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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 895386, 19 pages
http://dx.doi.org/10.1155/2011/895386
Research Article

A New Reduced Stabilized Mixed Finite-Element Method Based on Proper Orthogonal Decomposition for the Transient Navier-Stokes Equations

1School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
2Department of Mathematics, Baoji University of Arts and Sciences, Baoji 721007, China
3College of Global Change and Earth System Science, Beijing Normal University, Beijing 100009, China
4Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China

Received 31 May 2011; Accepted 15 August 2011

Academic Editor: Jan Sladek

Copyright © 2011 Aiwen Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A reduced stabilized mixed finite-element (RSMFE) formulation based on proper orthogonal decomposition (POD) for the transient Navier-Stokes equations is presented. An ensemble of snapshots is compiled from the transient solutions derived from a stabilized mixed finite-element (SMFE) method based on two local Gauss integrations for the two-dimensional transient Navier-Stokes equations by using the lowest equal-order pair of finite elements. Then, the optimal orthogonal bases are reconstructed by implementing POD techniques for the ensemble snapshots. Combining POD with the SMFE formulation, a new low-dimensional and highly accurate SMFE method for the transient Navier-Stokes equations is obtained. The RSMFE formulation could not only greatly reduce its degrees of freedom but also circumvent the constraint of inf-sup stability condition. Error estimates between the SMFE solutions and the RSMFE solutions are derived. Numerical tests confirm that the errors between the RSMFE solutions and the SMFE solutions are consistent with the the theoretical results. Conclusion can be drawn that RSMFE method is feasible and efficient for solving the transient Navier-Stokes equations.

1. Introduction

Mixed finite-element (MFE) methods are one of the most important approaches for solving the nonstationary Navier-Stokes equations [13]. However, some fully discrete MFE formulations for the nonstationary Navier-Stokes equations involve generally many degrees of freedom. In addition, the importance of ensuring the compatibility of the approximations for the velocity and pressure by satisfying the so-called inf-sup condition (LBB condition) is widely understood. Thus, an important problem is how to avoid the lack of the LBB stability and simplify the computational load by saving time-consuming calculations and resource demands in the actual computational process in a way that guarantees a sufficiently accurate numerical solution.

Proper orthogonal decomposition (POD) is an effective method for approximating a large amount of data. The method essentially finds a group of orthogonal bases from the given data to approximately represent them in a least squares optimal sense. In addition, as the POD is optimal in the least squares sense, it has the property that the model decomposition is completely dependent on the given data and does not require assuming any prior knowledge of the process. Combined with a Galerkin projection procedure, POD provides a powerful method for deriving lower dimensional models of dynamical systems from a high or even infinite dimensional space. A dynamic system is generally governed by related structures or the ensemble formed by a large number of different instantaneous solutions, and the POD method can capture the temporal and spatial structures of dynamic system by applying a statistical analysis to the ensemble of data. POD provides an adequate approximation for a large amount of data with a reduced number of degrees of freedom; it alleviates the computational load and provides substantial savings in memory requirements. POD has found widespread application in a variety of fields such as signal analysis and pattern recognition [4, 5], fluid dynamics and coherent structures [68], optimal flow control problems [9, 10], and land surface soil moisture data assimilation [11]. In fluid dynamics, Lumley first applied the POD method to capture the large eddy coherent structures in a turbulent boundary layer [12]. This method was further applied to study the relation between the turbulent structure and a chaotic dynamic system [13]. Sirovich introduced the method of snapshots and applied it to reduce the order of POD eigenvalue problem [14]. Kunisch and Volkwein presented Galerkin POD methods for parabolic problems and a general equation in fluid dynamics [15, 16]. More recently, a finite difference scheme (FDS) and a MFE formulation for the nonstationary Navier-Stokes equation based on POD were derived [17, 18], respectively. Finite-element formulation based on POD was also applied for parabolic equations and the Burgers equation [19, 20]. In other physical applications, an effective use of POD for a chemical vapor deposition reactor was demonstrated and some reduced-order FDS and MFE for the upper tropical Pacific Ocean model based on POD were presented [2125]. An optimizing reduced FDS based on POD for the chemical vapor deposit (CVD) equations was also presented in [26]. Except for POD, the empirical orthogonal function (EOF) analysis is another effective method to extract information from large datasets in time and space [27, 28].

In order to avoid the lack of LBB stability, some kinds of stabilized techniques for the lowest-order finite elements appear in [2944]. Luo et al.[45] has combined the POD method with a stabilize method [40] to deal with the non-stationary Navier-Stokes equations and obtained good results. But the stabilized mixed methods in [40, 45] are often developed using residuals of the momentum equation. These residual terms must be formulated using mesh-dependent parameters, whose optimal values are usually unknown. Particularly, for the lowest equal-order pairs of mixed elements such as 𝑃1𝑃1 and 𝑄1𝑄1, pressure and velocity derivatives in the residual either vanish or are poorly approximated, causing difficulties in the application of consistent stabilization.

In this paper, we mainly consider the two-dimensional transient Navier-Stokes equations by combining a new stabilized finite-element method [2931] based on two local Gauss’ integrations with POD method. This new stabilized finite element method has some prominent features: parameter-free, avoiding higher-order derivatives or edge-based data structures, and stabilization being completed locally at the element level. In this manner, we could not only ensure the stabilization of solutions of fully discrete stabilized mixed finite-element system but also greatly reduce degrees of freedom and save time-consuming calculations and resource demands in the actual computational process in a way that guarantees a sufficiently accurate numerical solution. we also derive the error estimates between the original SMFE solutions and the RSMFE solutions based on the POD technique. Numerical experiments show the errors between the original SMFE method and the RSMFE solutions are consistent with theoretical results.

