Mathematical Problems in Engineering

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Coupled Numerical Methods in Engineering Analysis

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Volume 2011 |Article ID 895386 | https://doi.org/10.1155/2011/895386

Aiwen Wang, Jian Li, Zhenhua Di, Xiangjun Tian, Dongxiu Xie, "A New Reduced Stabilized Mixed Finite-Element Method Based on Proper Orthogonal Decomposition for the Transient Navier-Stokes Equations", Mathematical Problems in Engineering, vol. 2011, Article ID 895386, 19 pages, 2011. https://doi.org/10.1155/2011/895386

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

Academic Editor: Jan Sladek
Received31 May 2011
Accepted15 Aug 2011
Published20 Oct 2011

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 [1–3]. 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 [6–8], 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 [21–25]. 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 [29–44]. 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 𝑃1βˆ’π‘ƒ1 and 𝑄1βˆ’π‘„1, 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 [29–31] 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 Ξ© be a bounded domain in 𝑅2, assumed to have a Lipschitz continuous boundary Ξ“ and to satisfy further assumptions below. The transient Navier-Stokes equations are considered as follows: 𝑒𝑑],βˆ’πœˆΞ”π‘’+(π‘’β‹…βˆ‡)𝑒+βˆ‡π‘=𝑓,div𝑒=0,(π‘₯,𝑑)βˆˆΞ©Γ—(0,𝑇(2.1)𝑒(π‘₯,0)=𝑒0(π‘₯),π‘₯∈Ω,𝑒(π‘₯,𝑑)|Ξ“[].=0,π‘‘βˆˆ0,𝑇(2.2) Here π‘’βˆΆΞ©β†’π‘…2 and π‘βˆΆΞ©β†’π‘… are the velocity and pressure, 𝜈>0 is the viscosity, and 𝑓 represents the body forces, 𝑇>0 the final time, and 𝑒𝑑=πœ•π‘’/πœ•π‘‘.

For the mathematical setting of problems (2.1)-(2.2), we introduce the following Sobolev spaces: 𝑋=𝐻10(Ξ©)2,𝑀=𝐿20ξ‚»(Ξ©)=π‘žβˆˆπΏ2ξ€œ(Ξ©);Ξ©ξ‚Όπ‘ž(π‘₯)𝑑π‘₯=0,(2.3)𝐷(𝐴)=𝐻2(Ξ©)2βˆ©π‘‹,𝑉={π‘£βˆˆπ‘‰βˆΆdiv𝑣=0}.(2.4) Furthermore, we make a regularity assumption on the Stokes problem as follows.

Assumption H1. For a given π‘”βˆˆπ‘Œ and the Stokes problem, βˆ’Ξ”π‘£+βˆ‡π‘ž=𝑔,inΞ©,div𝑣=0,inΞ©,π‘£βˆ£Ξ“=0,onπœ•Ξ©,(2.5) satisfying the following regularity result: ‖𝑣‖2+β€–π‘žβ€–1β‰€πœ…β€–π‘”β€–0,(2.6) where ‖⋅‖𝑖 is the norm of the Sobolev space 𝐻𝑖(Ξ©) or 𝐻𝑖(Ξ©)2, 𝑖=0,1,2, as appropriate, and πœ… is a positive constant depending only on Ξ©, which may stand for different value at its different occurrences. Subsequently, the positive constants πœ… and 𝑐 (with or without a subscript) will depend only on the data (𝜈,𝑇,Ξ©,𝑒0). Because the norm and seminorm are equivalent on 𝐻10(Ξ©)2, we use the same notation β€–β‹…β€–1 for them. It is well known that for each π‘£βˆˆπ‘‹ there hold the following inequalities: ‖𝑣‖𝐿4≀21/4‖𝑣‖01/2‖𝑣‖11/2.(2.7)

Assumption H2. The initial velocity 𝑒0∈𝐷(𝐴) and the body force 𝑓(π‘₯,𝑑)∈𝐿2(0,𝑇;𝐿2(Ξ©)2) are assumed to satisfy ‖‖𝑒0β€–β€–2+ξ‚΅ξ€œπ‘‡0‖𝑓‖20+‖‖𝑓𝑑‖‖20𝑑𝑑1/2≀𝑐.(2.8) Now, the bilinear forms π‘Ž(β‹…,β‹…) and 𝑑(β‹…,β‹…), on 𝑋×𝑋 and 𝑋×𝑀, are defined, respectively, by π‘Ž(𝑒,𝑣)=𝜈(βˆ‡π‘’,βˆ‡π‘£),βˆ€π‘’,π‘£βˆˆπ‘‹,𝑑(𝑣,π‘ž)=(π‘ž,div𝑣),βˆ€(𝑣,π‘ž)∈(𝑋,𝑀).(2.9) Also, a generalized bilinear form ℬ((β‹…,β‹…);(β‹…,β‹…)) on (𝑋,𝑀)Γ—(𝑋,𝑀) is defined by 𝐡((𝑒,𝑝);(𝑣,π‘ž))=π‘Ž(𝑒,𝑣)βˆ’π‘‘(𝑣,𝑝)+𝑑(𝑒,π‘ž).(2.10) Moreover, we define the trilinear form 1𝑏(𝑒,𝑣,𝑀)=((π‘’β‹…βˆ‡)𝑣,𝑀)+2=1((div𝑒)𝑣,𝑀)21((π‘’β‹…βˆ‡)𝑣,𝑀)βˆ’2((π‘’β‹…βˆ‡)𝑀,𝑣),βˆ€π‘’,𝑣,π‘€βˆˆπ‘‹.(2.11) By the above notations and the HΓΆlder inequality, there hold the following estimates: ||||≀1𝑏(𝑒,𝑣,𝑀)=βˆ’π‘(𝑒,𝑀,𝑣),βˆ€π‘’βˆˆπ‘‹,𝑣,π‘€βˆˆπ‘‹,𝑏(𝑒,𝑣,𝑀)2𝑐0‖𝑒‖01/2‖𝑒‖11/2‖𝑣‖1‖𝑀‖01/2‖𝑀‖11/2+‖𝑣‖01/2‖𝑣‖11/2‖𝑀‖1ξ€Έ||||+||||+||||,βˆ€π‘’,𝑣,π‘€βˆˆπ‘‹,𝑏(𝑒,𝑣,𝑀)𝑏(𝑣,𝑒,𝑀)𝑏(𝑀,𝑒,𝑣)≀𝑐1‖𝑒‖1‖𝑣‖2‖𝑀‖0,βˆ€π‘’βˆˆπ‘‹,π‘£βˆˆπ·(𝐴),π‘€βˆˆπ‘Œ.(2.12) Also, the Poincare inequality holds: ‖𝑣‖0≀𝛾0‖𝑣‖1,(2.13) where 𝑐0,𝑐1, and 𝛾0 are positive constants depending only on Ξ©.
For a given π‘“βˆˆπ‘Œ, the variational formulation of problem (2.1)-(2.2) reads as follows: find (𝑒,𝑝)∈(𝑋,𝑀),𝑑>0 such that𝑒𝑑,𝑣+ℬ((𝑒,𝑝);(𝑣,π‘ž))+𝑏(𝑒,𝑒,𝑣)=(𝑓,𝑣),βˆ€(𝑣,π‘ž)∈(𝑋,𝑀),𝑒(0)=𝑒0.(2.14)
For convenience, we recall the discrete Gronwall Lemma that will be frequently used.

