ni p ni 2 0] 2. A solution {ψj } of (3.3) an (3.4) is known as a POD basis of rank. We introuce te correlation matrix K = (K ij ) l l R l l corresponing to te snapsots {U i } l by (3.5) K ij = l (U i, U j ) ˆX. Te matrix K is positive semi-efinite an as rank l. Te solution of (3.3) an (3.4) can be foun in [0, 5, or 28], for example. Proposition 3.2. Let λ λ 2 λ l > 0 enote te positive eigenvalues of K an v, v 2,, v l te associate ortonormal eigenvectors. Ten a POD basis of rank l is given by (3.6) ψ i = λi (v i ) j U j, were (v i ) j enotes te j-t component of te eigenvector v i. Furtermore, te following error formula ols (3.7) l U i (U i, ψ j ) ˆXψ j 2ˆX = j=+ λ j. Let V = span {ψ, ψ 2,, ψ } an X M = V wit X X X an M M M. Set te Ritz-projection P : X X (if P is restricte to Ritz-projection from X to

9 MFE FORMULATION AND ERROR ESTIMATES BASED ON POD 9 X, it is written as P ) suc tat P X = P : X X an P : X\X X \X an L 2 -projection ρ : M M enote by, respectively, (3.8) a(p u, v ) = a(u, v ) v X an (3.9) (ρ p, q ) 0 = (p, q ) 0 q M, were u X an p M. Due to (3.8) an (3.9) te linear operators P an ρ are well-efine an boune: (3.0) (P u) 0 u 0, ρ p 0 p 0 u X an p M. Lemma 3.2. For every ( l) te projection operators P an ρ satisfy respectively (3.) l (u ni P u ni ) 2 0 j=+ λ j, (3.2) l u ni P u ni 2 0 C2 j=+ λ j, an (3.3) l p ni ρ p ni 2 0 j=+ λ j, were u ni Proof. = (uni, uni 2 ) an (uni, uni 2, pni )T V. For any u X we euce from (3.8) tat Terefore, we obtain tat ν (u P u) 2 0 = a(u P u, u P u) = a(u P u, u v ) ν (u P u) 0 (u v ) 0 v X. (3.4) (u P u) 0 (u v ) 0 v X. If u = u ni, an P is restricte to Ritz-projection from X to X, i.e., P u ni = P u ni X, taking v = (u ni, ψ uj) X ψ uj X X (were ψ uj is te component of ψ j corresponing to u) in (3.4), we can obtain (3.) from (3.7). In orer to prove (3.2), we consier te following variational problem: (3.5) ( w, v) = (u P u, v) v X.

10 0 Z. D. LUO, J. CHEN, AND I. M. NAVON Tus, w [H 0 (Ω) H2 (Ω)] 2 an satisfies w 2 C u P u 0. Taking v = u P u in (3.5), from (3.4) we obtain tat (3.6) u P u 2 0 = ( w, (u P u)) = ( (w P w), (u P u)) (w P w) 0 (u P u) 0 (w w ) 0 (u P u) 0 w X. Taking w = r w, from (2.4) an (3.6) we ave Tus, we obtain tat u P u 2 0 C w 2 (u P u) 0 C u P u 0 (u P u) 0. (3.7) u P u 0 C (u P u) 0. Terefore, if u = u ni an P is restricte to Ritz-projection from X to X, i.e., P u ni = P u ni X, by (3.7) an (3.) we obtain (3.2). Using Höler inequality an (3.9) can yiel consequently, p ni ρ p ni 2 0 = (p ni ρ p ni, pni ρ p ni ) = (pni ρ p ni, pni q ) (3.8) p ni Taking q = (p ni p ni ρ p ni 0 p ni q 0 q M, ρ p ni 0 p ni q 0 q M., ψ pj) 0 ψ pj (were ψ pj is te component of ψ j corresponing to p) in (3.8), from (3.7) we can obtain (3.3), wic completes te proof of Lemma 3.2. Tus, using V = X M, we can obtain te reuce formulation for Problem (III) as follows. (3.9) Problem (IV) were n L. Remark 3.3. Fin (u n, pn ) V suc tat (u n, v ) + ka(u n, v ) + ka (u n, u n, v ) kb(p n, v ) = k(f n, v ) + (u n, v ) v X, b(q, u n ) = 0 q M, u 0 = 0, Problem (IV) is a reuce MFE formulation base on POD tecnique for Problem (III), since it only inclues 3 ( l l L) egrees of freeom an is inepenent of te spatial gri scale, wile Problem (III) inclues 3N p + N K 5N p for Mini s element of Example 2.2 (were N p is te number of vertices in I an N K te number of elements in I ) an 3 5N p (for example in Section 5, 7, wile N p = = 024). Te number of

