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

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 𝑃₁−𝑃₁ and 𝑄₁−𝑄₁, 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 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: 𝑢_𝑡−𝜈𝑢Δ(𝑥𝑢,+0)(𝑢=⋅𝑢∇₀)𝑢+∇𝑝=𝑓,div𝑢=0,(𝑥,𝑡)∈Ω×(0,𝑇],(2.1)(𝑥),𝑥∈Ω,𝑢(𝑥,𝑡)|_Γ=0,𝑡∈[0,𝑇].(2.2) Here 𝑢∶Ω→𝑅² and 𝑝∶Ω→𝑅 are the velocity and pressure, 𝜈>0 is the viscosity, and 𝑓 represents the body forces, 𝑇>0 the final time, and 𝑢_𝑡=𝜕𝑢/𝜕𝑡.

For the mathematical setting of problems (2.1)-(2.2), we introduce the following Sobolev spaces: 𝐷(𝐴𝑋)==𝐻𝐻¹⁰₂(Ω)²,𝑀=𝐿²⁰(Ω)=𝑞∈𝐿²(Ω);_Ω𝑞(𝑥)𝑑𝑥=0,(2.3)(Ω)²∩𝑋,𝑉={𝑣∈𝑉∶div𝑣=0}.(2.4) Furthermore, we make a regularity assumption on the Stokes problem as follows.

Assumption H1. For a given 𝑔∈𝑌 and the Stokes problem, −Δ𝑣+d𝑣i∇v∣𝑣𝑞_Γ==𝑔0,,iinnΩΩ,,=0,on𝜕Ω,(2.5) satisfying the following regularity result: ‖𝑣‖₂+‖𝑞‖₁≤𝜅‖𝑔‖₀,(2.6) where ‖⋅‖_𝑖 is the norm of the Sobolev space 𝐻^𝑖(Ω) or 𝐻^𝑖(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 𝑣∈𝑋 there hold the following inequalities: ‖𝑣‖_𝐿⁴≤2^1/4‖𝑣‖^01/2‖𝑣‖^11/2.(2.7)

Assumption H2. The initial velocity 𝑢₀∈𝐷(𝐴) and the body force 𝑓(𝑥,𝑡)∈𝐿²(0,𝑇;𝐿²(Ω)²) are assumed to satisfy ‖‖𝑢₀‖‖₂+^𝑇0‖𝑓‖²⁰+‖‖𝑓_𝑡‖‖²⁰𝑑𝑡^1/2≤𝑐.(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,𝑣∈𝑋,𝑑(𝑣,𝑞)=(𝑞,div𝑣),∀(𝑣,𝑞)∈(𝑋,𝑀).(2.9) Also, a generalized bilinear form ℬ((⋅,⋅);(⋅,⋅)) 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 𝑏(𝑢,𝑣,𝑤)=((𝑢⋅∇)𝑣,𝑤)+1=12((div𝑢)𝑣,𝑤)2((𝑢⋅∇)𝑣,𝑤)−12((𝑢⋅∇)𝑤,𝑣),∀𝑢,𝑣,𝑤∈𝑋.(2.11) By the above notations and the Hölder inequality, there hold the following estimates: ||𝑏(𝑢,𝑣,𝑤)||≤1𝑏(𝑢,𝑣,𝑤)=−𝑏(𝑢,𝑤,𝑣),∀𝑢∈𝑋,𝑣,𝑤∈𝑋,2𝑐⁰||𝑏(𝑢,𝑣,𝑤)||+||𝑏(‖𝑣𝑢,‖𝑢,𝑤^01/2‖)||𝑢+‖||𝑏^11/2(𝑤‖,𝑣𝑢‖,𝑣₁‖)||𝑤≤‖𝑐^01/2₁‖𝑤‖^11/2+‖𝑣‖^01/2‖𝑣‖^11/2‖𝑤‖₁,∀𝑢,𝑣,𝑤∈𝑋,‖𝑢‖₁‖𝑣‖₂‖𝑤‖₀,∀𝑢∈𝑋,𝑣∈𝐷(𝐴),𝑤∈𝑌.(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 Ω.
For a given 𝑓∈𝑌, the variational formulation of problem (2.1)-(2.2) reads as follows: find (𝑢,𝑝)∈(𝑋,𝑀),𝑡>0 such that𝑢_𝑡,𝑣+ℬ((𝑢,𝑝);(𝑣,𝑞))+𝑏𝑢((𝑢0,)𝑢=,𝑣)𝑢₀=(𝑓,𝑣),∀(𝑣,𝑞)∈(𝑋,𝑀),.(2.14)
For convenience, we recall the discrete Gronwall Lemma that will be frequently used.

