 Research
 Open Access
A reducedorder extrapolating collocation spectral method based on POD for the 2D Sobolev equations
 Shiju Jin^{1} and
 Zhendong Luo^{2}Email authorView ORCID ID profile
 Received: 11 October 2018
 Accepted: 21 March 2019
 Published: 28 March 2019
Abstract
In this paper, we mainly use a proper orthogonal decomposition (POD) to reduce the order of the coefficient vectors of the solutions for the classical collocation spectral (CS) method of twodimensional (2D) Sobolev equations. We first establish a reducedorder extrapolating collocation spectral (ROECS) method for 2D Sobolev equations so that the ROECS method includes the same basis functions as the classic CS method and the superiority of the classic CS method. Then we use the matrix means to discuss the existence, stability, and convergence of the ROECS solutions so that the procedure of theoretical analysis becomes very concise. Lastly, we present two set of numerical examples to validate the effectiveness of theoretical conclusions and to illuminate that the ROECS method is far superior to the classic CS method, which shows that the ROECS method is quite valid to solve Sobolev equations. Therefore, both theory and method of this paper are completely different from the existing reducedorder methods.
Keywords
 Proper orthogonal decomposition
 Classic collocation spectral method
 Sobolev equations
 Reducedorder extrapolating collocation spectral method
 Existence, stability, and convergence
MSC
 65N30
 65N12
 65M15
1 Introduction
The system of Sobolev equations (1) is a class of significant partial differential equations (PDEs) with practical physical background, which favorably simulated many engineering problems (see [1, 2]). Particularly, it can be applied to simulate the porous phenomenons (see [3, 4]). Nevertheless, in actual applications, Sobolev equations frequently contain intricate boundary and initial values, complicated source items, or discontinuous constants. As a result, we cannot generally seek analytic solutions, so that we can only rest upon numerical methods.
Currently, finite difference (FD), finite volume element (FVE), finite element (FE), and spectral methods are four famous computational techniques. However, the spectral method gives the highest accuracy among the four computational methods since the unknown functions in the spectral methods are approximated with some sufficiently smooth functions, such as Chebyshev polynomials, trigonometric functions, Legendre polynomials, or Jacobi polynomials, whereas the unknown functions in the FVE and FE methods are commonly approximated with some standard polynomials, and the derivatives in the FD scheme are approximated with some difference quotients. The spectral method is commonly sorted into the collocation spectral (CS) method, Galerkin’s spectral method, and the spectral tau method. It has been applied to settle various PDEs including secondorder elliptic, parabolic, hyperbolic, telegraph, and hydromechanical equations (see [5–8]).
However, Sobolev equations are mainly solved with the FD scheme, the FE method, and the FVE method (see, e.g., [3, 4, 9–14]), except that onedimensional Sobolev equations have been settled by the Fourier spectral method [15], and 2D Sobolev equations have recently been settled by the classic CS method [16]. Though the classic CS method (see [16]) for 2D Sobolev equations can attain higher accuracy than the FD scheme, FE method, and FVE method, it also contains a lot of degrees of freedom (unknowns). In this way, because of the roundoff error accumulation in numerical calculations, after several computational steps, there generally occurs a floatingpoint overflow such that we cannot obtain the desired consequences. Hence, to ensure a sufficiently high precision of the classic CS solutions, the crucial question is how to lessen the unknowns (i.e., degrees of freedom) of the CS method to ease the roundoff error accumulation in the calculations, which is also central task in this paper.
Many examples have proven that the proper orthogonal decomposition (POD) can significantly reduce the order of numerical methods (see [17–20]). It can vastly decrease the degrees of freedom in the numerical methods. It has been applied in many fields including pattern recognition and signal analysis [21], statistical calculations [22], and computational fluid mechanics [23]. For the past few years, it has successfully been used to the order reduction for the Galerkin methods [24, 25], the FE methods [26, 27], the FD schemes [28–30], the FVE methods [31, 32], and the reduced basis methods for PDEs [33–35]. Nevertheless, the existing PODbased reducedorder methods (see [17–27, 29–35]) are mostly created with the POD bases produced by the classic solutions at all the time nodes on \([0, T]\), before repeatedly finding the order reduction solutions on the same time nodal points. In fact, they belong to some undesirable repeated computations. To get rid of the repeated computations, several reducedorder extrapolated approaches based on POD have been proposed [36–41].