11 MFE FORMULATION AND ERROR ESTIMATES BASED ON POD egrees of freeom of Example 2. is also approximately 5N p, but Example 2.3 an Example 2.4 are more. Wen one computes actual problems, one may obtain te ensemble of snapsots from pysical system trajectories by rawing samples from experiments an interpolation (or ata assimilation). For example, for weater forecast, one can use te previous weater preiction results to construct te ensemble of snapsots, ten restructure te POD basis for te ensemble of snapsots by above (3.3) (3.6), an finally combine it wit a Galerkin projection to erive a reuce orer ynamical system, i.e., one nees only to solve te above Problem (IV) wic as only 3 egrees of freeom, but it is unnecessary to solve Problem (III). Tus, te forecast of future weater cange can be quickly simulate, wic is a result of major importance for real-life applications. Since te evelopment an cange of a large number of future nature penomena are closely relate to previous results (for example, weater cange, biology anagenesis, an so on), using existing results as snapsots in orer to structure POD basis, by solving corresponing PDEs one may truly capture laws of cange of natural penomena. Terefore, tese POD metos provie useful an important application. 4. Existence an error analysis of solution of te reuce MFE formulation base on POD tecnique for te non-stationary Navier-Stokes equations. Tis section is evote to iscussing te existence an error estimates for Problem (IV). We see from (3.6) tat V = X M V X M X M, were X M is one of tose spaces in Example Terefore, we ave in te following result. Lemma 4.. Tere exists also an operator r : X X suc tat, for all v X, (4.) b(q, u r u ) = 0 q M, r u 0 c u 0, an, for every ( l), (4.2) l (u ni r u ni ) 2 0 C j=+ λ j. Proof. We use te Mini s an te secon finite element as examples. Noting tat for any q M an K I, q K P 0 (K), using Green formula, we ave b(q, u r u ) = q (u r u )xy = q K (u r u )xy. Ω K I K Define r as follows (4.3) r v K = P v K + γ K λ K λ K2 λ K3 v X an K I, were γ K = K (v P v )x/ K λ K λ K2 λ K3 x. Tus, te first equality of (4.) ols. Using (3.0) (3.2) yiels te inequality of (4.) Ten, if v = u ni, using (3.) (3.2), by simply computing we euce (4.2).