The remainder of this paper is organized as follows. In Section 2, an abstract functional setting for the two-dimensional Navier-Stokes equations is given, together with some basic notations. Section 3 is to state the fully discrete stabilized finite-element method and to generate snapshots from transient solutions computed from the equation system derived by the classical SMFE formulation. In Section 4, the optimal orthogonal bases are reconstructed from the elements of the snapshots with POD method and a reduced SMFE formulation with lower-dimensional number based on POD method for the transient Navier-Stokes equations is developed. In Section 5, error estimates between the classical SMFE solutions and the RSMFE solutions based on the POD method are derived. In Section 6, a series of numerical experiments are given to illustrate the theoretical results. We conclude with a few remarks in the final section.

2. Functional Setting of the Navier-Stokes Equations

Let Ω be a bounded domain in 𝑅2, assumed to have a Lipschitz continuous boundary Γ and to satisfy further assumptions below. The transient Navier-Stokes equations are considered as follows: 𝑢𝑡],𝜈Δ𝑢+(𝑢)𝑢+𝑝=𝑓,div𝑢=0,(𝑥,𝑡)Ω×(0,𝑇(2.1)𝑢(𝑥,0)=𝑢0(𝑥),𝑥Ω,𝑢(𝑥,𝑡)|Γ[].=0,𝑡0,𝑇(2.2) Here 𝑢Ω𝑅2 and 𝑝Ω𝑅 are the velocity and pressure, 𝜈>0 is the viscosity, and 𝑓 represents the body forces, 𝑇>0 the final time, and 𝑢𝑡=𝜕𝑢/𝜕𝑡.

For the mathematical setting of problems (2.1)-(2.2), we introduce the following Sobolev spaces: 𝑋=𝐻10(Ω)2,𝑀=𝐿20(Ω)=𝑞𝐿2(Ω);Ω𝑞(𝑥)𝑑𝑥=0,(2.3)𝐷(𝐴)=𝐻2(Ω)2𝑋,𝑉={𝑣𝑉div𝑣=0}.(2.4) Furthermore, we make a regularity assumption on the Stokes problem as follows.

Assumption H1. For a given 𝑔𝑌 and the Stokes problem, Δ𝑣+𝑞=𝑔,inΩ,div𝑣=0,inΩ,𝑣Γ=0,on𝜕Ω,(2.5) satisfying the following regularity result: 𝑣2+𝑞1𝜅𝑔0,(2.6) where 𝑖 is the norm of the Sobolev space 𝐻𝑖(Ω) or 𝐻𝑖(Ω)2, 𝑖=0,1,2, as appropriate, and 𝜅 is a positive constant depending only on Ω, which may stand for different value at its different occurrences. Subsequently, the positive constants 𝜅 and 𝑐 (with or without a subscript) will depend only on the data (𝜈,𝑇,Ω,𝑢0). Because the norm and seminorm are equivalent on 𝐻10(Ω)2, we use the same notation 1 for them. It is well known that for each 𝑣𝑋 there hold the following inequalities: 𝑣𝐿421/4𝑣01/2𝑣11/2.(2.7)

Assumption H2. The initial velocity 𝑢0𝐷(𝐴) and the body force 𝑓(𝑥,𝑡)𝐿2(0,𝑇;𝐿2(Ω)2) are assumed to satisfy 𝑢02+𝑇0𝑓20+𝑓𝑡20𝑑𝑡1/2𝑐.(2.8) Now, the bilinear forms 𝑎(,) and 𝑑(,), on 𝑋×𝑋 and 𝑋×𝑀, are defined, respectively, by 𝑎(𝑢,𝑣)=𝜈(𝑢,𝑣),𝑢,𝑣𝑋,𝑑(𝑣,𝑞)=(𝑞,div𝑣),(𝑣,𝑞)(𝑋,𝑀).(2.9) Also, a generalized bilinear form ((,);(,)) on (𝑋,𝑀)×(𝑋,𝑀) is defined by 𝐵((𝑢,𝑝);(𝑣,𝑞))=𝑎(𝑢,𝑣)𝑑(𝑣,𝑝)+𝑑(𝑢,𝑞).(2.10) Moreover, we define the trilinear form 1𝑏(𝑢,𝑣,𝑤)=((𝑢)𝑣,𝑤)+2=1((div𝑢)𝑣,𝑤)21((𝑢)𝑣,𝑤)2((𝑢)𝑤,𝑣),𝑢,𝑣,𝑤𝑋.(2.11) By the above notations and the Hölder inequality, there hold the following estimates: ||||1𝑏(𝑢,𝑣,𝑤)=𝑏(𝑢,𝑤,𝑣),𝑢𝑋,𝑣,𝑤𝑋,𝑏(𝑢,𝑣,𝑤)2𝑐0𝑢01/2𝑢11/2𝑣1𝑤01/2𝑤11/2+𝑣01/2𝑣11/2𝑤1||||+||||+||||,𝑢,𝑣,𝑤𝑋,𝑏(𝑢,𝑣,𝑤)𝑏(𝑣,𝑢,𝑤)𝑏(𝑤,𝑢,𝑣)𝑐1𝑢1𝑣2𝑤0,𝑢𝑋,𝑣𝐷(𝐴),𝑤𝑌.(2.12) Also, the Poincare inequality holds: 𝑣0𝛾0𝑣1,(2.13) where 𝑐0,𝑐1, and 𝛾0 are positive constants depending only on Ω.
For a given 𝑓𝑌, the variational formulation of problem (2.1)-(2.2) reads as follows: find (𝑢,𝑝)(𝑋,𝑀),𝑡>0 such that𝑢𝑡,𝑣+((𝑢,𝑝);(𝑣,𝑞))+𝑏(𝑢,𝑢,𝑣)=(𝑓,𝑣),(𝑣,𝑞)(𝑋,𝑀),𝑢(0)=𝑢0.(2.14)
For convenience, we recall the discrete Gronwall Lemma that will be frequently used.