Nevertheless, to our knowledge, there is no reducedorder extrapolating CS (ROECS) method for 2D Sobolev equations created by reducing the order for the coefficient vectors of the CS solutions of the classic CS method via POD. Hence, in this paper, utilizing POD to reduce the order of the coefficient vectors of the CS solutions for the classic CS method, we construct a ROECS method only holding few degrees of freedom. We employ matrix means to discuss the existence, convergence, and stability of the ROECS solutions so that the theoretical means here becomes very concise. In particular, we only employ the classic CS solutions on the first several time nodal points to form the snapshots, and then we use them to produce the POD bases and create the ROECS format so as to obtain the ROECS solutions on all the time nodal points. Thus, we avoid the repeated computations. Moreover, in this paper, we adopt the error estimates to serve as the suggestion of choice of POD bases. The ROECS format contains both advantages that the POD method can reduce the unknowns and the CS method has higher accuracy, so it is an innovation and development of the existing reducedorder methods.
The main merits of the ROECS method hare the following. First, we only reduce the order of the coefficient vectors of the solutions for the classic CS method by POD and have not altered the basis functions for the classic CS method so that the ROECS method holds simultaneously both virtues that POD can reduce the unknowns and the classic CS method has higher accuracy. Second, the classic CS method is totally different from the Galerkin spectral method, and the Sobolev equations not only include a firstorder derivative term of time and two 2ndorder derivative terms of spacial variables but also contain two mixed derivative terms of the first order with respect to time and of the second order with respect to spacial variables, that is, the 2D Sobolev equations are more complex than the hyperbolic and parabolic equations in [42, 43]. So the ROECS method is totally different from the methods in [42, 43], but 2D Sobolev equations have some special applications as stated before. Third, we use the matrix means to discuss the existence, convergence, and stability of the ROECS solutions so that theoretical analysis becomes very concise and our theory and methods are totally different from the other existing order reduction methods. Therefore, our method is totally new and superior over the existing order reduction methods.
The rest of this paper is organized the following. In Sect. 2, we first retrospect the classic CS format of the 2D Sobolev equations and gain snapshots from the initial few classic CS solutions. Then, in Sect. 3, we produce a cluster of POD basis from the snapshots, develop the ROECS format, prove the existence, convergence, and stability of the ROECS solutions by the matrix means, and supply the flowchart for settling the ROECS format. Next, in Sect. 4, we use two sets of numerical examples to illuminate that the ROECS format is distinctly superior to the classic CS model, to validate that the numeric computational conclusions accord with the theoretical ones and that the ROECS format is quite valid to solve Sobolev equations, and to confirm that the ROECS format can greatly lessen the unknowns (i.e., degrees of freedom), the calculation load, the CPU elapsed time, and the required storage volumes in numerical computations. Finally, in Sect. 5, we provide the chief conclusions and discussions.
2 The classic CS method for 2D Sobolev equations
Because any bounded closed domain Ω̅ in \(\mathbb{R}^{2}\) can be approximately covered with several rectangles \([a_{i}, b_{i}] \times [c_{i}, d_{i}]\) (\(i=1, 2, \ldots, I\)), for simplicity and without loss of generality, let \(\overline{\varOmega }=[a, b]\times [c, d] \subset \mathbb{R}^{2}\). Moreover, using the transforms \(x' = 1+{2(xa)}/ {(ba)}\) and \(y' = 1+{2(yc)}/{(dc)}\), we can ensure \([a,b]\leftrightarrow [1,1]\) and \([c,d]\leftrightarrow [1,1]\), respectively. Thus, for convenience, we can further assume that \(a = c = 1\) and \(b = d =1\).
2.1 The variational formulation for the 2D Sobolev equations
We consider the following variational formulation for 2D Sobolev equations.
Problem 1
The following result on the existence, uniqueness, and stability of the generalized solution for Problem 1 has been provided in [16].
Theorem 2
2.2 The classic CS method for the 2D Sobolev equations
Theorem 3
Now, we obtain the following CS format for 2D Sobolev equations.
Problem 4
The result on the existence, uniqueness, stability, and convergence about the CS solutions for Problem 4 is given in [16].