Lemma 2.1 (see [1, 45, 46]). Let {𝑎_𝑛},{𝑏_𝑛}, and {𝑐_𝑛} be three positive sequences, and let {𝑐_𝑛} be monotone and satisfy 𝑎_𝑛+𝑏_𝑛≤𝑐_𝑛+𝜆^𝑛−1_𝑖=0𝑎_𝑖,𝜆>0,𝑎₀+𝑏₀≤𝑐₀,(2.15) then 𝑎_𝑛+𝑏_𝑛≤𝑐_𝑛exp(𝑛𝜆),𝑛≥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 (𝑢,𝑝) satisfying the following regularities: sup_{0<𝑡≤𝑇}sup_{0<𝑡≤𝑇}‖𝑢(𝑡)‖²²+‖𝑝(𝑡)‖²¹+‖‖𝑢_𝑡(𝑡)‖‖²⁰≤𝑐,𝜏(𝑡)‖‖𝑢_𝑡‖‖²¹+^𝑇0𝜏(𝑡)‖‖𝑢_𝑡‖‖²²+‖‖𝑝_𝑡‖‖²¹+‖‖𝑢_𝑡𝑡‖‖²⁰𝑑𝑡≤𝑐,(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 ℎ>0 be a real positive parameter. Finite-element subspace (𝑋_ℎ,𝑀_ℎ) of (𝑋,𝑀) is characterized by 𝜏_ℎ=𝜏_ℎ(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, ||cos𝜃_𝑖𝐾ℎ_𝐾≤𝜎𝜌_𝐾,∀𝐾∈𝜏_ℎ,||≤𝜔,𝑖=1,2,3,4,∀𝐾∈𝜏_ℎ,(3.1) where ℎ_𝐾 is the diameter of element 𝐾, 𝜌_𝐾 is the diameter of the inscribed circle of element 𝐾, and 𝜃_𝑖𝐾 are the angles of 𝐾 in the case of a quadrilateral partitioning. The mesh parameter ℎ is given by ℎ=max_{𝑘∈𝜏ℎ}ℎ_𝐾. The finite-element subspaces of this paper are defined by setting 𝑅₁(𝐾)=𝑃¹𝑄(𝐾),ifKistriangular,₁(𝐾),ifKisthequadrilateral.(3.2) Then, the finite-element pairs are coupled as follows: 𝑀𝑋_ℎ=𝑣∈𝑋;𝑣_𝑖|_𝐾∈𝑅₁(𝐾),𝑖=1,2_ℎ,=𝑞∈𝑀∶𝑞|_𝐾∈𝑅₁(𝐾),∀𝐾∈𝜏_ℎ.(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]:𝐺𝑝_ℎ,𝑞_ℎ=𝑝^𝑇𝑖𝑀_𝑘−𝑀₁𝑞_𝑗=𝑝^𝑇𝑖𝑀_𝑘𝑞_𝑗−𝑝^𝑇𝑖𝑀₁𝑞_𝑗.(3.4) Here, we set 𝑝^𝑇𝑖=𝑝₀,𝑀𝑝₁,_𝑖𝑗…,𝑝_𝑁−1^𝑇,𝑞_𝑗=𝑞₀,𝑞₁,…,𝑞_𝑁−1,=𝜙_𝑖,𝜙_𝑗,𝑝_ℎ=^𝑁−1_𝑖=0𝑝_𝑖𝑝_𝑖𝜙_𝑖,=𝑝_ℎ𝑥_𝑖,∀𝑝_ℎ∈𝑀_ℎ,𝑖,𝑗=0,…,𝑁−1,(3.5) where 𝜙_𝑖 is the basis function of the pressure on the domain Ω such that its value is one at node 𝑥_𝑖 and zero at other nodes; the symmetric and positive 𝑀_𝑘,𝑘≥2 and 𝑀₁ are pressure mass matrix computed by using the k-order and 1-order Gauss integrations in each direction; respectively, also, 𝑝_𝑖 and 𝑞_𝑖,𝑖=0,1,…,𝑁, are the value of 𝑝_ℎ and 𝑞_ℎ at the node 𝑥_𝑖. 𝑝^𝑇𝑖 is the transpose of the matrix 𝑝_𝑖.