Lemma 2.1 (see [1, 45, 46]). Let {𝑎𝑛},{𝑏𝑛}, and {𝑐𝑛} be three positive sequences, and let {𝑐𝑛} be monotone and satisfy 𝑎𝑛+𝑏𝑛𝑐𝑛+𝜆𝑛1𝑖=0𝑎𝑖,𝜆>0,𝑎0+𝑏0𝑐0,(2.15) then 𝑎𝑛+𝑏𝑛𝑐𝑛exp(𝑛𝜆),𝑛0.(2.16)

The following existence and uniqueness result is classical (see [1, 46]).

Theorem 2.2. Assume that (H1) and (H2) hold. Then, for any given 𝑇>0, there exists a unique solution (𝑢,𝑝) satisfying the following regularities: sup0<𝑡𝑇𝑢(𝑡)22+𝑝(𝑡)21+𝑢𝑡(𝑡)20𝑐,sup0<𝑡𝑇𝜏𝑢(𝑡)𝑡21+𝑇0𝜏𝑢(𝑡)𝑡22+𝑝𝑡21+𝑢𝑡𝑡20𝑑𝑡𝑐,(2.17) where 𝜏(𝑡)=min{1,𝑡}.

3. Fully Discrete SMFE Method and Generation of Snapshots

In this section, we focus on the stabilized method proposed by [29] for the Stokes equations. Let >0 be a real positive parameter. Finite-element subspace (𝑋,𝑀) of (𝑋,𝑀) is characterized by 𝜏=𝜏(Ω), a partitioning of Ω into triangles or quadrilaterals 𝐾, assumed to be regular in the usual sense; that is, for some 𝜎 and 𝜔 with 𝜎>1 and 0<𝜔<1, 𝐾𝜎𝜌𝐾,𝐾𝜏,||cos𝜃𝑖𝐾||𝜔,𝑖=1,2,3,4,𝐾𝜏,(3.1) where 𝐾 is the diameter of element 𝐾, 𝜌𝐾 is the diameter of the inscribed circle of element 𝐾, and 𝜃𝑖𝐾 are the angles of 𝐾 in the case of a quadrilateral partitioning. The mesh parameter is given by =max𝑘𝜏𝐾. The finite-element subspaces of this paper are defined by setting 𝑅1𝑃(𝐾)=1𝑄(𝐾),ifKistriangular,1(𝐾),ifKisthequadrilateral.(3.2) Then, the finite-element pairs are coupled as follows: 𝑋=𝑣𝑋;𝑣𝑖|𝐾𝑅1,𝑀(𝐾),𝑖=1,2=𝑞𝑀𝑞|𝐾𝑅1(𝐾),𝐾𝜏.(3.3) It is well known that this lowest equal-order finite-element pair does not satisfy the inf-sup condition. We define the following local difference between a consistent and an under-integrated mass matrices the stabilized formulation [2931]:𝐺𝑝,𝑞=𝑝𝑇𝑖𝑀𝑘𝑀1𝑞𝑗=𝑝𝑇𝑖𝑀𝑘𝑞𝑗𝑝𝑇𝑖𝑀1𝑞𝑗.(3.4) Here, we set 𝑝𝑇𝑖=𝑝0,𝑝1,,𝑝𝑁1𝑇,𝑞𝑗=𝑞0,𝑞1,,𝑞𝑁1,𝑀𝑖𝑗=𝜙𝑖,𝜙𝑗,𝑝=𝑁1𝑖=0𝑝𝑖𝜙𝑖,𝑝𝑖=𝑝𝑥𝑖,𝑝𝑀,𝑖,𝑗=0,,𝑁1,(3.5) where 𝜙𝑖 is the basis function of the pressure on the domain Ω such that its value is one at node 𝑥𝑖 and zero at other nodes; the symmetric and positive 𝑀𝑘,𝑘2 and 𝑀1 are pressure mass matrix computed by using the k-order and 1-order Gauss integrations in each direction; respectively, also, 𝑝𝑖 and 𝑞𝑖,𝑖=0,1,,𝑁, are the value of 𝑝 and 𝑞 at the node 𝑥𝑖. 𝑝𝑇𝑖 is the transpose of the matrix 𝑝𝑖.

Let Π𝑀𝑅0 be the standard 𝐿2-projection with the following properties [2932]:𝑝,𝑞=Π𝑝,𝑞,𝑝𝑀,𝑞𝑅0,Π𝑝0𝑐𝑝0,𝑝𝑀,𝑝Π𝑝0𝑝1,𝑝𝐻1(Ω)𝑀,(3.6) where 𝑅0={𝑞𝑀𝑞|𝐾isaconstant,𝐾𝐾}. Then we can rewrite the bilinear form 𝐺(,) by𝐺(𝑝,𝑞)=(𝑝Π𝑝,𝑞Π𝑞).(3.7)

Remark 3.1. The bilinear form 𝐺(,) in (3.7) is a symmetric, semipositive definite form generated on each local set 𝐾. The term can alleviate and offset the inf-sup condition [29]. It differs from the stabilized term in [45]. It does not require a selection of mesh-dependent stabilization parameters or a calculation of higher-order derivatives. Its another valuable feature is that the action of stabilization operators can be performed locally at the element level with minimal additional cost. With the above notation, we begin by choosing an integer 𝑁 and defining the time step 𝜏=𝑇/𝑁 and discrete times 𝑡𝑛=𝑛𝜏, 𝑛=0,1,2,,𝑁. We obtain the fully discrete scheme as follows: find functions {𝑢𝑛}𝑛0𝑋 and {𝑝𝑛}𝑛1𝑀 as solutions of the recursive linear algebraic equations, 𝑑𝑡𝑢𝑛,𝑣+𝑢𝑛,𝑝𝑛;𝑣,𝑞𝑢+𝑏𝑛1,𝑢,𝑣=𝑓𝑡𝑛,𝑣,𝑢0=𝑢0.(3.8) for all (𝑣,𝑞)(𝑋,𝑀), where 𝑑𝑡𝑢𝑛=𝑢𝑛𝑢𝑛1𝜏,𝑢,𝑝;𝑣,𝑞𝑢=𝑎,𝑣𝑣𝑑,𝑝𝑢+𝑑,𝑞𝑝+𝐺,𝑞,(3.9) and 𝑢0 is the approximation of 𝑢0. The solutions {𝑢𝑛}𝑛0 and {𝑝𝑛}𝑛1 to (3.8)-(3.9) are expected to the approximations of {𝑢(𝑡𝑛)}𝑛0 and {𝑝(𝑡𝑛)}𝑛1 with 𝑝𝑡𝑛=1𝜏𝑡𝑛𝑡𝑛1𝑝(𝑡)𝑑𝑡.(3.10)