12 2 Z. D. LUO, J. CHEN, AND I. M. NAVON Set V = {v X; b(q, v) = 0 q M}, V = {v X ; b(q, v ) = 0 q M }, V = {v X ; b(q, v ) = 0 q M }. Using ual principle an equations (3.) an (3.2), we euce te following result (see [, 3-33]). Lemma 4.2. Tere exists an operator R : V V V suc tat, for all v V V, (v R v, v ) = 0 v V, R v 0 C v 0, an, for every ( l), (4.4) l u ni R u ni 2 C2 l (u ni R u ni ) 2 0 C 2 j=+ λ j, were enotes te normal of space H (Ω) 2 (see (2.7)). We ave te following result for solution of Problem (IV). Teorem 4.3. Uner te ypoteses of Teorem 2.2, Problem (IV) as a unique solution (u n, pn ) X M an satisfies n (4.5) u n kν u i 2 0 kν n f i 2. Proof. Using same tecnique as te proof of Teorem 2.2, we coul prove tat Problem (IV) as a unique solution (u n, pn ) X M an satisfies (4.5). In te following teorem, error estimates of solution for Problem (IV) are erive. Teorem 4.4. Uner te ypoteses of Teorem 2.2, if 2 = O(k), k = O(l 2 ), snapsots are equably taken, an f H (Ω) 2 satisfies 2ν 2 N n f i <, ten te error between te solution (u n, pn ) for Problem (IV) an te solution (un, pn ) for Problem (III) as te following error estimates, for n =, 2,, L, (4.6) u ni C uni k /2 0 + k /2 p ni pni 0 + k /2 (u ni uni ) 0 /2, i =, 2,, l; j=+ λ j u n un 0 + k /2 p n pn 0 + k /2 (u n un ) 0 /2 Ck + C k /2, n {n, n 2,, n l }. j=+ λ j Proof. M can yiel Subtracting Problem (IV) from Problem (III) taking v = v X an q = q (4.7) (u n un, v ) + ka(u n un, v ) kb(p n pn, v ) + ka (u n, u n, v ) ka (u n, u n, v ) = (u n, v ) v X,

13 MFE FORMULATION AND ERROR ESTIMATES BASED ON POD 3 (4.8) b(q, u n un ) = 0 q M, (4.9) u 0 u0 = 0. We obtain, from (2.3), (2.7), Teorem 2.2, an Teorem 4.3, by Höler inequality, tat (4.0) a (u n, u n, v ) a (u n, u n, v ) = a (u n, u n, v ) +a (u n, u n un, v ) C[ (u n ) 0 + (u n un ) 0] v 0, especially, if v = P u n un, ten (4.) a (u n, u n, P u n un ) a (u n, u n, P u n un ) = a (u n = a (u n, u n, P u n un ) + a (u n, u n u n, P u n u n ), u n, P u n un ) + a (u n, u n, un un ) +a (u n, u n un, P u n un ) C (u n P u n ) ε[ (u n +N u n 0 (u n ) (u n un ) 2 0] ) 0 (u n un ) 0, were ε is a small positive constant wic can be cosen arbitrarily. Write t u n = [un un ]/k an note tat t u n V an t R u n n V. From Lemma 4.2, (4.7), an (4.0), we ave tat (4.2) t u n t u n t u n t R u n + t R u n t u n t u n t R u n ( + sup t R u n t u n, v) v V v 0 = t u n t R u n ( + sup t u n t u n, R v) v V v 0 = t u n t R u n + sup v V a (u n, u n, R v) + a (u n, u n, R v)] = t u n t R u n + sup v V a (u n, u n, R v) + a (u n, u n, R v)] v 0 [b(p n pn, R v) a(u n un, R v) v 0 [b(p n ρ p n, R v) a(u n u n, R v) t u n t R u n + C[ p n ρ p n 0 + (u n By using (2.9), (4.7), (4.0), (4.2), an Lemma 4., we ave tat (4.3) ) 0 + (u n un ) 0]. β ρ p n pn b(ρ p n 0 sup pn, v ) b(p n = sup pn, r v ) v X v 0 v X v 0 = sup [( v X v t u n t u n, r v ) + a(u n u n, r v) 0 +a (u n, u n, r v) a (u n, u n, r v)] C[ t u n t u n + (u n C[ t u n t R u n + p n ρ p n 0 + (u n ) 0 + (u n u n ) 0 ]. ) 0 + (u n u n ) 0 ]