Theorem 5
Remark 1
The error estimates in Theorem 5 attain an optimal order. Theorem 5 shows that the classic CS format, that is, Problem 4 for 2D Sobolev equations has a unique series of solutions, which is stable and continuously depends on the initial value and source functions. This theoretically ensures that Problem 4 is effective and reliable for solving 2D Sobolev equations.
2.3 The matrix representation of the classic CS format
Remark 2
Because the classic CS format adopts the Chebyshev polynomials as basic functions, it has a higher accuracy than general numerical methods, such as the FE method, FD scheme, and FVE method, but it also contains as many unknowns as the general numerical methods, so that it has to bear a lot of computing load. Thus, reducing the order for the classic CS format is more significative than for other numerical methods. For this purpose, we extract the initial L coefficient vectors \({\boldsymbol{U}}_{N}^{1}, {\boldsymbol{U}}_{N}^{2}, \ldots, {\boldsymbol{U}}_{N}^{L}\) (\(L\ll K\)) in the series of coefficient vectors \(\{{\boldsymbol{U}}_{N}^{n}\}_{n=1}^{K}\) for the classic CNCS matrix format (11) to form a set of snapshots.
3 The ROECS method based on POD for 2D Sobolev equations
3.1 Formulation of POD basis
Remark 3
Since the order \((2N+1)^{2}\) of the matrix \({\boldsymbol{P}}{\boldsymbol{P}}^{T}\) is far larger than the order L of the matrix \({\boldsymbol{P}}^{T}{\boldsymbol{P}}\), the number of the nodes of spatial meshes \((2N+1)^{2}\) is far larger than that of extracted snapshots L. Nevertheless, both positive eigenvalues \(\lambda _{i}\) (\(i=1,2,\ldots,r\)) are the same, and thus we may first search out the eigenvalues \(\lambda _{i}\) (\(i=1,2,\ldots,r\)) of \({\boldsymbol{P}}^{T}{\boldsymbol{P}}\) and the associated eigenvectors \({\boldsymbol{\varphi }} _{i}\) (\(i=1,2,\ldots,r\)), and then by the formula \({\boldsymbol{\phi }}_{i}= {\boldsymbol{P}}{\boldsymbol{\varphi }}_{i}/\sqrt{{\lambda _{i}}}\) (\(i=1,2,\ldots,r\)) we can gain the eigenvectors \({\boldsymbol{\phi }}_{i}\) (\(i=1,2,\ldots,r\)) associated with the positive eigenvalues \(\lambda _{i}\) (\(i=1,2,\ldots,r\)) of \({\boldsymbol{P}}{\boldsymbol{P}}^{T}\) and such that we can expediently obtain the POD basis.
3.2 Establishment of the ROECS model
Remark 4
As equation (11) contains \((N+1)^{2}\) unknowns at each time node, but the ROECS model, that is, the format (15) at the same time node only involves d unknowns (\(d\leqslant L\ll (N+1)^{2}\), for example, \(d=6\), but \((N+1)^{2}=10\text{,}201\) in Sect. 4), the format (15) is obviously superior to equation (11). After we have gained \(\boldsymbol{U}_{d}^{n}=(u_{d,0,0}^{n}, u_{d,1,0}^{n},\ldots ,u _{d, N,0}^{n},u_{d,0,1}^{n}, u_{d,1,1}^{n},\ldots ,u_{d,N,1}^{n}, \ldots ,u_{d,0,N}^{n},u_{d,1,N}^{n},\ldots , u_{d,N,N}^{n+1})^{T}\) (\(1\leqslant n\leqslant K\)) via (15), we can obtain the ROECS solutions by the formula \(u_{d}^{n}(x,y) = \sum_{j=0}^{N}\sum_{k=0}^{N}u_{d,j,k} ^{n}h_{j}(x) h_{k}(y)\) (\(n=1, 2, \ldots, K\)).
3.3 The existence, stability, and convergence for the ROECS solutions
Lemma 6
We have the following main result of the existence, stability, and convergence of the ROECS solutions for the format (15).
Theorem 7
Proof
(1) The existence and stability for the ROECS solutions.