Theorem 3.2 (see [2932]). Let (𝑋,𝑀) be defined as above, then there exists a positive constant 𝛽, independent of , such that ||||((𝑢,𝑝);(𝑣,𝑞))𝑐𝑢1+𝑝0𝑣1+𝑞0𝛽𝑢,(𝑢,𝑝),(𝑣,𝑞)(𝑋,𝑀),1+𝑝0sup𝑣,𝑞𝑋,𝑀||𝑢,𝑝;𝑣,𝑞||𝑣1+𝑞0𝑢,,𝑝𝑋,𝑀,||||𝐺(𝑝,𝑞)𝑐𝑝Π𝑝0𝑞Π𝑞0,𝑝,𝑞𝑀.(3.11)

By using the same approaches as those in [45], we can prove the following result.

Theorem 3.3. Under the assumptions of Theorems 2.2 and 3.2, if and 𝜏 are sufficiently small and =𝑂(𝜏), then problem (3.8)-(3.9) has a unique solution (𝑢𝑛,𝑝𝑛)𝑋×𝑀 such that 𝑢𝑛20+𝑛𝑗=1𝜏𝜈𝑢𝑗20𝑝+𝜏𝐺𝑗,𝑝𝑗𝑢020+𝑐𝜏𝜈𝑛1𝑗=1𝑓𝑗20.(3.12)

Theorem 3.4 (see [32]). Under the assumptions of Theorem 3.3, the error (𝑢(𝑡𝑛)𝑢𝑛,𝑝(𝑡𝑛)𝑝𝑛) satisfies the following bound: 𝜏𝑁𝑛=1𝑢𝑡𝑛𝑢𝑛20𝑡+𝜏𝑚𝑢𝑡𝑚𝑢𝑚20𝑐4+𝜏2,𝜏𝑁𝑛=1𝑢𝑡𝑛𝑢𝑛21𝑡+𝜏𝑚𝑢𝑡𝑚𝑢𝑚21𝑐2+𝜏2,𝜏𝑁𝑛=1𝜏𝑡𝑛𝑝𝑡𝑛𝑝𝑛20𝑐2+𝜏2,(3.13) for all 𝑡𝑚(0,𝑇].
If 𝜈, the time step increment 𝜏, and the right-hand side 𝑓 are given, by solving problem(3.8)-(3.9), we can obtain solution ensemble {𝑢𝑛1,𝑢𝑛2,𝑝𝑛}𝐿𝑛=1. Then we choose 𝐿 (in general, 𝐿𝑁, e.g., 𝐿=20,𝑁=200) instantaneous solutions 𝑈𝑖(𝑥,𝑦)=(𝑢𝑖1,𝑢𝑖2,𝑝𝑖)(𝑖=1,2,,𝐿) from the 𝐿 group of solutions (𝑢𝑛,𝑝𝑛)(1𝑛𝐿) for problems (3.8), which are known as snapshots.

Remark 3.5. When one computes actual problems, one may obtain the ensemble of snapshots from physical system trajectories by drawing samples from experiments and interpolation (or data assimilation). For example, when one finds numerical solutions to PDES representing weather forecast, one can use the previous weather prediction results to construct the ensemble of snapshots, then restructure the POD optimal basis for the ensemble of snapshots by the following POD method, next replace finite element space (𝑋,𝑀) with the subspace spanned by the optimal POD basis. Numerical weather forecast equation is reduced to a fully discrete algebra equation with fewer degrees of freedom. Thus, the forecast of future weather change can be quickly simulated, which is a result of major importance for real-life applications.

4. Reduced SMFE Formulation Based on POD Method

The POD method has received much attention in recent years as a tool to analyze complex physical systems. In this section, we use POD technique to deal with the snapshots in Section 3 and then use the POD basis to develop an RSMFE formulation for the transient Navier-Stokes equations.

Let 𝑋=𝑋×𝑀, and let 𝑈𝑖(𝑥,𝑦)=(𝑢𝑖1,𝑢𝑖2,𝑝𝑖)(𝑖=1,2,,𝐿, see Section 3). Set𝑈𝑉=span1,𝑈2,,𝑈𝐿,(4.1) where 𝑉 is the ensemble consisting of the snapshots {𝑈𝑖}𝐿𝑖=1, at least one of which is supposed to nonzero. Let {Ψ𝑗}𝑙𝑗=1 denote an orthogonal basis of 𝑉 with 𝑙=dim𝑉(𝑙𝐿). Then each member of the ensemble is expressed as𝑈𝑖=𝑙𝑖=1𝑈𝑖,Ψ𝑗𝑋Ψ𝑗,for𝑖=1,2,,𝐿,(4.2) where (𝑈𝑖,Ψ𝑗)𝑋=(𝑢𝑖,Ψ𝑢𝑗)+(𝑝𝑖,Ψ𝑝𝑗), Ψ𝑢𝑗 and Ψ𝑝𝑗 are the orthogonal basis corresponding to 𝑢 and 𝑝, respectively.

The method of POD consists in finding the orthogonal basis such that, for every 𝑑(1𝑑𝐿), the mean square error between the elements 𝑈𝑖(1𝑖𝐿) and corresponding 𝑑th partial sum of (4.2) is minimized on averageminΨ𝑗𝑑𝑗=11𝐿𝐿𝑖=1𝑈𝑖𝑑𝑗=1𝑈𝑖,Ψ𝑗𝑋Ψ𝑗2𝑋,(4.3) subject toΨ𝑖,Ψ𝑗𝑋=𝛿𝑖𝑗,for1𝑖𝑑,1𝑗𝑖,(4.4) where 𝑈𝑖𝑋=(𝑢𝑖121+𝑢𝑖221+𝑝𝑖20)1/2. A solution sequence {Ψ𝑗}𝑑𝑗=1 of (4.3) and (4.4) is known as a POD basis of rank 𝑑.