Lemma 2.1 (see [1, 45, 46]). Let {π‘Žπ‘›},{𝑏𝑛}, and {𝑐𝑛} be three positive sequences, and let {𝑐𝑛} be monotone and satisfy π‘Žπ‘›+𝑏𝑛≀𝑐𝑛+πœ†π‘›βˆ’1𝑖=0π‘Žπ‘–,πœ†>0,π‘Ž0+𝑏0≀𝑐0,(2.15) then π‘Žπ‘›+𝑏𝑛≀𝑐𝑛exp(π‘›πœ†),𝑛β‰₯0.(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 (𝑒,𝑝) satisfying the following regularities: sup0<𝑑≀𝑇‖‖𝑒(𝑑)22+‖‖𝑝(𝑑)21+‖‖𝑒𝑑‖‖(𝑑)20≀𝑐,sup0<π‘‘β‰€π‘‡πœβ€–β€–π‘’(𝑑)𝑑‖‖21+ξ€œπ‘‡0πœξ‚€β€–β€–π‘’(𝑑)𝑑‖‖22+‖‖𝑝𝑑‖‖21+‖‖𝑒𝑑𝑑‖‖20𝑑𝑑≀𝑐,(2.17) where 𝜏(𝑑)=min{1,𝑑}.

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 β„Ž>0 be a real positive parameter. Finite-element subspace (π‘‹β„Ž,π‘€β„Ž) of (𝑋,𝑀) is characterized by πœβ„Ž=πœβ„Ž(Ξ©), a partitioning of Ξ© into triangles or quadrilaterals 𝐾, assumed to be regular in the usual sense; that is, for some 𝜎 and πœ” with 𝜎>1 and 0<πœ”<1, β„ŽπΎβ‰€πœŽπœŒπΎ,βˆ€πΎβˆˆπœβ„Ž,||cosπœƒπ‘–πΎ||β‰€πœ”,𝑖=1,2,3,4,βˆ€πΎβˆˆπœβ„Ž,(3.1) where β„ŽπΎ is the diameter of element 𝐾, 𝜌𝐾 is the diameter of the inscribed circle of element 𝐾, and πœƒπ‘–πΎ are the angles of 𝐾 in the case of a quadrilateral partitioning. The mesh parameter β„Ž is given by β„Ž=maxπ‘˜βˆˆπœβ„Žβ„ŽπΎ. The finite-element subspaces of this paper are defined by setting 𝑅1𝑃(𝐾)=1𝑄(𝐾),ifKistriangular,1(𝐾),ifKisthequadrilateral.(3.2) Then, the finite-element pairs are coupled as follows: π‘‹β„Ž=ξ€½π‘£βˆˆπ‘‹;𝑣𝑖|πΎβˆˆπ‘…1ξ€Ύ,𝑀(𝐾),𝑖=1,2β„Ž=ξ€½π‘žβˆˆπ‘€βˆΆπ‘ž|πΎβˆˆπ‘…1(𝐾),βˆ€πΎβˆˆπœβ„Žξ€Ύ.(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 [29–31]:πΊξ€·π‘β„Ž,π‘žβ„Žξ€Έ=π‘π‘‡π‘–ξ€·π‘€π‘˜βˆ’π‘€1ξ€Έπ‘žπ‘—=π‘π‘‡π‘–π‘€π‘˜π‘žπ‘—βˆ’π‘π‘‡π‘–π‘€1π‘žπ‘—.(3.4) Here, we set 𝑝𝑇𝑖=𝑝0,𝑝1,…,π‘π‘βˆ’1𝑇,π‘žπ‘—=ξ€Ίπ‘ž0,π‘ž1,…,π‘žπ‘βˆ’1ξ€»,𝑀𝑖𝑗=ξ€·πœ™π‘–,πœ™π‘—ξ€Έ,π‘β„Ž=π‘βˆ’1𝑖=0π‘π‘–πœ™π‘–,𝑝𝑖=π‘β„Žξ€·π‘₯𝑖,βˆ€π‘β„Žβˆˆπ‘€β„Ž,𝑖,𝑗=0,…,π‘βˆ’1,(3.5) where πœ™π‘– is the basis function of the pressure on the domain Ξ© such that its value is one at node π‘₯𝑖 and zero at other nodes; the symmetric and positive π‘€π‘˜,π‘˜β‰₯2 and 𝑀1 are pressure mass matrix computed by using the k-order and 1-order Gauss integrations in each direction; respectively, also, 𝑝𝑖 and π‘žπ‘–,𝑖=0,1,…,𝑁, are the value of π‘β„Ž and π‘žβ„Ž at the node π‘₯𝑖. 𝑝𝑇𝑖 is the transpose of the matrix 𝑝𝑖.

