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

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 NavierStokes 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.


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 meshdependent parameters, whose optimal values are usually unknown.Particularly, for the lowest equal-order pairs of mixed elements such as P 1 − P 1 and Q 1 − Q 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.

Functional Setting of the Navier-Stokes Equations
Let Ω be a bounded domain in R 2 , assumed to have a Lipschitz continuous boundary Γ and to satisfy further assumptions below.The transient Navier-Stokes equations are considered as follows: Here u : Ω → R 2 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: Furthermore, we make a regularity assumption on the Stokes problem as follows.
Assumption H1.For a given g ∈ Y and the Stokes problem, satisfying the following regularity result: 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, u 0 .Because the norm and seminorm are equivalent on H 1 0 Ω 2 , we use the same notation • 1 for them.It is well known that for each v ∈ X there hold the following inequalities:

2.7
Assumption H2.The initial velocity u 0 ∈ D A and the body force f x, t ∈ L 2 0, T; L 2 Ω 2 are assumed to satisfy 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, div v , ∀ v, q ∈ X, M .

2.9
Also, a generalized bilinear form B •, • ; •, • on X, M × X, M is defined by Moreover, we define the trilinear form

2.11
By the above notations and the H ölder inequality, there hold the following estimates: Also, the Poincare inequality holds: where c 0 , c 1 , and γ 0 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

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 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:

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,

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∈τ h h K .The finite-element subspaces of this paper are defined by setting Then, the finite-element pairs are coupled as follows:

3.3
It is well known that this lowest equal-order finite-element pair does not satisfy the infsup condition.We define the following local difference between a consistent and an underintegrated mass matrices the stabilized formulation 29-31 : Here, we set 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 M 1 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 T i is the transpose of the matrix p i .
Let Π h : M → R 0 be the standard L 2 -projection with the following properties 29-32 : 3.8 for all v h , q h ∈ X h , M h , where

3.11
By using the same approaches as those in 45 , we can prove the following result.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.

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 where V is the ensemble consisting of the snapshots {U i } L i 1 , at least one of which is supposed to nonzero.Let {Ψ j } l j 1 denote an orthogonal basis of V with l dim V l ≤ L .Then each member of the ensemble is expressed as where U i , Ψ j X u i h , Ψ uj p i h , 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 average min {Ψj} where . A solution sequence {Ψ j } d j 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 } L i 1 by

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 .
where v i j denotes the jth component of the eigenvector v i .Furthermore, the following error formula holds: 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: where

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 finiteelement 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.

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 Using the same approaches as proving Theorem 3.3, we could prove the following existence result for solutions of problem 4.14 .

5.1
In the following theorem, the errors between the solution u n d , p n d to Problem 4.14 -4. 15

5.5
Using 2.12 -2.13 , the Hölder inequality and the Young inequality, we see that

Mathematical Problems in Engineering
Noting that a a − b a 2 − b 2 a − b 2 /2 for a ≥ 0 and b ≥ 0 , owing to 5.6 , we obtain that

5.8
By using the discrete Gronwall inequality, we obtain that

5.9
If h and τ are sufficiently small, τ O L −2 , and noting that n i τ ≤ n i N ≤ T , we find that λ j , n 1, 2, . . ., N. 5.16

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 and f is determined by 2.1 .
All the numerical experiments have been performed using the conforming Q 1 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  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 H 1 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.

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 P 1 − P 1 over triangles or Q 1 − Q 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.

Theorem 5 . 1 .
Under the assumptions of Theorems 2.2 and 3.3, Problem 4.14 has a unique solution sequence u n d , p n d ∈ X d × M d and satisfies, for 1 ≤ n ≤ N,
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 n h } n≥0 ⊂ X h and {p n h } n≥1 ⊂ M h as solutions of the recursive linear algebraic equations, Then we can rewrite the bilinear form G •, • by G p, q p − Πp, q − Πq .3.7 9and u 0h is the approximation of u 0 .The solutions {u n h } n≥0 and {p n h } 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 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 Under the assumptions of Theorem 3.3, the error u t n − u n h , p t n − p n h satisfies the following bound: we can obtain the reduced SMFE formulation for problems 3.8 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 n d , p n d to Problem 4.14 , and the solutions u n h , p n h to Problem 3.8 have the following error estimates, for 1 ≤ n ≤ N: and the solution u n h , p n h to Problem 3.8 are derived.Theorem 5.2.
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 n i −1 , t n i and t n is the nearest point to t n i .u n h and p n h are expanded into the Taylor series expansion at point t n i . is the number of time steps from t n to t n i .If the snapshots are equably taken, then s ≤ N/L.Summing 5.7 from n 1 to n i , n, and using 5.12 , if |∂u h ξ 1 /∂t| and |∂p h ξ 2 /∂t| are bounded, by discrete Gronwall inequality and Lemma 4.2 and 3.12 , we obtain that

Table 1 :
Numerical results for the SMFE method.

Table 2 :
Numerical results for the RSMFE method.