We introduce the correlation matrix 𝐸=(𝐸𝑖𝑗)𝐿×𝐿𝑅𝐿×𝐿 corresponding to the snapshots {𝑈𝑖}𝐿𝑖=1 by𝐸𝑖𝑗=1𝐿𝑈𝑖,𝑈𝑗𝑋.(4.5) The matrix 𝐸 is positive semidefinite and has rank 𝑙. The solution of (4.3) and (4.4) can be found in [45].

Proposition 4.1. Let 𝜆1𝜆2𝜆𝑙>0 denote the positive eigenvalues of 𝐸, and let 𝑣1,𝑣2,,𝑣𝑙 be the associated eigenvectors. Then a POD basis of rank 𝑑𝑙 is given by Ψ𝑖=1𝜆𝑖𝐿𝑗=1𝑣𝑖𝑗,𝑈𝑗,𝑖=1,2,,𝑑𝑙,(4.6) where (𝑣𝑖)𝑗 denotes the 𝑗th component of the eigenvector 𝑣𝑖. Furthermore, the following error formula holds: 1𝐿𝐿𝑖=1𝑈𝑖𝑑𝑗=1𝑈𝑖,Ψ𝑗𝑋Ψ𝑗2𝑋=𝑙𝑗=𝑑+1𝜆𝑗.(4.7)
Let 𝑉𝑑={Ψ1,Ψ2,,Ψ𝑑}, and let 𝑋𝑑×𝑀𝑑=𝑉𝑑 with 𝑋𝑑X𝑋, and let 𝑀𝑑𝑀𝑀. Set the Ritz-projection 𝑃𝑋𝑋 (if 𝑃 is restricted to the Ritz-projection from 𝑋 to 𝑋𝑑, it is written as 𝑃𝑑) such that 𝑃|𝑋=𝑃𝑑𝑋𝑋𝑑 and 𝑃𝑋𝑋𝑋𝑋𝑑 and 𝐿2-projection 𝜌𝑑𝑀𝑀𝑑 denoted by, respectively, 𝑎𝑃𝑢,𝑣=𝑎𝑢,𝑣,𝑣𝑋,𝜌(4.8)𝑑𝑝,𝑞𝑑0=𝑝,𝑞𝑑0,𝑞𝑑𝑀𝑑,(4.9) where 𝑢𝑋 and 𝑝𝑀. Owing to (4.8)-(4.9) the linear operators 𝑃 and 𝜌𝑑 are well defined and bounded: 𝑃𝑑𝑢1𝑢1,𝜌𝑑𝑝0,𝑝0,𝑢𝑋,𝑝𝑀.(4.10)

Lemma 4.2 (see [45]). For every 𝑑(1𝑑𝑙), the projection operators 𝑃𝑑 and 𝜌𝑑 satisfy, respectively, 1𝐿𝐿𝑖=1𝑢𝑛𝑖𝑃𝑑𝑢𝑛𝑖21𝑙𝑗=𝑑+1𝜆𝑗,1(4.11)𝐿𝐿𝑖=1𝑢𝑛𝑖𝑃d𝑢𝑛𝑖20𝐶2𝑙𝑗=𝑑+1𝜆𝑗1,(4.12)𝐿𝐿𝑖=1𝑝𝑛𝑖𝜌𝑑𝑝𝑛𝑖20𝑙𝑗=𝑑+1𝜆𝑗,(4.13) where 𝑢𝑛𝑖=(𝑢𝑛𝑖1,𝑢𝑛𝑖2) and (𝑢𝑛𝑖1,𝑢𝑛𝑖2,𝑝𝑛𝑖)𝑉.
Thus, using 𝑉𝑑=𝑋𝑑×𝑀𝑑, we can obtain the reduced SMFE formulation for problems (3.8) as follows. Find ̂𝑢𝑛𝑑=(𝑢𝑛𝑑,𝑝𝑛𝑑)𝑉𝑑 such that 𝑑𝑡𝑢𝑛𝑑,𝑣𝑑+𝐵𝑢𝑛𝑑,𝑝𝑛𝑑;𝑣𝑑,𝑞𝑑𝑢+𝑏𝑑𝑛1,𝑢𝑑,𝑣𝑑=𝑓𝑡𝑛,𝑣𝑑𝑢,1𝑛𝑁,0𝑑=𝑢0.(4.14)

Remark 4.3. Problem (3.8) includes 𝑁 (𝑁 is the number of triangles or quadrilaterals vertex in 𝜏) freedom degrees, while problem (4.14) includes 𝑑(𝑑𝑙𝐿𝑁) freedom degrees. For actual science and engineering problems, the number of the vertex in 𝜏 are tens of thousands, even hundreds of millions, but 𝑑 is the number of the largest eigenvalues of 𝑙 snapshots from 𝐿 transient solutions; it is very small. For numerical example in Section 6, 𝑑=7, but 𝑁=32×32×3=3072. Thus, problem (4.14) is a simplified stabilized finite-element scheme. In addition, the future development of many natural phenomena is affected by previous information, such as biological evolution and weather change. Here, we use the existing data to construct the POD basis, which contains the information on past data. Thus, this method can not only save computational load, but also make better use of the existing information to capture the law of the future development of natural phenomena.

5. Existence and Error Analysis of Solution to the Optimizing RSMFE Formulation

This section is devoted to discussing the existence and error estimates of solutions to problem (4.14). We see from (4.6) that 𝑉𝑑=𝑋𝑑×𝑀𝑑𝑉𝑋×𝑀𝑋×𝑀. Using the same approaches as proving Theorem 3.3, we could prove the following existence result for solutions of problem (4.14).