14 4 Z. D. LUO, J. CHEN, AND I. M. NAVON Tus, we obtain tat (4.4) p n pn 0 p n ρ p n 0 + ρ p n pn 0 C[ (u n + (u n u n ) 0 + t u n t R u n + p n ρ p n 0 ]. ) 0 Taking v = P u n un in (4.7), it follows from (4.8) tat (4.5) (u n u n, u n u n ) (u n = (u n u n (u n, u n u n ) + ka(u n u n, u n u n ) ), u n P u n ) + ka(u n P u n, u n P u n ) +kb(p n ρ p n, un un ) + kb(pn pn, un P u n ) ka (u n, u n, P u n u n ) + ka (u n, u n, P u n u n ). Tus, noting tat a(a b) = [a 2 b 2 + (a b) 2 ]/2 (for a 0 an b 0), by (4.), (4.4), Höler inequality, Caucy inequality, an Proposition 3.2, we obtain tat (4.6) [ u n u n 2 0 u n 2 +νk (u n un ) un un (un u n u n (u n ) 2 ] 0 ) un P u n 2 0 +Ck (u n P u n ) Ck p n ρ p n C t u n t R u n 2 +(ε + Cε 2 + ε)k[ (u n un ) (un + 2 k[n 2 γ u n 2 0 (u n ) 2 0 ] ) γ (u n u n ) 2 0], were ε an ε 2 are two small positive constants wic can be cosen arbitrarily. ε + ε + Cε 2 = ν/4, it follows from (4.6) tat Taking (4.7) [ u n u n 2 0 u n 2 0] + νk (u n u n ) 2 0 u n P u n Ck (u n P u n ) Ck p n ρ p n 2 0 +C t u n t R u n kγ (un +kn 2 γ u n 2 0 (u n ) 2 0 ) 2 0, n L. If 2 = O(k), 2ν 2 N n f j <, n = n i (i =, 2,, l), summing (4.7) from n = n, n 2,, n i (i =, 2,, l), let n 0 = 0, an noting tat u 0 u0 = 0 an l L, from Lemma 3.2, Lemma , we obtain tat (4.8) u ni +Ck +C C uni νk (u ni uni ) 2 0 C i [ (u nj i [ u nj i u nj P u nj ) p nj ρ p nj 2 0] R u nj 2 + u nj R u nj 2 ] λ j, i =, 2,, l. j=+ P u nj 2 0

15 MFE FORMULATION AND ERROR ESTIMATES BASED ON POD 5 Tus, we obtain tat (4.9) u ni uni 0 + (νk) /2 (u ni uni ) 0 C k /2 j=+ λ j /2, i =, 2,, l. Combining (4.9) an (4.4), by Lemma 3.2, Lemma , we obtain te first inequality of (4.6). If n n i (i =, 2,, l), we may as well let t (n) (t (ni ), t (ni) ) an t (n) be te nearest point to t (ni). Expaning u n an pn into Taylor series wit respect to t(ni) yiels tat (4.20) u n = uni η ik u (ξ ), t (n) ξ t (ni) ; p n t = pni η ik p (ξ 2 ), t (n) ξ 2 t (ni), t were η i is te step number from t (n) to t (ni). If 2 = O(k), 2ν 2 N n f j <, k = O(l 2 ), summing (4.7) for n,, n i, n, let n 0 = 0, an noting tat u 0 u0 = 0, from Lemma an Lemma 3.2, we obtain tat (4.2) u n u n kγ (u n u n ) 2 0 Cη 2 k 3 + Ck /2 j=+ Since snapsots are equably taken, η i L/(2l). If k = O(l 2 ), we obtain tat (4.22) u n u n 0 + k /2 (u n u n ) 0 Ck + C k /2 j=+ Combining (4.22) an (4.4), by Lemma 3.2, Lemma , we obtain te secon inequality of (4.6). Combining Teorem 2.2 an Teorem 4.4 yiels te following result. Teorem 4.5. Uner Teorem 2.2 an Teorem 4.4 ypoteses, te error estimate between te solutions for Problem (II) an te solutions for te reuce orer basic Problem (IV) is, for n =, 2,, L, m =, 2, 3, λ j λ j. /2. (4.23) u n u n 0 + k /2 p n p n 0 + k /2 (u n u n ) 0 /2 Ck + C m + C k /2. j=+ λ j Remark 4.6. Te conition k = O(l 2 ), wic implies L = O(l 2 ), in Teorem 4.4 sows te relation between te number l of snapsots an te number L at all time instances. Terefore, it is unnecessary to take total transient solutions at all time instances t (n) as snapsots (see for instance in [27-29]). Teorems 4.4 an 4.5 ave presente te error estimates between te solution of te reuce MFE formulation Problem (IV) an te solution of usual MFE formulation Problem (III) an Problem (II), respectively. Since our metos employ some MFE solutions (u n, pn ) (n =, 2,, L) for Problem (III) as assistant analysis, te error estimates