Due to the reversibility of the matrix \((\boldsymbol{A} + \varepsilon \boldsymbol{B} +\gamma \Delta t\boldsymbol{B} )\), from the format (15) and Remark 4 we can conclude that the format (15) has a unique series of the ROECS solutions.
Write \(\boldsymbol{H}(x,y)=(h_{0}(x)h_{0}(y),h_{1}(x)h_{0}(y),\ldots ,h_{N}(x)h _{0}(y),h_{0}(x)h_{1}(y), h_{1}(x)h_{1}(y), \ldots , h_{N}(x) h _{1}(y), \ldots , h_{0}(x)h_{N}(y), h_{1}(x)h_{N}(y), \ldots , h_{N}(x)h _{N}(y))^{T}\). Then we denote the solutions for Problem 4 by \(u_{N}^{n} =(\boldsymbol{U}_{N}^{n})^{T}\boldsymbol{H}(x,y)=\boldsymbol{U} _{N}^{n}\cdot \boldsymbol{H}(x,y)\). Similarly, \(u_{d}^{n} = (\boldsymbol{U}_{d}^{n})^{T} \boldsymbol{H}(x,y) = \boldsymbol{U}_{d}^{n}\cdot \boldsymbol{H}(x,y)\).
(2) Error estimates (17).
Remark 5
 (1)
The factor \(\sqrt{\lambda _{d+1}}\) in Theorem 7 is caused by the order reduction for the CS format and can serve as the criterion of choice of the POD basis, that is, it is necessary to choose the number of the POD basis d and L satisfying \(\sqrt{\lambda _{d+1}}\leqslant \max \{\Delta t^{2},N^{2}\}\).
 (2)
We clearly can get that the matrix representation of the classic CS format (11) contains \((N+1)^{2}\) unknowns at each time node; nevertheless, the ROECS format (15) has only d unknowns (\(d\leqslant L\ll (N+1)^{2}\), for instance, \(d=6\), but \((N+1)^{2}=10\text{,}201\) in Sect. 4) at the same time node. Therefore, in comparison with the classic CS format, the ROECS format can greatly lessen unknowns, so that it can alleviate the calculation load and save the CPU consuming time and the storage requirements in the computational process for solving 2D Sobolev equations.
3.4 The flowchart for solving the ROECS format
 Step 1.:

For given parameters ε and γ, the source term \(f(x,y,t)\) and the initial function \(u_{0}(x,y)\), the number of nodes N in the direction of x or y, the nodes \(\{x_{m}\}_{m=0}^{N}=\cos (m\pi /N)\) and \(\{y_{l}\}_{l=0} ^{N}=\cos (l\pi /N)\), the time increment Δt. Solving the classic CS format (11) on the first L steps obtains the numerical solutions \(\boldsymbol{U}^{n}_{N}\) (\(1\leqslant n\leqslant L\)).
 Step 2.:

Put \(\boldsymbol{P}=(\boldsymbol{U}^{1}_{N}, \boldsymbol{U}^{2}_{N}, \ldots, \boldsymbol{U}^{L}_{N})_{(N+1)^{2}\times L}\) and seek the positive eigenvalues \(\lambda _{1}\geqslant \lambda _{2}\geqslant \cdots \geqslant \lambda _{\kappa }> 0\) (\(r=\dim \{u_{N}^{n}:1\leqslant n\leqslant L\}\)) and the associated eigenvectors \(\boldsymbol{\varphi }_{i}\) (\(i=1,2,\ldots ,r\)) of \(\boldsymbol{P}^{T}\boldsymbol{P}\).
 Step 3.:

Determine the number d of POD basis by means of the inequality \(\lambda _{d+1}\leqslant \max \{\Delta t^{4},N^{2q}\}\) and produce the POD basis \(\boldsymbol{\varPhi }= (\phi _{1},\phi _{2},\ldots ,\phi _{d})\) by the formula \({\boldsymbol{\phi }}_{i}={\boldsymbol{P}}{\boldsymbol{\varphi }}_{i}/\sqrt{ {\lambda _{i}}}\) (\(1\leqslant i\leqslant d\)).