Theorem 5.1. Under the assumptions of Theorems 2.2 and 3.3, Problem (4.14) has a unique solution sequence (𝑢𝑛𝑑,𝑝𝑛𝑑)𝑋𝑑×𝑀𝑑 and satisfies, for 1𝑛𝑁, 𝑢𝑛𝑑20+𝑛𝑗=1𝑢𝜏𝜈𝑗𝑑21𝑝+𝜏𝐺𝑗𝑑,𝑝𝑗𝑑𝑢020+𝑐𝜏𝜈𝑛1𝑗=1𝑓𝑗20.(5.1)

In the following theorem, the errors between the solution (𝑢𝑛𝑑,𝑝𝑛𝑑) to Problem (4.14)-(4.15) and the solution (𝑢𝑛,𝑝𝑛) to Problem (3.8) are derived.

Theorem 5.2. Under the assumptions of Theorem 5.1, if and 𝜏 are sufficiently small, =𝑂(𝜏), and 𝜏=𝑂(𝐿2), then the errors between the solutions (𝑢𝑛𝑑,𝑝𝑛𝑑) to Problem (4.14), and the solutions (𝑢𝑛,𝑝𝑛) to Problem (3.8) have the following error estimates, for 1𝑛𝑁: 𝑢𝑛𝑢𝑛𝑑20+𝜏𝜈𝑛𝑖𝑗=𝑛1𝑢𝑗𝑢𝑗𝑑21+𝜏𝑛𝑖𝑗=𝑛1𝑝𝑗𝑝𝑗𝑑20𝐶𝜏𝑙1/2𝑗=𝑑+1𝜆𝑗,if𝑛=𝑛𝑖𝑛1,𝑛2,,𝑛𝐿,𝑢(5.2)𝑛𝑢𝑛𝑑20𝑢+𝜏𝜈𝑛𝑢𝑛𝑑21+𝑛𝑖𝑗=𝑛1𝑢𝑗𝑢𝑗𝑑21𝑝+𝜏𝑛𝑝𝑛𝑑20+𝑛𝑖𝑗=𝑛1𝑝𝑗𝑝𝑗𝑑20𝐶𝜏𝑙1/2𝑗=𝑑+1𝜆𝑗+𝐶𝜏2,if𝑛=𝑛𝑖𝑛1,𝑛2,,𝑛𝐿.(5.3)

Proof. Let 𝑤𝑛𝑑=𝑃𝑑𝑢𝑛𝑢𝑛𝑑, 𝑟𝑛𝑑=𝜌𝑑𝑝𝑛𝑝𝑛𝑑. Subtracting (3.8) from (4.14) yields that 1𝜏𝑢𝑛𝑢𝑛𝑑,𝑣𝑑1𝜏𝑢𝑛1𝑢𝑑𝑛1,𝑣𝑑𝑢+𝑎𝑛𝑢𝑛𝑑,𝑣𝑑𝑝𝑏𝑛𝑝𝑛𝑑,𝑣𝑑𝑢+𝑏𝑛1𝑢𝑑𝑛1,𝑢𝑛,𝑣𝑑𝑢+𝑏𝑑𝑛1,𝑢𝑛𝑢𝑛𝑑,𝑣𝑑𝑢+𝑏𝑛𝑢𝑛𝑑,𝑞𝑑𝑝+𝐺𝑛𝑝𝑛𝑑,𝑞𝑑=0.(5.4) Taking (𝑣𝑑,𝑞𝑑)=2𝜏(𝑤𝑛𝑑,𝑟𝑛𝑑) in (5.4), since 𝑎(𝑢𝑛𝑃𝑑𝑢𝑛,𝑤𝑛𝑑)=0, 𝑏(𝑢𝑛𝑢𝑛𝑑,𝑟𝑛𝑑)+𝐺(𝑝𝑛𝑝𝑛𝑑,𝑟𝑛𝑑)=0, we deduce 2𝑤𝑛𝑑,𝑤𝑛𝑑𝑤2𝑑𝑛1,𝑤𝑛𝑑𝑤+2𝜏𝑎𝑛𝑑,𝑤𝑛𝑑𝑢=2𝑛1𝑝𝑑𝑢𝑛1,𝑤𝑛𝑑𝑝+2𝑑𝑢𝑛𝑢𝑛,𝑤𝑛𝑑𝑝+2𝜏𝑏𝑛𝜌𝑑𝑝𝑛,𝑤𝑛𝑑𝑃+2𝜏𝑏𝑑𝑢𝑛1𝑢𝑛1,𝑢𝑛,𝑤𝑛𝑑𝑤2𝜏𝑏𝑑𝑛1,𝑢𝑛,𝑤𝑛𝑑𝑢+2𝜏𝑏𝑑𝑛1,𝑃𝑑𝑢𝑛𝑢𝑛,𝑤𝑛𝑑.(5.5) Using (2.12)-(2.13), the Hölder inequality and the Young inequality, we see that ||𝑢𝑛1𝑝𝑑𝑢𝑛1,𝑤𝑛𝑑+𝑝𝑑𝑢𝑛𝑢𝑛,𝑤𝑛𝑑||𝐶1𝑢𝑛1𝑝𝑑𝑢𝑛121+𝑢𝑛𝑝𝑑𝑢𝑛21+𝜈𝑤10𝑑21,||𝑏𝑝𝑛𝜌𝑑𝑝𝑛,𝑤𝑛𝑑||𝐶4𝑝𝑛𝜌𝑑𝑝𝑛20+𝜈𝑤10𝑛𝑑21,||𝑏𝑃𝑑𝑢𝑛1𝑢𝑛1,𝑢𝑛,𝑤𝑛𝑑||𝜈𝑤10𝑛𝑑21+𝐶5𝑢𝑛1𝑃𝑑𝑢𝑛121,||𝑏𝑢𝑑𝑛1,𝑃𝑑𝑢𝑛𝑢𝑛,𝑤𝑛𝑑||𝜈𝑤10𝑛𝑑21+𝐶6𝑢𝑛𝑃𝑑𝑢𝑛21,||𝑤𝑏𝑑𝑛1,𝑢𝑛,𝑤𝑛𝑑||𝐶7𝑤𝑑𝑛120+𝜈𝑤10𝑛𝑑21.(5.6) Noting that 𝑎(𝑎𝑏)=[𝑎2𝑏2+(𝑎𝑏)2]/2 (for (𝑎0 and 𝑏0)), owing to (5.6), we obtain that 𝑤𝑛𝑑20𝑤𝑑𝑛120𝑤+𝜈𝜏𝑑21𝐶7𝜏𝑤𝑑𝑛120𝑢+𝐶𝜏𝑛𝑃𝑑𝑢𝑛21+𝑢𝑛1𝑃𝑑𝑢𝑛121+𝑝𝑛𝜌𝑑𝑝𝑛20.(5.7) First, we consider the case of 𝑛{𝑛1,𝑛2,,𝑛𝐿}. Summing (5.7) from 𝑛1 to 𝑛𝑖, 𝑖=1,2,,𝐿, and noting that 𝑢0𝑢0𝑑=0, using Lemma 4.2, we can derive that 𝑤𝑛𝑖𝑑20+𝜈𝜏𝑛𝑖𝑗=𝑛1𝑤𝑗𝑑21𝐶𝜏𝐿𝑙𝑗=𝑑+1𝜆𝑗+𝐶7𝜏𝑛𝑖1𝑗=𝑛0𝑤𝑗𝑑20.(5.8) By using the discrete Gronwall inequality, we obtain that 𝑤𝑛𝑖𝑑20+𝜈𝜏𝑛𝑖𝑗=𝑛1𝑤𝑗𝑑21𝐶𝜏𝐿𝑙𝑗=𝑑+1𝜆𝑗𝐶exp7𝜏𝑛𝑖.(5.9) If and 𝜏 are sufficiently small, 𝜏=𝑂(𝐿2), and noting that 𝑛𝑖𝜏𝑛𝑖𝑁𝑇, we find that 𝑤𝑛𝑖𝑑20+𝜈𝜏𝑛𝑖𝑗=𝑛1𝑤𝑗𝑑21𝐶𝜏𝑙1/2𝑗=𝑑+1𝜆𝑗.(5.10) Thanks to 𝑏(𝑢𝑛𝑢𝑛𝑑,𝑟𝑛𝑑)+𝐺(𝑝𝑛𝑝𝑛𝑑,𝑟𝑛𝑑)=0, we obtain 𝑝𝑛𝑝𝑛𝑑0𝑢𝐶𝑛𝑢𝑛𝑑1.(5.11) By combining (5.10)-(5.11) with Lemma 4.2, we obtain the error estimate result (5.2).