16 6 Z. D. LUO, J. CHEN, AND I. M. NAVON in Teorem 4.5 are correlate to te spatial gri scale an time step size k. However, wen one computes actual problems, one may obtain te ensemble of snapsots from pysical system trajectories by rawing samples from experiments an interpolation (or ata assimilation). Terefore, te assistant (u n, pn ) (n =, 2,, L) coul be replace wit te interpolation functions of experimental an previous results, tus renering it unnecessary to solve Problem (III), an requiring only to irectly solve Problem (IV) suc tat Teorem 4.4 is satisfie. 5. Some numerical experiments. In tis section, we present some numerical examples of te pysical moel of cavity flows for Mini s element an ifferent Reynols numbers by te reuce formulation Problem (IV) tus valiating te feasibility an efficiency of te POD meto. Figure. Pysical moel of te cavity flows: t = 0 i.e., n = 0 initial values on bounary Figure 2. Wen Re=750, velocity stream line figure for usual MFE solutions (on left-an sie figure) an = 6 te solution of te reuce MFE formulation (on rigt-an sie figure) Let te sie lengt of te cavity be (see Figure ). We first ivie te cavity into = 024 small squares wit sie lengt x = y = 32, an ten link te iagonal of te square to ivie eac square into two triangles in te same irection wic consists of triangularization I. Take time step increment as t = Except tat u is equal to on upper bounary,

17 MFE FORMULATION AND ERROR ESTIMATES BASED ON POD 7 all oter initial value, bounary values, an (f, f 2 ) are all taken as 0 (see Figure ). We obtain 20 values (i.e., snapsots) at time t = 0, 20, 30,, 200 by solving te usual MFE formulation, i.e., Problem (III). It is sown by computing tat eigenvalues satisfy [k /2 20 i=7 λ i] / Wen t = 200, we obtain te solutions of te reuce formulation Problem (IV) base on POD meto of MFE epicte grapically in Figure 2 to Figure 5 on te rigt-an sie employe 6 POD bases for Re = 750 an require 6 POD bases for Re = 500, wile te solutions obtaine wit usual MFE formulation Problem (III) are epicte grapically in Figure 2 to Figure 5 on left-an sie (Since tese figures are equal to solutions obtaine wit 20 bases, tey are also referre to as te figures of te solution wit full bases). Figure 3. Wen Re=500, velocity stream line figure for usual MFE solution (on left-an sie figure) an = 6 solution of te reuce MFE formulation (on rigt-an sie figure) Figure 4. Wen Re=750, pressure figure for usual MFE solution (on left-an sie figure) an = 6 solution of reuce MFE formulation (on rigt-an sie figure) Figure 6 sows te errors between solutions obtaine wit ifferent number of POD bases an solutions obtaine wit full bases. Comparing te usual MFE formulation Problem (III) wit te reuce MFE formulation Problem (IV) containing 6 POD bases implementing 3000 times te numerical simulation computations, we fin tat for usual MFE formulation Problem