Let Ξ β„ŽβˆΆπ‘€β†’π‘…0 be the standard 𝐿2-projection with the following properties [29–32]:𝑝,π‘žβ„Žξ€Έ=ξ€·Ξ β„Žπ‘,π‘žβ„Žξ€Έ,βˆ€π‘βˆˆπ‘€,π‘žβ„Žβˆˆπ‘…0,β€–β€–Ξ β„Žπ‘β€–β€–0≀𝑐‖𝑝‖0β€–β€–,βˆ€π‘βˆˆπ‘€,π‘βˆ’Ξ β„Žπ‘β€–β€–0≀‖𝑝‖1,βˆ€π‘βˆˆπ»1(Ξ©)βˆ©π‘€,(3.6) where 𝑅0={π‘žβ„Žβˆˆπ‘€βˆΆπ‘žβ„Ž|𝐾isaconstant,βˆ€πΎβˆˆπΎβ„Ž}. Then we can rewrite the bilinear form 𝐺(β‹…,β‹…) by𝐺(𝑝,π‘ž)=(π‘βˆ’Ξ π‘,π‘žβˆ’Ξ π‘ž).(3.7)

Remark 3.1. The bilinear form 𝐺(β‹…,β‹…) 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 𝑑𝑛=π‘›πœ, 𝑛=0,1,2,…,𝑁. We obtain the fully discrete scheme as follows: find functions {π‘’π‘›β„Ž}𝑛β‰₯0βŠ‚π‘‹β„Ž and {π‘π‘›β„Ž}𝑛β‰₯1βŠ‚π‘€β„Ž as solutions of the recursive linear algebraic equations, ξ€·π‘‘π‘‘π‘’π‘›β„Ž,π‘£β„Žξ€Έ+β„¬β„Žπ‘’ξ€·ξ€·π‘›β„Ž,π‘π‘›β„Žξ€Έ;ξ€·π‘£β„Ž,π‘žβ„Žξ€·π‘’ξ€Έξ€Έ+π‘β„Žπ‘›βˆ’1,π‘’β„Ž,π‘£β„Žξ€Έ=𝑓𝑑𝑛,π‘£β„Žξ€Έ,𝑒0β„Ž=𝑒0β„Ž.(3.8) for all (π‘£β„Ž,π‘žβ„Ž)∈(π‘‹β„Ž,π‘€β„Ž), where π‘‘π‘‘π‘’π‘›β„Ž=π‘’π‘›β„Žβˆ’π‘’β„Žπ‘›βˆ’1𝜏,β„¬β„Žπ‘’ξ€·ξ€·β„Ž,π‘β„Žξ€Έ;ξ€·π‘£β„Ž,π‘žβ„Žξ€·π‘’ξ€Έξ€Έ=π‘Žβ„Ž,π‘£β„Žξ€Έξ€·π‘£βˆ’π‘‘β„Ž,π‘β„Žξ€Έξ€·π‘’+π‘‘β„Ž,π‘žβ„Žξ€Έξ€·π‘+πΊβ„Ž,π‘žβ„Žξ€Έ,(3.9) and 𝑒0β„Ž is the approximation of 𝑒0. The solutions {π‘’π‘›β„Ž}𝑛β‰₯0 and {π‘π‘›β„Ž}𝑛β‰₯1 to (3.8)-(3.9) are expected to the approximations of {π‘’β„Ž(𝑑𝑛)}𝑛β‰₯0 and {π‘β„Ž(𝑑𝑛)}𝑛β‰₯1 with π‘β„Žξ€·π‘‘π‘›ξ€Έ=1πœξ€œπ‘‘π‘›π‘‘π‘›βˆ’1π‘β„Ž(𝑑)𝑑𝑑.(3.10)

Theorem 3.2 (see [29–32]). Let (π‘‹β„Ž,π‘€β„Ž) be defined as above, then there exists a positive constant 𝛽, independent of β„Ž, such that ||||‖ℬ((𝑒,𝑝);(𝑣,π‘ž))≀𝑐𝑒‖1+‖𝑝‖0‖𝑣‖1+β€–π‘žβ€–0𝛽‖‖𝑒,(𝑒,𝑝),(𝑣,π‘ž)∈(𝑋,𝑀),β„Žβ€–β€–1+β€–β€–π‘β„Žβ€–β€–0≀supξ€·π‘£β„Ž,π‘žβ„Žξ€Έβˆˆξ€·π‘‹β„Ž,π‘€β„Žξ€Έ||β„¬β„Žπ‘’ξ€·ξ€·β„Ž,π‘β„Žξ€Έ;ξ€·π‘£β„Ž,π‘žβ„Ž||ξ€Έξ€Έβ€–β€–π‘£β„Žβ€–β€–1+β€–β€–π‘žβ„Žβ€–β€–0𝑒,βˆ€β„Ž,π‘β„Žξ€Έβˆˆξ€·π‘‹β„Ž,π‘€β„Žξ€Έ,||||‖‖𝐺(𝑝,π‘ž)β‰€π‘π‘βˆ’Ξ β„Žπ‘β€–β€–0β€–β€–π‘žβˆ’Ξ β„Žπ‘žβ€–β€–0,βˆ€π‘,π‘žβˆˆπ‘€.(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 β„Ž and 𝜏 are sufficiently small and β„Ž=𝑂(𝜏), then problem (3.8)-(3.9) has a unique solution (π‘’π‘›β„Ž,π‘π‘›β„Ž)βˆˆπ‘‹β„ŽΓ—π‘€β„Ž such that β€–β€–π‘’π‘›β„Žβ€–β€–20+π‘›βˆ‘π‘—=1ξ‚€β€–β€–πœπœˆβˆ‡π‘’π‘—β„Žβ€–β€–20𝑝+πœπΊπ‘—β„Ž,π‘π‘—β„Žξ€Έξ‚β‰€β€–β€–π‘’0β€–β€–20+π‘πœπœˆπ‘›βˆ’1βˆ‘π‘—=1‖‖𝑓𝑗‖‖20.(3.12)