Next, we consider the case of 𝑛{1,2,,𝐿}; we assume that 𝑡𝑛(𝑡𝑛𝑖1,𝑡𝑛𝑖) and 𝑡𝑛 is the nearest point to 𝑡𝑛𝑖. 𝑢𝑛 and 𝑝𝑛 are expanded into the Taylor series expansion at point 𝑡𝑛𝑖.𝑢𝑛=𝑢𝑛𝑖𝑠𝜏𝜕𝑢𝜉1𝜕𝑡,𝜉1𝑡𝑛𝑖,𝑡𝑛,𝑝(5.12)𝑛=𝑝𝑛𝑖𝑠𝜏𝜕𝑝𝜉2𝜕𝑡,𝜉2𝑡𝑛𝑖,𝑡𝑛,(5.13) where 𝑠 is the number of time steps from 𝑡𝑛 to 𝑡𝑛𝑖. If the snapshots are equably taken, then 𝑠𝑁/𝐿. Summing (5.7) from 𝑛1 to 𝑛𝑖, n, and using (5.12), if |𝜕𝑢(𝜉1)/𝜕𝑡| and |𝜕𝑝(𝜉2)/𝜕𝑡| are bounded, by discrete Gronwall inequality and Lemma 4.2 and (3.12), we obtain that 𝑤𝑛𝑑20𝑤+𝜈𝜏𝑛𝑑21+𝑛𝑖𝑗=𝑛1𝑤𝑗𝑑21𝐶𝜏𝐿𝑙𝑗=𝑑+1𝜆𝑗+𝐶𝜏2.(5.14)

If 𝜏=𝑂(𝐿2), by (5.14) we obtain that𝑤𝑛𝑑20𝑤+𝜈𝜏𝑛𝑑21+𝑛𝑖𝑗=𝑛1𝑤𝑗𝑑21𝐶𝜏𝑙1/2𝑗=𝑑+1𝜆𝑗+𝐶𝜏2.(5.15) Hence, combining (5.11), (5.13), and (5.15) with Lemma 4.2 yields (5.3).

Theorem 5.3. Under hypotheses of Theorems 3.4 and 5.2, the error estimates between the solution (𝑢(𝑡),𝑝(𝑡)) to Problem (2.1)-(2.2) and the solutions (𝑢𝑛𝑑,𝑝𝑛𝑑) to Problem (4.14) are as follows: 𝑢𝑡𝑛𝑢𝑛𝑑20+𝜏𝜈𝑛𝑗=1u𝑗𝑢𝑗𝑑21+𝜏𝑛𝑗=1𝑝𝑗𝑝𝑗𝑑20𝑐2+𝜏2+𝐶𝜏𝑙1/2𝑗=𝑑+1𝜆𝑗,𝑛=1,2,,𝑁.(5.16)

6. Numerical Examples

In order to illustrate and verify the theoretical results of Theorem 5.3, we present the results obtained in a simple test case. We set Ω is the unit square [0,1]×[0,1] and viscosity 𝜈=0.05. The velocity and pressure are designed on the same uniform triangulation of Ω. The exact solution is given by 𝑢𝑢=1(𝑥,𝑦),𝑢2𝑢(𝑥,𝑦),𝑝(𝑥,𝑦)=10(2𝑥1)(2𝑦1)cos(𝑡),1(𝑥,𝑦)=10𝑥2(1𝑥)2𝑢𝑦(1𝑦)(12𝑦)cos(𝑡),2(𝑥,𝑦)=10𝑥(1𝑥)(12𝑥)𝑦2(1𝑦)2cos(𝑡),(6.1) and 𝑓 is determined by (2.1).