Let Π_ℎ∶𝑀→𝑅₀ be the standard 𝐿²-projection with the following properties [29–32]:𝑝,𝑞_ℎ‖‖=Π_ℎΠ_ℎ𝑝,𝑞_ℎ,∀𝑝∈𝑀,𝑞_ℎ∈𝑅₀,‖‖𝑝−Π_ℎ𝑝‖‖₀≤𝑐‖𝑝‖₀,∀𝑝∈𝑀,𝑝‖‖₀≤‖𝑝‖₁,∀𝑝∈𝐻¹(Ω)∩𝑀,(3.6) where 𝑅₀={𝑞_ℎ∈𝑀∶𝑞_ℎ|_𝐾isaconstant,∀𝐾∈𝐾_ℎ}. Then we can rewrite the bilinear form 𝐺(⋅,⋅) by𝐺(𝑝,𝑞)=(𝑝−Π𝑝,𝑞−Π𝑞).(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 𝑡_𝑛=𝑛𝜏, 𝑛=0,1,2,…,𝑁. We obtain the fully discrete scheme as follows: find functions {𝑢^𝑛ℎ}_𝑛≥0⊂𝑋_ℎ and {𝑝^𝑛ℎ}_𝑛≥1⊂𝑀_ℎ as solutions of the recursive linear algebraic equations, 𝑑_𝑡𝑢^𝑛ℎ,𝑣_ℎ+ℬ_ℎ𝑢^𝑛ℎ,𝑝^𝑛ℎ;𝑣_ℎ𝑢,_0ℎ𝑞_ℎ+𝑏𝑢^ℎ𝑛−1,𝑢_ℎ,𝑣_ℎ=𝑓𝑡_𝑛,𝑣_ℎ,=𝑢_0ℎ.(3.8) for all (𝑣_ℎ,𝑞_ℎ)∈(𝑋_ℎ,𝑀_ℎ), where 𝑑_𝑡𝑢^𝑛ℎ=𝑢^𝑛ℎ−𝑢^ℎ𝑛−1ℬ_ℎ𝜏,𝑢_ℎ,𝑝_ℎ;𝑣_ℎ,𝑞_ℎ=𝑎𝑢_ℎ,𝑣_ℎ−𝑑𝑣_ℎ,𝑝_ℎ+𝑑𝑢_ℎ,𝑞_ℎ+𝐺𝑝_ℎ,𝑞_ℎ,(3.9) and 𝑢_0ℎ is the approximation of 𝑢₀. The solutions {𝑢^𝑛ℎ}_𝑛≥0 and {𝑝^𝑛ℎ}_𝑛≥1 to (3.8)-(3.9) are expected to the approximations of {𝑢_ℎ(𝑡_𝑛)}_𝑛≥0 and {𝑝_ℎ(𝑡_𝑛)}_𝑛≥1 with 𝑝_ℎ𝑡_𝑛=1𝜏^𝑡𝑛_𝑡𝑛−1𝑝_ℎ(𝑡)𝑑𝑡.(3.10)

Theorem 3.2 (see [29–32]). Let (𝑋_ℎ,𝑀_ℎ) be defined as above, then there exists a positive constant 𝛽, independent of ℎ, such that 𝛽‖‖||𝑢ℬ((𝑢,𝑝);(𝑣,𝑞))||≤𝑐‖𝑢‖₁+‖𝑝‖₀‖𝑣‖₁+‖𝑞‖₀,(𝑢,𝑝),(𝑣,𝑞)∈(𝑋,𝑀),_ℎ‖‖₁+‖‖𝑝_ℎ‖‖₀≤sup_{𝑣ℎ,𝑞ℎ∈𝑋ℎ,𝑀ℎ}||ℬ_ℎ𝑢_ℎ,𝑝_ℎ;𝑣_ℎ,𝑞_ℎ||‖‖𝑣_ℎ‖‖₁+‖‖𝑞_ℎ‖‖₀||𝐺(𝑝,𝑞)||≤𝑐‖‖𝑝−Π_ℎ,∀𝑢_ℎ,𝑝_ℎ∈𝑋_ℎ,𝑀_ℎ,𝑝‖‖₀‖‖𝑞−Π_ℎ𝑞‖‖₀,∀𝑝,𝑞∈𝑀.(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 ℎ and 𝜏 are sufficiently small and ℎ=𝑂(𝜏), then problem (3.8)-(3.9) has a unique solution (𝑢^𝑛ℎ,𝑝^𝑛ℎ)∈𝑋_ℎ×𝑀_ℎ such that ‖‖𝑢^𝑛ℎ‖‖²⁰+∑^𝑛_𝑗=1𝜏𝜈‖‖∇𝑢^𝑗ℎ‖‖²⁰+𝜏𝐺𝑝^𝑗ℎ,𝑝^𝑗ℎ≤‖‖𝑢₀‖‖²⁰+𝑐𝜏𝜈^𝑛−1∑_𝑗=1‖‖𝑓^𝑗‖‖²⁰.(3.12)