Theorem 3.4 (see [32]). Under the assumptions of Theorem 3.3, the error (𝑒(𝑑𝑛)βˆ’π‘’π‘›β„Ž,𝑝(𝑑𝑛)βˆ’π‘π‘›β„Ž) satisfies the following bound: πœπ‘ξ“π‘›=1β€–β€–π‘’ξ€·π‘‘π‘›ξ€Έβˆ’π‘’π‘›β„Žβ€–β€–20𝑑+πœπ‘šξ€Έβ€–β€–π‘’ξ€·π‘‘π‘šξ€Έβˆ’π‘’π‘šβ„Žβ€–β€–20ξ€·β„Žβ‰€π‘4+𝜏2ξ€Έ,πœπ‘ξ“π‘›=1β€–β€–π‘’ξ€·π‘‘π‘›ξ€Έβˆ’π‘’π‘›β„Žβ€–β€–21𝑑+πœπ‘šξ€Έβ€–β€–π‘’ξ€·π‘‘π‘šξ€Έβˆ’π‘’π‘šβ„Žβ€–β€–21ξ€·β„Žβ‰€π‘2+𝜏2ξ€Έ,πœπ‘ξ“π‘›=1πœξ€·π‘‘π‘›ξ€Έβ€–β€–π‘ξ€·π‘‘π‘›ξ€Έβˆ’π‘π‘›β„Žβ€–β€–20ξ€·β„Žβ‰€π‘2+𝜏2ξ€Έ,(3.13) for all π‘‘π‘šβˆˆ(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 {𝑒𝑛1β„Ž,𝑒𝑛2β„Ž,π‘π‘›β„Ž}𝐿𝑛=1. Then we choose 𝐿 (in general, 𝐿≀𝑁, e.g., 𝐿=20,𝑁=200) instantaneous solutions π‘ˆπ‘–(π‘₯,𝑦)=(𝑒𝑖1β„Ž,𝑒𝑖2β„Ž,π‘π‘–β„Ž)(𝑖=1,2,…,𝐿) from the 𝐿 group of solutions (π‘’π‘›β„Ž,π‘π‘›β„Ž)(1≀𝑛≀𝐿) 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 (π‘‹β„Ž,π‘€β„Ž) 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 π‘ˆπ‘–(π‘₯,𝑦)=(𝑒𝑖1β„Ž,𝑒𝑖2β„Ž,π‘π‘–β„Ž)(𝑖=1,2,…,𝐿, see Section 3). Setξ€½π‘ˆπ‘‰=span1,π‘ˆ2,…,π‘ˆπΏξ€Ύ,(4.1) where 𝑉 is the ensemble consisting of the snapshots {π‘ˆπ‘–}𝐿𝑖=1, at least one of which is supposed to nonzero. Let {Ψ𝑗}𝑙𝑗=1 denote an orthogonal basis of 𝑉 with 𝑙=dim𝑉(𝑙≀𝐿). Then each member of the ensemble is expressed asπ‘ˆπ‘–=𝑙𝑖=1ξ€·π‘ˆπ‘–,Ψ𝑗𝑋Ψ𝑗,for𝑖=1,2,…,𝐿,(4.2) where (π‘ˆπ‘–,Ψ𝑗)𝑋=(π‘’π‘–β„Ž,Ψ𝑒𝑗)+(π‘π‘–β„Ž,Ψ𝑝𝑗), Ψ𝑒𝑗 and Ψ𝑝𝑗 are the orthogonal basis corresponding to 𝑒 and 𝑝, respectively.