 Step 4.:

First, obtain the ROECS solutions \(\boldsymbol{U}_{d}^{n}=(u _{d,0,0}^{n}, u_{d,1,0}^{n},\ldots ,u_{d, N,0}^{n},u_{d,0,1}^{n}, u_{d,1,1}^{n},\ldots , u_{d,N,1}^{n},\ldots , u_{d,0,N}^{n},u _{d,1,N}^{n},\ldots , u_{d,N,N}^{n+1})^{T}\) (\(1\leqslant n\leqslant K\)) by solving the ROECS format, that is, the format (15), and then we can obtain the ROECS solutions by the formula \(u_{d}^{n}(x,y) = \sum_{j=0} ^{N}\sum_{k=0}^{N}u_{d,j,k}^{n}h_{j}(x)h_{k}(y)\) (\(n=1, 2, \ldots, K\)).
 Step 5.:

If \(\u_{d}^{n}u_{d}^{n+1}\_{0,\omega }\leqslant \max \{\Delta t^{2},N^{q}\}\) (\(n=L,L+1,\ldots ,K1\)), then end. Else, let \(\boldsymbol{U}_{N}^{i}=\boldsymbol{U}_{d}^{nLi}\) (\(i=1,2,\ldots ,L\)) and return to Step 2.
4 Some numerical examples
5 Conclusions and discussions
In this study, we have studied the reducedorder of the coefficient vectors of the solutions for the classic CS method of 2D Sobolev equations. We have established the ROECS format in matrix form for 2D Sobolev equations via the POD technique, proven the existence, uniqueness, stability, and convergence of the ROECS solutions by the matrix means, and also given the flowchart for solving the ROECS format of 2D Sobolev equations. Moreover, we have supplied two numerical examples to verify the correctness of the theoretical analysis to explain that the ROECS format is far superior to the classic CS format because the unknowns of the ROECS format are far fewer than those of the classic CS format, so that, compared to the classic CS format, the ROECS format can greatly lessen the computational load, retard the roundoff error accumulation, and save the CPU consuming time in the operational process.
Especially, the ROECS format for the 2D Sobolev equations is first presented in this paper and is a development and improvement over the existing reducedorder methods because the ROECS format has higher accuracy than other reducedorder methods, such as the reducedorder FE method, FVE method, and FD scheme. Both theory and method of this paper are new and completely different from the existing reducedorder methods.
Although we restrict our ROECS method to Sobolev equations on rectangular domain \(\overline{\varOmega }=[a, b]\times [c, d]\), our technique can be extended to more general domains and applied in more complex engineering problems. Therefore, our technique has important applied prospect.
Declarations
Acknowledgements
The authors are thankful to the honorable reviewers and editors for their valuable suggestions and comments, which improved the paper.
Availability of data and materials
The authors declare that all data and material in the paper are available and veritable.
Funding
This research was supported by the National Science Foundation of China grant 11671106 and Qian Science Cooperation Platform Talent grant [2017]572641.
Authors’ contributions
Both authors contributed equally and significantly in writing this article. Both authors wrote, read, and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Ting, T.W.: A cooling process according to twotemperature theory of heat conduction. J. Math. Anal. Appl. 45(1), 23–31 (1974) MathSciNetMATHView ArticleGoogle Scholar
 Shi, D.M.: On the initial boundary value problem of nonlinear equation of the migration of the moisture in soil. Acta Math. Appl. Sin. 13(1), 31–38 (1990) MathSciNetMATHGoogle Scholar
 Liu, Y., Li, H., He, S., Gao, W., Mu, S.: A new mixed scheme based on variation of constants for Sobolev equation with nonlinear convection term. Appl. Math. J. Chin. Univ. 28(2), 158–172 (2013) MathSciNetMATHView ArticleGoogle Scholar
 Shi, D.Y., Wang, H.H.: Nonconforming H1Galerkin mixed FEM for Sobolev equations on anisotropic meshes. Acta Math. Appl. Sin. 25(2), 335–344 (2009) MathSciNetMATHView ArticleGoogle Scholar