18 8 Z. D. LUO, J. CHEN, AND I. M. NAVON (III) te require CPU time is 6 minutes, wile for te reuce MFE formulation Problem (IV) wit 6 POD bases te corresponing time is only tree secons, i.e., te usual MFE formulation Problem (III) require a CPU time wic is by a factor of 20 larger tan tat require by te reuce MFE formulation Problem (IV) wit 6 POD bases, wile te error between teir respective solutions oes not excee 0 3. It is also sown tat fining te approximate solutions for te non-stationary Navier Stokes equations wit te reuce MFE formulation Problem (IV) is computationally very effective. An te results for numerical examples are consistent wit tose obtaine for te teoretical case. Figure 5. Wen Re=500, pressure figure for usual MFE solutions (on left-an sie figure) an = 6 solution of reuce MFE formulation (on rigt-an sie figure) Figure 6. Error for Re=750 on left an sie, error for Re=500 on rigt-an sie 6. Conclusions. In tis paper, we ave employe te POD tecnique to erive a reuce formulation for te non-stationary Navier Stokes equations. We first reconstruct optimal ortogonal bases of ensembles of ata wic are compile from transient solutions erive by using usual MFE equation system, wile in actual applications, one may obtain te ensemble of snapsots from pysical system trajectories by rawing samples from experiments an interpolation (or ata assimilation). For example, for weater forecast, one may use previous weater preiction results to construct te ensemble of snapsots to restructure te POD basis for te ensemble of snapsots by metos of above Section 3. We ave also combine te opti-

19 MFE FORMULATION AND ERROR ESTIMATES BASED ON POD 9 mal ortogonal bases wit a Galerkin projection proceure, tus yieling a new reuce MFE formulation of lower imensional orer an of ig accuracy for te non-stationary Navier Stokes equations. We ave ten proceee to erive error estimates between our reuce MFE approximate solutions an te usual MFE approximate solutions, an ave sown using numerical examples tat te error between te reuce MFE approximate solution an te usual MFE solution is consistent wit te teoretical error results, tus valiating bot feasibility an efficiency of our reuce MFE formulation. Future researc work in tis area will aim to exten te reuce MFE formulation, applying it to a realistic operational atmosperic numerical weater forecast system an to more complicate PDEs. We ave sown bot by teoretical analysis as well as by numerical examples tat te reuce MFE formulation presente erein as extensive potential applications. Toug Kunisc an Volkwein ave presente some Galerkin proper ortogonal ecomposition metos for a general equation in flui ynamics, i.e., for te nonstationary Navier Stokes equations in [28], our meto is ifferent from teir approaces, wose metos consist of Galerkin projection approaces were te original variables are substitute for linear combination of POD basis an te error estimates of te velocity fiel terein are only erive, teir POD basis being generate wit te solutions of te pysical system at all time instances, wile our POD basis is generate wit only few solutions of te pysical system wic are useful an of interest for us. Especially, only te velocity fiel is approximate in Reference [28], wile bot te velocity fiel an te pressure are all syncronously approximate in our present meto, an error estimates of velocity fiel an pressure approximate solutions are also syncronously erive. Tus our meto appears to be more optimal tan tat in [28]. references [] V. Girault an P. A. Raviart, Finite Element Metos for Navier Stokes Equations, Teorem an Algoritms, Springer Verlag, 986. [2] J. G. Heywoo, R. Rannacer, Finite element approximation of te nonstationary Navier Stokes problem, I. Regularity of solutions an secon orer estimates for spatial iscretization, SIAM J. Numer. Anal., 9 (982), pp [3] Z. D. Luo, Te tir orer estimate of mixe finite element for te Navier Stokes problems, Cinese Quarterly Journal of Matematics, 0:3 (995), pp [4] K. Fukunaga, Introuction to Statistical Recognition, Acaemic Press, 990. [5] I. T. Jolliffe, Principal Component Analysis, Springer Verlag, [6] D. T. Crommelin, an A. J. Maja, Strategies for moel reuction: comparing ifferent optimal bases, J. Atmos. Sci., 6 (2004), pp [7] A. J. Maja, I. Timofeyev, an E. Vanen Eijnen, Systematic strategies for stocastic moe reuction in climate, J. Atmos. Sci., 60 (2003), pp [8] F. Selten, Baroclinic empirical ortogonal functions as basis functions in an atmosperic moel, J. Atmos. Sci., 54 (997), pp

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