The method of POD consists in finding the orthogonal basis such that, for every 𝑑(1≀𝑑≀𝐿), the mean square error between the elements π‘ˆπ‘–(1≀𝑖≀𝐿) and corresponding 𝑑th partial sum of (4.2) is minimized on averageminΨ𝑗𝑑𝑗=11𝐿𝐿𝑖=1β€–β€–β€–β€–π‘ˆπ‘–βˆ’π‘‘ξ“π‘—=1ξ€·π‘ˆπ‘–,Ψ𝑗𝑋Ψ𝑗‖‖‖‖2𝑋,(4.3) subject toΨ𝑖,Ψ𝑗𝑋=𝛿𝑖𝑗,for1≀𝑖≀𝑑,1≀𝑗≀𝑖,(4.4) where β€–π‘ˆπ‘–β€–ξπ‘‹=(‖𝑒𝑖1β„Žβ€–21+‖𝑒𝑖2β„Žβ€–21+β€–π‘π‘–β„Žβ€–20)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)πΏΓ—πΏβˆˆπ‘…πΏΓ—πΏ corresponding to the snapshots {π‘ˆπ‘–}𝐿𝑖=1 by𝐸𝑖𝑗=1πΏξ€·π‘ˆπ‘–,π‘ˆπ‘—ξ€Έξπ‘‹.(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 πœ†1β‰₯πœ†2β‰₯β‹―β‰₯πœ†π‘™>0 denote the positive eigenvalues of 𝐸, and let 𝑣1,𝑣2,…,𝑣𝑙 be the associated eigenvectors. Then a POD basis of rank 𝑑≀𝑙 is given by Ψ𝑖=1βˆšπœ†π‘–πΏξ“π‘—=1𝑣𝑖𝑗,π‘ˆπ‘—,𝑖=1,2,…,𝑑≀𝑙,(4.6) where (𝑣𝑖)𝑗 denotes the 𝑗th component of the eigenvector 𝑣𝑖. Furthermore, the following error formula holds: 1𝐿𝐿𝑖=1β€–β€–β€–β€–π‘ˆπ‘–βˆ’π‘‘ξ“π‘—=1ξ€·π‘ˆπ‘–,Ψ𝑗𝑋Ψ𝑗‖‖‖‖2𝑋=𝑙𝑗=𝑑+1πœ†π‘—.(4.7)
Let 𝑉𝑑={Ξ¨1,Ξ¨2,…,Ψ𝑑}, and let 𝑋𝑑×𝑀𝑑=𝑉𝑑 with π‘‹π‘‘βŠ‚Xβ„ŽβŠ‚π‘‹, and let π‘€π‘‘βŠ‚π‘€β„ŽβŠ‚π‘€. Set the Ritz-projection π‘ƒβ„ŽβˆΆπ‘‹β†’π‘‹β„Ž (if π‘ƒβ„Ž is restricted to the Ritz-projection from π‘‹β„Ž to 𝑋𝑑, it is written as 𝑃𝑑) such that π‘ƒβ„Ž|π‘‹β„Ž=π‘ƒπ‘‘βˆΆπ‘‹β„Žβ†’π‘‹π‘‘ and π‘ƒβ„ŽβˆΆπ‘‹β§΅π‘‹β„Žβ†’π‘‹β„Žβ§΅π‘‹π‘‘ and 𝐿2-projection πœŒπ‘‘βˆΆπ‘€β†’π‘€π‘‘ denoted by, respectively, π‘Žξ€·π‘ƒβ„Žπ‘’,π‘£β„Žξ€Έξ€·=π‘Žπ‘’,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,ξ€·πœŒ(4.8)𝑑𝑝,π‘žπ‘‘ξ€Έ0=𝑝,π‘žπ‘‘ξ€Έ0,π‘žπ‘‘βˆˆπ‘€π‘‘,(4.9) where π‘’βˆˆπ‘‹ and π‘βˆˆπ‘€. Owing to (4.8)-(4.9) the linear operators π‘ƒβ„Ž and πœŒπ‘‘ are well defined and bounded: ‖‖𝑃𝑑𝑒‖‖1≀‖𝑒‖1,β€–β€–πœŒπ‘‘π‘β€–β€–0,≀‖𝑝‖0,βˆ€π‘’βˆˆπ‘‹,π‘βˆˆπ‘€.(4.10)

Lemma 4.2 (see [45]). For every 𝑑(1≀𝑑≀𝑙), the projection operators 𝑃𝑑 and πœŒπ‘‘ satisfy, respectively, 1𝐿𝐿𝑖=1β€–β€–π‘’π‘›π‘–β„Žβˆ’π‘ƒπ‘‘π‘’π‘›π‘–β„Žβ€–β€–21≀𝑙𝑗=𝑑+1πœ†π‘—,1(4.11)𝐿𝐿𝑖=1β€–β€–π‘’π‘›π‘–β„Žβˆ’π‘ƒdπ‘’π‘›π‘–β„Žβ€–β€–20β‰€πΆβ„Ž2𝑙𝑗=𝑑+1πœ†π‘—1,(4.12)𝐿𝐿𝑖=1β€–β€–π‘π‘›π‘–β„Žβˆ’πœŒπ‘‘π‘π‘›π‘–β„Žβ€–β€–20≀𝑙𝑗=𝑑+1πœ†π‘—,(4.13) where π‘’π‘›π‘–β„Ž=(𝑒𝑛𝑖1β„Ž,𝑒𝑛𝑖2β„Ž) and (𝑒𝑛𝑖1β„Ž,𝑒𝑛𝑖2β„Ž,π‘π‘›π‘–β„Ž)βˆˆπ‘‰.
Thus, using 𝑉𝑑=𝑋𝑑×𝑀𝑑, we can obtain the reduced SMFE formulation for problems (3.8) as follows. Find ̂𝑒𝑛𝑑=(𝑒𝑛𝑑,𝑝𝑛𝑑)βˆˆπ‘‰π‘‘ such that 𝑑𝑑𝑒𝑛𝑑,𝑣𝑑+π΅β„Žπ‘’ξ€·ξ€·π‘›π‘‘,𝑝𝑛𝑑;𝑣𝑑,π‘žπ‘‘ξ€·π‘’ξ€Έξ€Έ+π‘π‘‘π‘›βˆ’1,𝑒𝑑,𝑣𝑑=𝑓𝑑𝑛,𝑣𝑑𝑒,1≀𝑛≀𝑁,0𝑑=𝑒0β„Ž.(4.14)

Remark 4.3. Problem (3.8) includes π‘β„Ž (π‘β„Ž is the number of triangles or quadrilaterals vertex in πœβ„Ž) freedom degrees, while problem (4.14) includes 𝑑(𝑑β‰ͺ𝑙≀𝐿≀𝑁) freedom degrees. For actual science and engineering problems, the number of the vertex in πœβ„Ž 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 π‘β„Ž=32Γ—32Γ—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=π‘‹π‘‘Γ—π‘€π‘‘βŠ‚π‘‰βŠ‚π‘‹β„ŽΓ—π‘€β„ŽβŠ‚π‘‹Γ—π‘€. 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 (𝑒𝑛𝑑,𝑝𝑛𝑑)βˆˆπ‘‹π‘‘Γ—π‘€π‘‘ and satisfies, for 1≀𝑛≀𝑁, ‖‖𝑒𝑛𝑑‖‖20+𝑛𝑗=1ξ‚€β€–β€–π‘’πœπœˆπ‘—π‘‘β€–β€–21𝑝+πœπΊπ‘—π‘‘,𝑝𝑗𝑑≀‖‖𝑒0β€–β€–20+π‘πœπœˆπ‘›βˆ’1𝑗=1‖‖𝑓𝑗‖‖20.(5.1)

In the following theorem, the errors between the solution (𝑒𝑛𝑑,𝑝𝑛𝑑) to Problem (4.14)-(4.15) and the solution (π‘’π‘›β„Ž,π‘π‘›β„Ž) to Problem (3.8) are derived.