Theorem 3.4 (see [32]). Under the assumptions of Theorem 3.3, the error (𝑢(𝑡_𝑛)−𝑢^𝑛ℎ,𝑝(𝑡_𝑛)−𝑝^𝑛ℎ) satisfies the following bound: 𝜏^𝑁_𝑛=1𝜏_𝑁‖‖𝑢𝑡_𝑛−𝑢^𝑛ℎ‖‖²⁰+𝜏𝑡_𝑚‖‖𝑢𝑡_𝑚−𝑢^𝑚ℎ‖‖²⁰≤𝑐ℎ⁴+𝜏²,_𝑛=1‖‖𝑢𝑡𝜏_𝑛_𝑁−𝑢^𝑛ℎ‖‖²¹+𝜏𝑡_𝑚‖‖𝑢𝑡_𝑚−𝑢^𝑚ℎ‖‖²¹≤𝑐ℎ²+𝜏²,_𝑛=1𝜏𝑡_𝑛‖‖𝑝𝑡_𝑛−𝑝^𝑛ℎ‖‖²⁰≤𝑐ℎ²+𝜏²,(3.13) for all 𝑡_𝑚∈(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 {𝑢^𝑛1ℎ,𝑢^𝑛2ℎ,𝑝^𝑛ℎ}^𝐿𝑛=1. Then we choose 𝐿 (in general, 𝐿≤𝑁, e.g., 𝐿=20,𝑁=200) instantaneous solutions 𝑈_𝑖(𝑥,𝑦)=(𝑢^𝑖1ℎ,𝑢^𝑖2ℎ,𝑝^𝑖ℎ)(𝑖=1,2,…,𝐿) from the 𝐿 group of solutions (𝑢^𝑛ℎ,𝑝^𝑛ℎ)(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 (𝑋_ℎ,𝑀_ℎ) 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 𝑈_𝑖(𝑥,𝑦)=(𝑢^𝑖1ℎ,𝑢^𝑖2ℎ,𝑝^𝑖ℎ)(𝑖=1,2,…,𝐿, see Section 3). Set𝑉=span𝑈₁,𝑈₂,…,𝑈_𝐿,(4.1) where 𝑉 is the ensemble consisting of the snapshots {𝑈_𝑖}^𝐿𝑖=1, at least one of which is supposed to nonzero. Let {Ψ_𝑗}^𝑙𝑗=1 denote an orthogonal basis of 𝑉 with 𝑙=dim𝑉(𝑙≤𝐿). Then each member of the ensemble is expressed as𝑈_𝑖=^𝑙_𝑖=1𝑈_𝑖,Ψ_𝑗𝑋Ψ_𝑗,for𝑖=1,2,…,𝐿,(4.2) where (𝑈_𝑖,Ψ_𝑗)𝑋=(𝑢^𝑖ℎ,Ψ_𝑢𝑗)+(𝑝^𝑖ℎ,Ψ_𝑝𝑗), Ψ_𝑢𝑗 and Ψ_𝑝𝑗 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(1≤𝑑≤𝐿), the mean square error between the elements 𝑈_𝑖(1≤𝑖≤𝐿) and corresponding 𝑑th partial sum of (4.2) is minimized on averagemin_{Ψ𝑗𝑑𝑗=1}1𝐿^𝐿_𝑖=1‖‖‖‖𝑈_𝑖−^𝑑_𝑗=1𝑈_𝑖,Ψ_𝑗𝑋Ψ_𝑗‖‖‖‖²𝑋,(4.3) subject toΨ_𝑖,Ψ_𝑗𝑋=𝛿_𝑖𝑗,for1≤𝑖≤𝑑,1≤𝑗≤𝑖,(4.4) where ‖𝑈_𝑖‖𝑋=(‖𝑢^𝑖1ℎ‖²¹+‖𝑢^𝑖2ℎ‖²¹+‖𝑝^𝑖ℎ‖²⁰)^1/2. A solution sequence {Ψ_𝑗}^𝑑𝑗=1 of (4.3) and (4.4) is known as a POD basis of rank 𝑑.

We introduce the correlation matrix 𝐸=(𝐸_𝑖𝑗)_𝐿×𝐿∈𝑅^𝐿×𝐿 corresponding to the snapshots {𝑈_𝑖}^𝐿𝑖=1 by𝐸_𝑖𝑗=1𝐿𝑈^𝑖,𝑈_𝑗𝑋.(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 𝜆₁≥𝜆₂≥⋯≥𝜆_𝑙>0 denote the positive eigenvalues of 𝐸, and let 𝑣₁,𝑣₂,…,𝑣_𝑙 be the associated eigenvectors. Then a POD basis of rank 𝑑≤𝑙 is given by Ψ_𝑖=1√𝜆_{𝑖𝐿𝑗=1}𝑣_𝑖_𝑗,𝑈_𝑗,𝑖=1,2,…,𝑑≤𝑙,(4.6) where (𝑣_𝑖)_𝑗 denotes the 𝑗th component of the eigenvector 𝑣_𝑖. Furthermore, the following error formula holds: 1𝐿^𝐿_𝑖=1‖‖‖‖𝑈_𝑖−^𝑑_𝑗=1𝑈_𝑖,Ψ_𝑗𝑋Ψ_𝑗‖‖‖‖²𝑋=^𝑙_𝑗=_𝑑+1𝜆_𝑗.(4.7)
Let 𝑉^𝑑={Ψ₁,Ψ₂,…,Ψ_𝑑}, and let 𝑋^𝑑×𝑀^𝑑=𝑉^𝑑 with 𝑋^𝑑⊂X_ℎ⊂𝑋, and let 𝑀^𝑑⊂𝑀_ℎ⊂𝑀. Set the Ritz-projection 𝑃^ℎ∶𝑋→𝑋_ℎ (if 𝑃^ℎ is restricted to the Ritz-projection from 𝑋_ℎ to 𝑋^𝑑, it is written as 𝑃^𝑑) such that 𝑃^ℎ|_𝑋ℎ=𝑃^𝑑∶𝑋_ℎ→𝑋^𝑑 and 𝑃^ℎ∶𝑋⧵𝑋_ℎ→𝑋_ℎ⧵𝑋^𝑑 and 𝐿²-projection 𝜌^𝑑∶𝑀→𝑀^𝑑 denoted by, respectively, 𝑎𝜌𝑃^ℎ𝑢,𝑣_ℎ=𝑎𝑢,𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,(4.8)_𝑑𝑝,𝑞_𝑑₀=𝑝,𝑞_𝑑₀,𝑞_𝑑∈𝑀^𝑑,(4.9) where 𝑢∈𝑋 and 𝑝∈𝑀. Owing to (4.8)-(4.9) the linear operators 𝑃^ℎ and 𝜌^𝑑 are well defined and bounded: ‖‖𝑃^𝑑𝑢‖‖₁≤‖𝑢‖₁,‖‖𝜌^𝑑𝑝‖‖₀,≤‖𝑝‖₀,∀𝑢∈𝑋,𝑝∈𝑀.(4.10)