 Guo, B.Y.: Some progress in spectral methods. Sci. China Math. 56(12), 2411–2438 (2013) MathSciNetMATHView ArticleGoogle Scholar
 Guo, B.Y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998) MATHView ArticleGoogle Scholar
 Zhou, Y.J., Luo, Z.D.: A Crank–Nicolson collocation spectral method for the twodimensional telegraph equations. J. Inequal. Appl. 2018, 137 (2018) MathSciNetView ArticleGoogle Scholar
 Shen, J., Tang, T.: Spectral and HighOrder Methods with Applications. Science Press, Beijing (2006) MATHGoogle Scholar
 Luo, Z.D., Teng, F.: A reducedorder extrapolated finite difference iterative scheme based on POD method for 2D Sobolev equation. Appl. Math. Comput. 329, 374–383 (2018) MathSciNetGoogle Scholar
 Gao, F.Z., Qiu, J.X., Zhang, Q.: Local discontinuous Galerkin finite element method and error estimates for one class of Sobolev equation. J. Sci. Comput. 41, 436–460 (2009) MathSciNetMATHView ArticleGoogle Scholar
 Jiang, Z.W., Chen, H.Z.: Error estimates for mixed finite element methods for Sobolev equation. Northeast. Math. J. 17(3), 301–314 (2001) MathSciNetMATHGoogle Scholar
 Li, H., Luo, Z.D., An, J.: A fully discrete finite volume element formulation for Sobolev equation and numerical simulations. Math. Numer. Sin. 34(2), 163–172 (2010) MathSciNetMATHGoogle Scholar
 Shi, D.Y., Wang, H.H., Guo, C.: Anisotropic rectangular nonconforming finite element analysis for Sobolev equations. Appl. Math. Mech. 29(9), 1203–1214 (2008) MathSciNetMATHView ArticleGoogle Scholar
 Luo, Z.D., Teng, F., Chen, J.: A PODbased reducedorder Crank–Nicolson finite volume element extrapolating algorithm for 2D Sobolev equations. Math. Comput. Simul. 146, 118–133 (2018) MathSciNetView ArticleGoogle Scholar
 Lu, W.J., Zhang, F.Y.: Longtime behavior of completely discrete Fourier spectral method of solutions to Sobolev equations. J. Nat. Sci. Heilongjiang Univ. 18(2), 5–8 (2001) MATHGoogle Scholar
 Jin, S.J., Luo, Z.D.: A collocation spectral method for the twodimensional Sobolev equations. Bound. Value Probl. 2018, 53 (2018) MathSciNetView ArticleGoogle Scholar
 Cazemier, W., Verstappen, R.W.C.P., Veldman, A.E.P.: Proper orthogonal decomposition and lowdimensional models for driven cavity flows. Phys. Fluids 10(7), 1685–1699 (1998) View ArticleGoogle Scholar
 Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996) MATHView ArticleGoogle Scholar
 Ly, H.V., Tran, H.T.: Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. Q. Appl. Math. 60(4), 631–656 (1989) MathSciNetMATHView ArticleGoogle Scholar
 Sirovich, L.: Turbulence and the dynamics of coherent structures: parts I–III. Q. Appl. Math. 45(3), 561–590 (1987) MATHView ArticleGoogle Scholar
 Fukunaga, K.: Introduction to Statistical Recognition. Academic Press, New York (1990) MATHGoogle Scholar
 Jolliffe, I.T.: Principal Component Analysis. Springer, Berlin (2002) MATHGoogle Scholar
 Selten, F.M.: Baroclinic empirical orthogonal functions as basis functions in an atmospheric model. J. Atmos. Sci. 54(16), 2099–2114 (1997) View ArticleGoogle Scholar
 Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90(1), 117–148 (2001) MathSciNetMATHView ArticleGoogle Scholar
 Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamischs. SIAM J. Numer. Anal. 40(2), 492–515 (2002) MathSciNetMATHView ArticleGoogle Scholar
 Luo, Z.D., Chen, J., Navon, I.M., Yang, X.Z.: Mixed finite element formulation and error estimates based on proper orthogonal decomposition for the nonstationary Navier–Stokes equations. SIAM J. Numer. Anal. 47(1), 1–19 (2008) MathSciNetMATHView ArticleGoogle Scholar
 Luo, Z.D., Li, H., Zhou, Y.J., Xie, Z.H.: A reduced finite element formulation based on POD method for twodimensional solute transport problems. J. Math. Anal. Appl. 385(1), 371–383 (2012) MathSciNetMATHView ArticleGoogle Scholar