Theorem 5.2. Under the assumptions of Theorem 5.1, if β„Ž and 𝜏 are sufficiently small, β„Ž=𝑂(𝜏), and 𝜏=𝑂(πΏβˆ’2), then the errors between the solutions (𝑒𝑛𝑑,𝑝𝑛𝑑) to Problem (4.14), and the solutions (π‘’π‘›β„Ž,π‘π‘›β„Ž) to Problem (3.8) have the following error estimates, for 1≀𝑛≀𝑁: β€–β€–π‘’π‘›β„Žβˆ’π‘’π‘›π‘‘β€–β€–20+πœπœˆπ‘›π‘–ξ“π‘—=𝑛1β€–β€–π‘’π‘—β„Žβˆ’π‘’π‘—π‘‘β€–β€–21+πœπ‘›π‘–ξ“π‘—=𝑛1β€–β€–π‘π‘—β„Žβˆ’π‘π‘—π‘‘β€–β€–20β‰€πΆπœπ‘™1/2𝑗=𝑑+1πœ†π‘—,if𝑛=π‘›π‘–βˆˆξ€½π‘›1,𝑛2,…,𝑛𝐿,‖‖𝑒(5.2)π‘›β„Žβˆ’π‘’π‘›π‘‘β€–β€–20‖‖𝑒+πœπœˆπ‘›β„Žβˆ’π‘’π‘›π‘‘β€–β€–21+𝑛𝑖𝑗=𝑛1β€–β€–π‘’π‘—β„Žβˆ’π‘’π‘—π‘‘β€–β€–21‖‖𝑝+πœπ‘›β„Žβˆ’π‘π‘›π‘‘β€–β€–20+𝑛𝑖𝑗=𝑛1β€–β€–π‘π‘—β„Žβˆ’π‘π‘—π‘‘β€–β€–20ξƒ­β‰€πΆπœπ‘™1/2𝑗=𝑑+1πœ†π‘—+𝐢𝜏2,if𝑛=π‘›π‘–βˆ‰ξ€½π‘›1,𝑛2,…,𝑛𝐿.(5.3)