Lemma 4.2 (see [45]). For every 𝑑(1≤𝑑≤𝑙), the projection operators 𝑃^𝑑 and 𝜌^𝑑 satisfy, respectively, 1𝐿^𝐿_𝑖=1‖‖𝑢^𝑛𝑖_ℎ−𝑃^𝑑𝑢^𝑛𝑖_ℎ‖‖²¹≤^𝑙_𝑗=_𝑑+11𝜆_𝑗,(4.11)𝐿^𝐿_𝑖=1‖‖𝑢^𝑛𝑖_ℎ−𝑃^d𝑢^𝑛𝑖_ℎ‖‖²⁰≤𝐶ℎ^2𝑙_𝑗=_𝑑+11𝜆_𝑗,(4.12)𝐿^𝐿_𝑖=1‖‖𝑝^𝑛𝑖_ℎ−𝜌^𝑑𝑝^𝑛𝑖_ℎ‖‖²⁰≤^𝑙_𝑗=_𝑑+1𝜆_𝑗,(4.13) where 𝑢^𝑛𝑖_ℎ=(𝑢^𝑛𝑖_1ℎ,𝑢^𝑛𝑖_2ℎ) and (𝑢^𝑛𝑖_1ℎ,𝑢^𝑛𝑖_2ℎ,𝑝^𝑛𝑖_ℎ)∈𝑉.
Thus, using 𝑉^𝑑=𝑋^𝑑×𝑀^𝑑, we can obtain the reduced SMFE formulation for problems (3.8) as follows. Find 𝑢^𝑛𝑑̂=(𝑢^𝑛𝑑,𝑝^𝑛𝑑)∈𝑉^𝑑 such that 𝑑_𝑡𝑢^𝑛𝑑,𝑣_𝑑+𝐵_ℎ𝑢^𝑛𝑑,𝑝^𝑛𝑑;𝑣_𝑑,𝑞_𝑑+𝑢𝑏𝑢^𝑑𝑛−1,𝑢_𝑑,𝑣_𝑑=𝑓𝑡_𝑛,𝑣_𝑑,1≤𝑛≤𝑁,_0𝑑=𝑢_0ℎ.(4.14)

Remark 4.3. Problem (3.8) includes 𝑁_ℎ (𝑁_ℎ is the number of triangles or quadrilaterals vertex in 𝜏_ℎ) freedom degrees, while problem (4.14) includes 𝑑(𝑑≪𝑙≤𝐿≤𝑁) freedom degrees. For actual science and engineering problems, the number of the vertex in 𝜏_ℎ 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_ℎ=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 𝑉^𝑑=𝑋^𝑑×𝑀^𝑑⊂𝑉⊂𝑋_ℎ×𝑀_ℎ⊂𝑋×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 (𝑢^𝑛𝑑,𝑝^𝑛𝑑)∈𝑋^𝑑×𝑀^𝑑 and satisfies, for 1≤𝑛≤𝑁, ‖‖𝑢^𝑛𝑑‖‖²⁰+^𝑛_𝑗=1𝜏𝜈‖‖𝑢^𝑗𝑑‖‖²¹+𝜏𝐺𝑝^𝑗𝑑,𝑝^𝑗𝑑≤‖‖𝑢₀‖‖²⁰+𝑐𝜏𝜈^𝑛−1_𝑗=1‖‖𝑓^𝑗‖‖²⁰.(5.1)

In the following theorem, the errors between the solution (𝑢^𝑛𝑑,𝑝^𝑛𝑑) to Problem (4.14)-(4.15) and the solution (𝑢^𝑛ℎ,𝑝^𝑛ℎ) to Problem (3.8) are derived.