All the numerical experiments have been performed using the conforming 𝑄1 finite element for both velocity and pressure. The implicit (backward) Euler’s scheme is used for the time discretization. For simplicity, the unit square is divided into 𝑛×𝑛 small squares with side length =1/𝑛. In order to make 𝜏=𝑂(), we take time step increment as 𝜏=1/𝑛.

We obtain 20 values (i.e., snapshots) outputting at time 𝑡=10𝜏,20𝜏,30𝜏,,200𝜏 by solving the SMFE formulation. We use 7 optimal POD bases to obtain the solutions of the reduced formulation problem (4.14) as 𝑡=200𝜏. In Table 1, we present the velocity and pressure relative error estimates, convergence rates, and CPU times using the SMFE method, and, in Table 2, we give the corresponding results obtained using the RSMFE method. In particular, as 𝑛=32, the SMFE solutions (𝑢200𝑖, 𝑖=1,2)(c), the exact solutions (𝑢𝑖200,𝑖=1,2) (b), and the RSMFE solutions (𝑢200𝑑𝑖,𝑖=1,2) (a) are depicted, respectively, in Figures 1 and 3. Moreover, the difference (𝑢𝑖200𝑢200𝑑𝑖, 𝑖=1,2) (a) between the exact solutions and the RSMFE solutions and the difference (𝑢200𝑖𝑢200𝑑𝑖,𝑖=1,2) (b) between the SMFE solutions and RSMFE solutions are depicted in Figures 2 and 4, respectively. From Tables 1, 2, and Figures 14, we can find that the RSMFE solutions has the same accuracy as the reduced SMFE solutions and the exact solutions. As 𝑛=32, for the SMFE Problem (3.8)-(3.9), there are 3×32×32=3072 freedom degrees; the performing time required is 2906 seconds, while the reduced SMFE Problem (4.14) with 7 POD bases only has 7 freedom degrees and the corresponding time is only 14 seconds, that is, the required implementing time to solve the usual SMFE Problem (3.8) is as 207 times as that to do the reduced SMFE problem (4.14) with 7 POD bases, while the errors between their respective solutions do not exceed 3×103. As 𝑛=56, Figure 5 shows the velocity 𝐻1 relative errors between solutions with different number of optimal POD bases and solutions obtained with full bases at 𝑡=100𝜏 and 𝑡=200𝜏. It is shown that the reduced SMFE problem (4.14) is very effective and feasible. In addition, the results obtained for the numerical examples are consistent with the theoretical ones.

tab1
Table 1: Numerical results for the SMFE method.
tab2
Table 2: Numerical results for the RSMFE method.
fig1
Figure 1: RSMFE solutions (a), the exact solutions (b), and SMFE solutions (c).
fig2
Figure 2: Difference between the RSMFE solutions and the exact solutions (a) and difference between the SMFE solutions and the RSMFE solutions (b).
fig3
Figure 3: RSMFE solutions (a), the exact solutions (b), and SMFE solutions (c).
fig4
Figure 4: Difference between the RSMFE solutions and the exact solutions (a) and difference between the SMFE solutions and the RSMFE solutions (b).
895386.fig.005
Figure 5: The velocity 𝐻1-error changing with the number of POD basis as 𝑛=56.

7. Conclusions

In this paper, we have combined the POD techniques with a SMFE formulation based on two local Gauss’ integrations to derive a reduced SMFE method for the transient Navier-Stokes equations. The discretization uses a pair of spaces of finite elements 𝑃1𝑃1 over triangles or 𝑄1𝑄1 over quadrilateral elements. This SMFE method differs from that in [45]. It has some prominent features: parameter-free, avoiding higher-order derivatives or edge-based data structures, and stabilization being completed locally at the element level. We have also analyzed the errors between the solutions of their usual SMFE formulation and the solutions of the reduced SMFE based on POD basis and discussed theoretically the relation of the number of snapshots and the number of solutions at all time instances, which have shown that our present method has improved and innovated the existing methods. We have validated the correctness of our theoretical results with numerical examples. Though snapshots and the POD basis of our numerical examples are constructed with the solutions of the usual SMFE formulation, when one computes actual problems, one may structure the snapshots and the POD basis with interpolation or data assimilation by drawing samples from experiments, then solve Problem (4.14), while it is unnecessary to solve Problem (3.8). Thus, the time-consuming calculations and resource demands in the computational process are greatly saved, and the computational efficiency is vastly improved. Therefore, the method in this paper holds a good prospect of extensive applications.

Future research work in this area aims at addressing some practical engineering problems arising in the fluid dynamics and more complicated PDES, extending the optimizing reduced SMFE formulation, applying it to a realistic atmosphere quality forecast system, and to a set of more complicated nonlinear PDES, for instance, 3D realistic model equations coupling strongly nonlinear properties, nonhomogeneous variable flux, and boundary.

Acknowledgments

This work is supported by the Scientific Research Common Program of the Beijing Municipal Commission of Education (Grant no. KM201110772019), the Knowledge Innovation Program of the Chinese Academy of Sciences (Grant no. KZCX2-EW-QN207), the National Natural Science Foundation of China (Grant no. 11071193 and 41075076), the Academic Human Resources Development in the Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (Grant no. PHR201006116), the Natural Science New Star of Science and Technologies Research Plan in Shaanxi Province of China (Grant no. 2011kjxx12), Research Program of the Education Department of the Shaanxi Province (Grant no. 11JK0490). The author would like to thank the referees for their valuable comments and suggestions.

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