Proof. Let 𝑀𝑛𝑑=π‘ƒπ‘‘π‘’π‘›β„Žβˆ’π‘’π‘›π‘‘, π‘Ÿπ‘›π‘‘=πœŒπ‘‘π‘π‘›β„Žβˆ’π‘π‘›π‘‘. Subtracting (3.8) from (4.14) yields that 1πœξ€·π‘’π‘›β„Žβˆ’π‘’π‘›π‘‘,π‘£π‘‘ξ€Έβˆ’1πœξ€·π‘’β„Žπ‘›βˆ’1βˆ’π‘’π‘‘π‘›βˆ’1,𝑣𝑑𝑒+π‘Žπ‘›β„Žβˆ’π‘’π‘›π‘‘,π‘£π‘‘ξ€Έξ€·π‘βˆ’π‘π‘›β„Žβˆ’π‘π‘›π‘‘,𝑣𝑑𝑒+π‘β„Žπ‘›βˆ’1βˆ’π‘’π‘‘π‘›βˆ’1,π‘’π‘›β„Ž,𝑣𝑑𝑒+π‘π‘‘π‘›βˆ’1,π‘’π‘›β„Žβˆ’π‘’π‘›π‘‘,𝑣𝑑𝑒+π‘π‘›β„Žβˆ’π‘’π‘›π‘‘,π‘žπ‘‘ξ€Έξ€·π‘+πΊπ‘›β„Žβˆ’π‘π‘›π‘‘,π‘žπ‘‘ξ€Έ=0.(5.4) Taking (𝑣𝑑,π‘žπ‘‘)=2𝜏(𝑀𝑛𝑑,π‘Ÿπ‘›π‘‘) in (5.4), since π‘Ž(π‘’π‘›β„Žβˆ’π‘ƒπ‘‘π‘’π‘›β„Ž,𝑀𝑛𝑑)=0, 𝑏(π‘’π‘›β„Žβˆ’π‘’π‘›π‘‘,π‘Ÿπ‘›π‘‘)+𝐺(π‘π‘›β„Žβˆ’π‘π‘›π‘‘,π‘Ÿπ‘›π‘‘)=0, we deduce 2𝑀𝑛𝑑,π‘€π‘›π‘‘ξ€Έξ€·π‘€βˆ’2π‘‘π‘›βˆ’1,𝑀𝑛𝑑𝑀+2πœπ‘Žπ‘›π‘‘,𝑀𝑛𝑑𝑒=2β„Žπ‘›βˆ’1βˆ’π‘π‘‘π‘’β„Žπ‘›βˆ’1,𝑀𝑛𝑑𝑝+2π‘‘π‘’π‘›β„Žβˆ’π‘’π‘›β„Ž,𝑀𝑛𝑑𝑝+2πœπ‘π‘›β„Žβˆ’πœŒπ‘‘π‘π‘›β„Ž,𝑀𝑛𝑑𝑃+2πœπ‘π‘‘π‘’β„Žπ‘›βˆ’1βˆ’π‘’β„Žπ‘›βˆ’1,π‘’π‘›β„Ž,π‘€π‘›π‘‘ξ€Έξ€·π‘€βˆ’2πœπ‘π‘‘π‘›βˆ’1,π‘’π‘›β„Ž,𝑀𝑛𝑑𝑒+2πœπ‘π‘‘π‘›βˆ’1,π‘ƒπ‘‘π‘’π‘›β„Žβˆ’π‘’π‘›β„Ž,𝑀𝑛𝑑.(5.5) Using (2.12)-(2.13), the HΓΆlder inequality and the Young inequality, we see that ||ξ€·π‘’β„Žπ‘›βˆ’1βˆ’π‘π‘‘π‘’β„Žπ‘›βˆ’1,𝑀𝑛𝑑+ξ€·π‘π‘‘π‘’π‘›β„Žβˆ’π‘’π‘›β„Ž,𝑀𝑛𝑑||≀𝐢1ξ‚€β€–β€–π‘’β„Žπ‘›βˆ’1βˆ’π‘π‘‘π‘’β„Žπ‘›βˆ’1β€–β€–21+β€–β€–π‘’π‘›β„Žβˆ’π‘π‘‘π‘’π‘›β„Žβ€–β€–21+πœˆβ€–β€–π‘€10𝑑‖‖21,||π‘ξ€·π‘π‘›β„Žβˆ’πœŒπ‘‘π‘π‘›β„Ž,𝑀𝑛𝑑||≀𝐢4β€–β€–π‘π‘›β„Žβˆ’πœŒπ‘‘π‘π‘›β„Žβ€–β€–20+πœˆβ€–β€–π‘€10𝑛𝑑‖‖21,||π‘ξ€·π‘ƒπ‘‘π‘’β„Žπ‘›βˆ’1βˆ’π‘’β„Žπ‘›βˆ’1,π‘’π‘›β„Ž,𝑀𝑛𝑑||β‰€πœˆβ€–β€–π‘€10𝑛𝑑‖‖21+𝐢5β€–β€–ξ€·π‘’β„Žπ‘›βˆ’1βˆ’π‘ƒπ‘‘π‘’β„Žπ‘›βˆ’1ξ€Έβ€–β€–21,||π‘ξ€·π‘’π‘‘π‘›βˆ’1,π‘ƒπ‘‘π‘’π‘›β„Žβˆ’π‘’π‘›β„Ž,𝑀𝑛𝑑||β‰€πœˆβ€–β€–π‘€10𝑛𝑑‖‖21+𝐢6β€–β€–π‘’π‘›β„Žβˆ’π‘ƒπ‘‘π‘’π‘›β„Žβ€–β€–21,||ξ€·π‘€βˆ’π‘π‘‘π‘›βˆ’1,π‘’π‘›β„Ž,𝑀𝑛𝑑||≀𝐢7β€–β€–π‘€π‘‘π‘›βˆ’1β€–β€–20+πœˆβ€–β€–π‘€10𝑛𝑑‖‖21.(5.6) Noting that π‘Ž(π‘Žβˆ’π‘)=[π‘Ž2βˆ’π‘2+(π‘Žβˆ’π‘)2]/2 (for (π‘Žβ‰₯0 and 𝑏β‰₯0)), owing to (5.6), we obtain that ‖‖𝑀𝑛𝑑‖‖20βˆ’β€–β€–π‘€π‘‘π‘›βˆ’1β€–β€–20‖‖𝑀+πœˆπœπ‘‘β€–β€–21≀𝐢7πœβ€–β€–π‘€π‘‘π‘›βˆ’1β€–β€–20‖‖𝑒+πΆπœπ‘›β„Žβˆ’π‘ƒπ‘‘π‘’π‘›β„Žξ€Έβ€–β€–21+β€–β€–ξ€·π‘’β„Žπ‘›βˆ’1βˆ’π‘ƒπ‘‘π‘’β„Žπ‘›βˆ’1ξ€Έβ€–β€–21+β€–β€–π‘π‘›β„Žβˆ’πœŒπ‘‘π‘π‘›β„Žβ€–β€–20.(5.7) First, we consider the case of π‘›βˆˆ{𝑛1,𝑛2,…,𝑛𝐿}. Summing (5.7) from 𝑛1 to 𝑛𝑖, 𝑖=1,2,…,𝐿, and noting that 𝑒0β„Žβˆ’π‘’0𝑑=0, using Lemma 4.2, we can derive that ‖‖𝑀𝑛𝑖𝑑‖‖20+πœˆπœπ‘›π‘–ξ“π‘—=𝑛1‖‖𝑀𝑗𝑑‖‖21β‰€πΆπœπΏπ‘™ξ“π‘—=𝑑+1πœ†π‘—+𝐢7πœπ‘›π‘–βˆ’1𝑗=𝑛0‖‖𝑀𝑗𝑑‖‖20.(5.8) By using the discrete Gronwall inequality, we obtain that ‖‖𝑀𝑛𝑖𝑑‖‖20+πœˆπœπ‘›π‘–ξ“π‘—=𝑛1‖‖𝑀𝑗𝑑‖‖21β‰€πΆπœπΏπ‘™ξ“π‘—=𝑑+1πœ†π‘—ξ€·πΆexp7πœπ‘›π‘–ξ€Έ.(5.9) If β„Ž and 𝜏 are sufficiently small, 𝜏=𝑂(πΏβˆ’2), and noting that π‘›π‘–πœβ‰€π‘›π‘–π‘β‰€π‘‡, we find that ‖‖𝑀𝑛𝑖𝑑‖‖20+πœˆπœπ‘›π‘–ξ“π‘—=𝑛1‖‖𝑀𝑗𝑑‖‖21β‰€πΆπœπ‘™1/2𝑗=𝑑+1πœ†π‘—.(5.10) Thanks to 𝑏(π‘’π‘›β„Žβˆ’π‘’π‘›π‘‘,π‘Ÿπ‘›π‘‘)+𝐺(π‘π‘›β„Žβˆ’π‘π‘›π‘‘,π‘Ÿπ‘›π‘‘)=0, we obtain β€–β€–π‘π‘›β„Žβˆ’π‘π‘›π‘‘β€–β€–0β€–β€–π‘’β‰€πΆπ‘›β„Žβˆ’π‘’π‘›π‘‘β€–β€–1.(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 π‘›βˆ‰{1,2,…,𝐿}; we assume that π‘‘π‘›βˆˆ(π‘‘π‘›π‘–βˆ’1,𝑑𝑛𝑖) and 𝑑𝑛 is the nearest point to 𝑑𝑛𝑖. π‘’π‘›β„Ž and π‘π‘›β„Ž are expanded into the Taylor series expansion at point 𝑑𝑛𝑖.π‘’π‘›β„Ž=π‘’π‘›π‘–β„Žβˆ’π‘ πœπœ•π‘’β„Žξ€·πœ‰1ξ€Έπœ•π‘‘,πœ‰1βˆˆξ€Ίπ‘‘π‘›π‘–,𝑑𝑛,𝑝(5.12)π‘›β„Ž=π‘π‘›π‘–β„Žβˆ’π‘ πœπœ•π‘β„Žξ€·πœ‰2ξ€Έπœ•π‘‘,πœ‰2βˆˆξ€Ίπ‘‘π‘›π‘–,𝑑𝑛,(5.13) where 𝑠 is the number of time steps from 𝑑𝑛 to 𝑑𝑛𝑖. If the snapshots are equably taken, then 𝑠≀𝑁/𝐿. Summing (5.7) from 𝑛1 to 𝑛𝑖, n, and using (5.12), if |πœ•π‘’β„Ž(πœ‰1)/πœ•π‘‘| and |πœ•π‘β„Ž(πœ‰2)/πœ•π‘‘| are bounded, by discrete Gronwall inequality and Lemma 4.2 and (3.12), we obtain that ‖‖𝑀𝑛𝑑‖‖20‖‖𝑀+πœˆπœπ‘›π‘‘β€–β€–21+𝑛𝑖𝑗=𝑛1‖‖𝑀𝑗𝑑‖‖21ξƒ­β‰€πΆπœπΏπ‘™ξ“π‘—=𝑑+1πœ†π‘—+𝐢𝜏2.(5.14)