Theorem 5.2. Under the assumptions of Theorem 5.1, if ℎ and 𝜏 are sufficiently small, ℎ=𝑂(𝜏), and 𝜏=𝑂(𝐿⁻²), then the errors between the solutions (𝑢^𝑛𝑑,𝑝^𝑛𝑑) to Problem (4.14), and the solutions (𝑢^𝑛ℎ,𝑝^𝑛ℎ) to Problem (3.8) have the following error estimates, for 1≤𝑛≤𝑁: ‖‖𝑢^𝑛ℎ−𝑢^𝑛𝑑‖‖²⁰+𝜏𝜈^𝑛𝑖_𝑗_=𝑛1‖‖𝑢^𝑗ℎ−𝑢^𝑗𝑑‖‖²¹+𝜏^𝑛𝑖_𝑗_=𝑛1‖‖𝑝^𝑗ℎ−𝑝^𝑗𝑑‖‖²⁰≤𝐶𝜏^𝑙1/2_𝑗=_𝑑+1if𝑛=𝑛_𝑖𝜆_𝑗,‖‖𝑢∈𝑛₁,𝑛₂,…,𝑛_𝐿,(5.2)_𝑛ℎ−𝑢^𝑛𝑑‖‖²⁰+𝜏𝜈‖‖𝑢^𝑛ℎ−𝑢^𝑛𝑑‖‖²¹+^𝑛𝑖_𝑗_=𝑛1‖‖𝑢^𝑗ℎ−𝑢^𝑗𝑑‖‖²¹+𝜏‖‖𝑝_𝑛ℎ−𝑝^𝑛𝑑‖‖²⁰+^𝑛𝑖_𝑗_=𝑛1‖‖𝑝^𝑗ℎ−𝑝^𝑗𝑑‖‖²⁰≤𝐶𝜏^𝑙1/2_𝑗=𝑑+1if𝑛=𝑛_𝑖𝜆_𝑗+𝐶𝜏²,∉𝑛₁,𝑛₂,…,𝑛_𝐿.(5.3)