 Cao, Y.H., Luo, Z.D.: A reducedorder extrapolating Crank–Nicolson finite difference scheme for the Riesz space fractional order equations with a nonlinear source function and delay. J. Nonlinear Sci. Appl. 11, 672–682 (2018) MathSciNetView ArticleGoogle Scholar
 Luo, Z.D., Yang, X.Z., Zhou, Y.J.: A reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for Burgers equation. J. Comput. Appl. Math. 229(1), 97–107 (2009) MathSciNetMATHView ArticleGoogle Scholar
 Sun, P., Luo, Z.D., Zhou, Y.J.: Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations. Appl. Numer. Math. 60(1–2), 154–164 (2010) MathSciNetMATHView ArticleGoogle Scholar
 Luo, Z.D., Li, H., Zhou, Y.J., Huang, X.M.: A reduced FVE formulation based on POD method and error analysis for twodimensional viscoelastic problem. J. Math. Anal. Appl. 385(1), 310–321 (2012) MathSciNetMATHView ArticleGoogle Scholar
 Luo, Z.D., Xie, Z.H., Shang, Y.Q., Chen, J.: A reduced finite volume element formulation and numerical simulations based on POD for parabolic problems. J. Comput. Appl. Math. 235(8), 2098–2111 (2011) MathSciNetMATHView ArticleGoogle Scholar
 Benner, P., Cohen, A., Ohlberger, M., Willcox, A.K.: Model Reduction and Approximation: Theory and Algorithm. Computational Science and Engineering. SIAM, Philadelphia (2017) MATHView ArticleGoogle Scholar
 Hesthaven, J.S., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer, Berlin (2016) MATHView ArticleGoogle Scholar
 Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations. Springer, Berlin (2016) MATHView ArticleGoogle Scholar
 Luo, Z.D., Chen, G.: Proper Orthogonal Decomposition Methods for Partial Differential Equations. Mathematics in Science and Engineering. Elsevier, Amsterdam (2018). https://www.elsevier.com/books/properorthogonaldecompositionmethodsforpartialdifferentialequations/luo/9780128167984 Google Scholar
 Luo, Z.D., Gao, J.Q.: A PODbased reducedorder finite difference timedomain extrapolating scheme for the 2D Maxwell equations in a lossy medium. J. Math. Anal. Appl. 444, 433–451 (2016) MathSciNetMATHView ArticleGoogle Scholar
 Luo, Z.D., Li, H.: A POD reducedorder SPDMFE extrapolating algorithm for hyperbolic equations. Acta Math. Sci. Ser. B Engl. Ed. 34(3), 872–890 (2014) MathSciNetMATHView ArticleGoogle Scholar
 Luo, Z.D., Li, H., Sun, P., Gao, J.Q.: A reducedorder finite difference extrapolation algorithm based on POD technique for the nonstationary Navier–Stokes equations. Appl. Math. Model. 37(7), 5464–5473 (2013) MathSciNetMATHView ArticleGoogle Scholar
 Xia, H., Luo, Z.D.: An optimized finite difference iterative scheme based on POD technique for the 2D viscoelastic wave equation. Appl. Math. Mech. 38(12), 1721–1732 (2017) MathSciNetMATHView ArticleGoogle Scholar
 Luo, Z.D., Teng, F.: Reducedorder proper orthogonal decomposition extrapolating finite volume element format for twodimensional hyperbolic equations. Appl. Math. Mech. 38(2), 289–310 (2017) MathSciNetMATHView ArticleGoogle Scholar
 An, J., Luo, Z.D., Li, H., Sun, P.: Reducedorder extrapolation spectralfinite difference scheme based on POD method and error estimation for threedimensional parabolic equation. Front. Math. China 10(5), 1025–1040 (2015) MathSciNetMATHView ArticleGoogle Scholar
 Luo, Z.D., Jin, S.J.: A reducedorder extrapolation spectralfinite difference scheme based on the POD method for 2D secondorder hyperbolic equations. Math. Model. Anal. 22(5), 569–586 (2017) MathSciNetView ArticleGoogle Scholar
 Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975) MATHGoogle Scholar
 Zhang, W.S.: Finite Difference Methods for Partial Differential Equations in Science Computation. Higher Education Press, Beijing (2006) (in Chinese) Google Scholar
 Luo, Z.D.: Mixed Finite Element Methods and Applications. Science Press, Beijing (2006) (in Chinese) Google Scholar