If 𝜏=𝑂(πΏβˆ’2), by (5.14) we obtain that‖‖𝑀𝑛𝑑‖‖20‖‖𝑀+πœˆπœπ‘›π‘‘β€–β€–21+𝑛𝑖𝑗=𝑛1‖‖𝑀𝑗𝑑‖‖21ξƒ­β‰€πΆπœπ‘™1/2𝑗=𝑑+1πœ†π‘—+𝐢𝜏2.(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 (𝑒(𝑑),𝑝(𝑑)) to Problem (2.1)-(2.2) and the solutions (𝑒𝑛𝑑,𝑝𝑛𝑑) to Problem (4.14) are as follows: β€–β€–π‘’ξ€·π‘‘π‘›ξ€Έβˆ’π‘’π‘›π‘‘β€–β€–20+πœπœˆπ‘›ξ“π‘—=1β€–β€–uπ‘—βˆ’π‘’π‘—π‘‘β€–β€–21+πœπ‘›ξ“π‘—=1β€–β€–π‘π‘—βˆ’π‘π‘—π‘‘β€–β€–20ξ€·β„Žβ‰€π‘2+𝜏2ξ€Έ+πΆπœπ‘™1/2𝑗=𝑑+1πœ†π‘—,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 Ξ© is the unit square [0,1]Γ—[0,1] and viscosity 𝜈=0.05. The velocity and pressure are designed on the same uniform triangulation of Ξ©. The exact solution is given by 𝑒𝑒=1(π‘₯,𝑦),𝑒2𝑒(π‘₯,𝑦),𝑝(π‘₯,𝑦)=10(2π‘₯βˆ’1)(2π‘¦βˆ’1)cos(𝑑),1(π‘₯,𝑦)=10π‘₯2(1βˆ’π‘₯)2𝑒𝑦(1βˆ’π‘¦)(1βˆ’2𝑦)cos(𝑑),2(π‘₯,𝑦)=βˆ’10π‘₯(1βˆ’π‘₯)(1βˆ’2π‘₯)𝑦2(1βˆ’π‘¦)2cos(𝑑),(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 β„Ž=1/𝑛. In order to make 𝜏=𝑂(β„Ž), we take time step increment as 𝜏=1/𝑛.

We obtain 20 values (i.e., snapshots) outputting at time 𝑑=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 𝑑=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 (𝑒200β„Žπ‘–, 𝑖=1,2)(c), the exact solutions (𝑒𝑖200,𝑖=1,2) (b), and the RSMFE solutions (𝑒200𝑑𝑖,𝑖=1,2) (a) are depicted, respectively, in Figures 1 and 3. Moreover, the difference (𝑒𝑖200βˆ’π‘’200𝑑𝑖, 𝑖=1,2) (a) between the exact solutions and the RSMFE solutions and the difference (𝑒200β„Žπ‘–βˆ’π‘’200𝑑𝑖,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 1–4, 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 3Γ—32Γ—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 3Γ—10βˆ’3. 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 𝑑=100𝜏 and 𝑑=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.


β„Ž βˆ’ 1 β€– 𝑒 βˆ’ 𝑒 β„Ž β€– 1 β€– 𝑒 β€– 1 𝑒 𝐻 1 rate β€– 𝑝 βˆ’ 𝑝 β„Ž β€– 0 β€– 𝑝 β€– 0 𝑝 𝐿 2 rateCPU time (seconds)

24 0.1443070 β€”0.0175033β€”1081
32 0.09730511.36990.01045061.79272906
40 0.07528751.14970.00721361.6612 11403
48 0.06193471.07080.0054000 1.588225375
56 0.05100661.25930.0042171 1.6040 42132
64 0.04375411.14860.00348051.4375 61320


β„Ž βˆ’ 1 β€– 𝑒 βˆ’ 𝑒 𝑑 β€– 1 β€– 𝑒 β€– 1 𝑒 𝐻 1 rate β€– 𝑝 βˆ’ 𝑝 𝑑 β€– 0 β€– 𝑝 β€– 0 𝑝 𝐿 2 rateCPU time (seconds)

24 0.1486190β€”0.0174335β€”3
32 0.10692301.14460.01076431.6760 14
40 0.0797401.31460.00719301.806636
48 0.06375861.22680.00537591.5971 83
56 0.05258421.25000.00418781.6202152
64 0.04459771.23370.0034562 1.4379 216

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 𝑃1βˆ’π‘ƒ1 over triangles or 𝑄1βˆ’π‘„1 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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