Proof. Let 𝑤^𝑛𝑑=𝑃^𝑑𝑢^𝑛ℎ−𝑢^𝑛𝑑, 𝑟^𝑛𝑑=𝜌^𝑑𝑝^𝑛ℎ−𝑝^𝑛𝑑. Subtracting (3.8) from (4.14) yields that 1𝜏𝑢^𝑛ℎ−𝑢^𝑛𝑑,𝑣_𝑑−1𝜏𝑢^ℎ𝑛−1+𝑏𝑢_ℎ𝑛−1−𝑢^𝑑𝑛−1,𝑣_𝑑+𝑎𝑢^𝑛ℎ−𝑢^𝑛𝑑,𝑣_𝑑−𝑏𝑝^𝑛ℎ−𝑝^𝑛𝑑,𝑣_𝑑−𝑢^𝑑𝑛−1,𝑢^𝑛ℎ,𝑣_𝑑+𝑏𝑢^𝑑𝑛−1,𝑢^𝑛ℎ−𝑢^𝑛𝑑,𝑣_𝑑+𝑏𝑢^𝑛ℎ−𝑢^𝑛𝑑,𝑞_𝑑+𝐺𝑝^𝑛ℎ−𝑝^𝑛𝑑,𝑞_𝑑=0.(5.4) Taking (𝑣_𝑑,𝑞_𝑑)=2𝜏(𝑤^𝑛𝑑,𝑟^𝑛𝑑) in (5.4), since 𝑎(𝑢^𝑛ℎ−𝑃^𝑑𝑢^𝑛ℎ,𝑤^𝑛𝑑)=0, 𝑏(𝑢^𝑛ℎ−𝑢^𝑛𝑑,𝑟^𝑛𝑑)+𝐺(𝑝^𝑛ℎ−𝑝^𝑛𝑑,𝑟^𝑛𝑑)=0, we deduce 2𝑤+^𝑛𝑑,2𝑤^𝑛𝑑𝑝_𝑑−2𝑤^𝑑𝑛−1,𝑤^𝑛𝑑+2𝜏𝑎𝑤^𝑛𝑑,𝑤^𝑛𝑑=2𝑢^ℎ𝑛−1−𝑝^𝑑𝑢^ℎ𝑛−1,𝑤^𝑛𝑑−2𝜏𝑏𝑢𝑤^𝑛ℎ−_𝑑𝑛−1𝑢^𝑛ℎ,𝑤^𝑛𝑑+2𝜏𝑏𝑝^𝑛ℎ−𝜌^𝑑𝑝^𝑛ℎ,𝑤^𝑛𝑑+2𝜏𝑏𝑃^𝑑𝑢^ℎ𝑛−1−𝑢^ℎ𝑛−1,𝑢^𝑛ℎ,𝑤^𝑛𝑑,𝑢^𝑛ℎ,𝑤^𝑛𝑑+2𝜏𝑏𝑢^𝑑𝑛−1,𝑃^𝑑𝑢^𝑛ℎ−𝑢^𝑛ℎ,𝑤^𝑛𝑑.(5.5) Using (2.12)-(2.13), the Hölder inequality and the Young inequality, we see that ≤||𝑢𝐶^ℎ𝑛−1₁−𝑝^𝑑𝑢^ℎ𝑛−1,𝑤^𝑛𝑑+𝑝^𝑑𝑢^𝑛ℎ−𝑢^𝑛ℎ,𝑤^𝑛𝑑||‖‖𝑢^ℎ𝑛−1−𝑝^𝑑𝑢^ℎ𝑛−1‖‖²¹+‖‖𝑢^𝑛ℎ−𝑝^𝑑𝑢^𝑛ℎ‖‖²¹||𝑏𝑝_𝑛ℎ+1𝜈0‖‖𝑤^𝑑‖‖²¹,−𝜌^𝑑𝑝^𝑛ℎ,𝑤^𝑛𝑑||≤𝐶₄‖‖𝑝^𝑛ℎ−𝜌^𝑑𝑝^𝑛ℎ‖‖²⁰+𝜈10‖‖𝑤^𝑛𝑑||𝑏𝑃_𝑑‖‖²¹,𝑢^ℎ𝑛−1−𝑢^ℎ𝑛−1,𝑢^𝑛ℎ,𝑤^𝑛𝑑||≤𝜈10‖‖𝑤^𝑛𝑑||𝑏𝑢_𝑑𝑛−1‖‖²¹+𝐶₅‖‖𝑢^ℎ𝑛−1−𝑃^𝑑𝑢^ℎ𝑛−1‖‖²¹,,𝑃^𝑑𝑢^𝑛ℎ−𝑢^𝑛ℎ,𝑤^𝑛𝑑||≤𝜈10‖‖𝑤^𝑛𝑑||−𝑏𝑤_𝑑𝑛−1‖‖²¹+𝐶₆‖‖𝑢^𝑛ℎ−𝑃^𝑑𝑢^𝑛ℎ‖‖²¹,,𝑢^𝑛ℎ,𝑤^𝑛𝑑||≤𝐶₇‖‖𝑤^𝑑𝑛−1‖‖²⁰+𝜈10‖‖𝑤^𝑛𝑑‖‖²¹.(5.6) Noting that 𝑎(𝑎−𝑏)=[𝑎²−𝑏²+(𝑎−𝑏)²]/2 (for (𝑎≥0 and 𝑏≥0)), owing to (5.6), we obtain that ‖‖𝑤^𝑛𝑑‖‖²⁰−‖‖𝑤^𝑑𝑛−1‖‖²⁰+𝜈𝜏‖‖𝑤_𝑑‖‖²¹≤𝐶₇𝜏‖‖𝑤^𝑑𝑛−1‖‖²⁰+𝐶𝜏‖‖𝑢^𝑛ℎ−𝑃^𝑑𝑢^𝑛ℎ‖‖²¹+‖‖𝑢^ℎ𝑛−1−𝑃^𝑑𝑢^ℎ𝑛−1‖‖²¹+‖‖𝑝^𝑛ℎ−𝜌^𝑑𝑝^𝑛ℎ‖‖²⁰.(5.7) First, we consider the case of 𝑛∈{𝑛₁,𝑛₂,…,𝑛_𝐿}. Summing (5.7) from 𝑛₁ to 𝑛_𝑖, 𝑖=1,2,…,𝐿, and noting that 𝑢^0ℎ−𝑢^0𝑑=0, using Lemma 4.2, we can derive that ‖‖𝑤^𝑛𝑖_𝑑‖‖²⁰+𝜈𝜏^𝑛𝑖_𝑗_=𝑛1‖‖𝑤^𝑗𝑑‖‖²¹≤𝐶𝜏𝐿^𝑙_𝑗=_𝑑+1𝜆_𝑗+𝐶₇𝜏^𝑛𝑖−1_𝑗_=𝑛0‖‖𝑤^𝑗𝑑‖‖²⁰.(5.8) By using the discrete Gronwall inequality, we obtain that ‖‖𝑤^𝑛𝑖_𝑑‖‖²⁰+𝜈𝜏^𝑛𝑖_𝑗_=𝑛1‖‖𝑤^𝑗𝑑‖‖²¹≤𝐶𝜏𝐿^𝑙_𝑗=_𝑑+1𝜆_𝑗exp𝐶₇𝜏𝑛_𝑖.(5.9) If ℎ and 𝜏 are sufficiently small, 𝜏=𝑂(𝐿⁻²), and noting that 𝑛_𝑖𝜏≤𝑛_𝑖𝑁≤𝑇, we find that ‖‖𝑤^𝑛𝑖_𝑑‖‖²⁰+𝜈𝜏^𝑛𝑖_𝑗_=𝑛1‖‖𝑤^𝑗𝑑‖‖²¹≤𝐶𝜏^𝑙1/2_𝑗=_𝑑+1𝜆_𝑗.(5.10) Thanks to 𝑏(𝑢^𝑛ℎ−𝑢^𝑛𝑑,𝑟^𝑛𝑑)+𝐺(𝑝^𝑛ℎ−𝑝^𝑛𝑑,𝑟^𝑛𝑑)=0, we obtain ‖‖𝑝^𝑛ℎ−𝑝^𝑛𝑑‖‖₀≤𝐶‖‖𝑢^𝑛ℎ−𝑢^𝑛𝑑‖‖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 𝑛∉{1,2,…,𝐿}; we assume that 𝑡_𝑛∈(𝑡_𝑛𝑖−1,𝑡_𝑛𝑖) and 𝑡_𝑛 is the nearest point to 𝑡_𝑛𝑖. 𝑢^𝑛ℎ and 𝑝^𝑛ℎ are expanded into the Taylor series expansion at point 𝑡_𝑛𝑖.𝑢^𝑛ℎ=𝑢^𝑛𝑖_ℎ−𝑠𝜏𝜕𝑢^ℎ𝜉₁𝜕𝑡,𝜉¹𝑝∈𝑡_𝑛𝑖,𝑡_𝑛,(5.12)_𝑛ℎ=𝑝^𝑛𝑖_ℎ−𝑠𝜏𝜕𝑝^ℎ𝜉₂𝜕𝑡,𝜉²∈𝑡_𝑛𝑖,𝑡_𝑛,(5.13) where 𝑠 is the number of time steps from 𝑡_𝑛 to 𝑡_𝑛𝑖. If the snapshots are equably taken, then 𝑠≤𝑁/𝐿. Summing (5.7) from 𝑛₁ to 𝑛_𝑖, n, and using (5.12), if |𝜕𝑢_ℎ(𝜉₁)/𝜕𝑡| and |𝜕𝑝_ℎ(𝜉₂)/𝜕𝑡| are bounded, by discrete Gronwall inequality and Lemma 4.2 and (3.12), we obtain that ‖‖𝑤^𝑛𝑑‖‖²⁰+𝜈𝜏‖‖𝑤^𝑛𝑑‖‖²¹+^𝑛𝑖_𝑗_=𝑛1‖‖𝑤^𝑗𝑑‖‖²¹≤𝐶𝜏𝐿^𝑙_𝑗=𝑑+1𝜆_𝑗+𝐶𝜏².(5.14)

If 𝜏=𝑂(𝐿⁻²), by (5.14) we obtain that‖‖𝑤^𝑛𝑑‖‖²⁰+𝜈𝜏‖‖𝑤^𝑛𝑑‖‖²¹+^𝑛𝑖_𝑗_=𝑛1‖‖𝑤^𝑗𝑑‖‖²¹≤𝐶𝜏^𝑙1/2_𝑗=𝑑+1𝜆_𝑗+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 (𝑢^𝑛𝑑,𝑝^𝑛𝑑) to Problem (4.14) are as follows: ‖‖𝑢𝑡_𝑛−𝑢^𝑛𝑑‖‖²⁰+𝜏𝜈^𝑛_𝑗=1‖‖u^𝑗−𝑢^𝑗𝑑‖‖²¹+𝜏^𝑛_𝑗=1‖‖𝑝^𝑗−𝑝^𝑗𝑑‖‖²⁰≤𝑐ℎ²+𝜏²+𝐶𝜏^𝑙1/2_𝑗=_𝑑+1𝜆_𝑗,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 𝑢=𝑢₁(𝑥𝑢,𝑦),𝑢₂(𝑥,𝑦),𝑝(𝑥,𝑦)=10(2𝑥−1)(2𝑦−1)cos(𝑡),₁𝑢(𝑥,𝑦)=10𝑥²(1−𝑥)²𝑦(1−𝑦)(1−2𝑦)cos(𝑡),₂(𝑥,𝑦)=−10𝑥(1−𝑥)(1−2𝑥)𝑦²(1−𝑦)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 ℎ=1/𝑛. In order to make 𝜏=𝑂(ℎ), 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 (𝑢^200ℎ𝑖, 𝑖=1,2)(c), the exact solutions (𝑢^𝑖200,𝑖=1,2) (b), and the RSMFE solutions (𝑢^200𝑑𝑖,𝑖=1,2) (a) are depicted, respectively, in Figures 1 and 3. Moreover, the difference (𝑢^𝑖200−𝑢^200𝑑𝑖, 𝑖=1,2) (a) between the exact solutions and the RSMFE solutions and the difference (𝑢^200ℎ𝑖−𝑢^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 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 3×10⁻³. 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 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.

Table 1 

Numerical results for the SMFE method.

Table 2 

Numerical results for the RSMFE method.

(a)

(b)

(c)

Figure 1 

RSMFE solutions (a), the exact solutions (b), and SMFE solutions (c).

(a)

(b)

Figure 2 

Difference between the RSMFE solutions and the exact solutions (a) and difference between the SMFE solutions and the RSMFE solutions (b).

(a)

(b)

(c)

Figure 3 

RSMFE solutions (a), the exact solutions (b), and SMFE solutions (c).

(a)

(b)

Figure 4 

Difference between the RSMFE solutions and the exact solutions (a) and difference between the SMFE solutions and the RSMFE solutions (b).

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 𝑃₁−𝑃₁ over triangles or 𝑄₁−𝑄₁ 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.