A stabilized MFE reduced-order extrapolation model based on POD for the 2D unsteady conduction-convection problem

In this study, we devote ourselves to establishing a stabilized mixed finite element (MFE) reduced-order extrapolation (SMFEROE) model holding seldom unknowns for the two-dimensional (2D) unsteady conduction-convection problem via the proper orthogonal decomposition (POD) technique, analyzing the existence and uniqueness and the stability as well as the convergence of the SMFEROE solutions and validating the correctness and dependability of the SMFEROE model by means of numerical simulations.


Introduction
Let ⊂ R  be an interconnected bounded domain. We are concerned with the following two-dimensional (D) unsteady conduction-convection problem (see, e.g., [-]).
Problem I Seek u = (u  , u y ) τ , p, and Q that satisfy where u = (u x , u y ) τ represents the unknown velocity vector, p represents the unknown pressure, Q represents the unknown heat energy, T is the final moment, j = (, ) τ , μ = √ Pr/Re, Pr is the Prandtl number, Re is the Reynolds, γ  = √ RePr, and f  (x, y, t), g  (x, y), (x, y, t) and ω(x, y) are four known functions. In order to facilitate theoretical analysis and not to lose universality, we assume that f  (x, y, t)=g  (x, y) = 0 and (x, y, t) =  in the following study.
Because the D unsteady conduction-convection problem is a system of nonlinear PDEs, it usually has no analytic solution so as to have to depend on approximate solutions. Until present, there have been many numerical methods for the D unsteady conductionconvection problem (see, e.g., [-]), but the stabilized mixed finite element (SMFE) method based on a parameter-free and two local Gauss integrals in [] is considered as one of the most efficient approaches to solving the D unsteady conduction-convection problem. However, the SMFE method includes a lot of unknowns so as to amass a lot of truncated errors and bear very large computational load in the real-world engineering applications. Thus, a key issue is how to decrease the unknowns of the SMFE method so as to ease the truncated error amassing and save the consuming time in the numerical computation but keeping sufficiently high accuracy of numerical solutions.
A number of numerical experiments (see, e.g., [-]) have shown that the proper orthogonal decomposition (POD) is a very useful approach to decrease the unknowns for numerical models and ease the truncated error amassing in the numerical computations. But the now available reduced-order numerical methods as stated above were built by means of the POD basis formulated by the classical numerical solutions on all time nodes, before calculating the reduced-order numerical solutions on the same time nodes, which are some vain reduplicated computations. Since , the reduced-order extrapolation MFE models based on POD for the D hyperbolic equations, unsteady parabolized Navier-Stokes (NS) equations, and viscoelastic wave equation have been proposed by Luo's team (see, e.g., [-]) to avert the vain reduplicated calculations.
However, as far as we know, there has not been any study where the POD technique is used to establish the SMFE reduced-order extrapolation (SMFEROE) model for the D unsteady conduction-convection problem. Therefore, in this article, we devote ourselves to establishing the SMFEROE model via the POD method for the D unsteady conductionconvection problem, analyzing the existence and uniqueness and the stability as well as the convergence of the SMFEROE solutions and validating the correctness and dependability of the SMFEROE model by means of numerical simulations.
The major differences between the SMFEROE model and the now available reducedorder extrapolation MFE models based on POD, as stated above, consist in the fact that the conduction-convection problem not only includes the unknown velocity and the unknown pressure, but also has the unknown heat energy coupled nonlinearly with the unknown velocity vector so that it is more complicated than the hyperbolic equations, unsteady parabolized NS equations, and viscoelastic wave equation. Thus, both the modeling of the SMFEROE method and the demonstration of the existence and uniqueness and the stability as well as the convergence of the SMFEROE solutions encounter more difficulties and require more techniques than the now available reduced-order extrapolation MFE models as stated above, but the SMFEROE model has some specific applications. Especially, the SMFEROE model is built by means of the POD basis generated by the SMFE solutions on the initial seldom time nodes, before finding out the SMFEROE solutions at all time nodes by means of the extrapolation iteration so that it does not have reduplicated computation. Consequently, it is development and improvement over the existing models as mentioned above.
The rest of the article is scheduled as follows. In Section , we review the SMFE model and the corresponding results for the D unsteady conduction-convection problem. In Section , we constitute the POD basis by means of the SMFE solutions on the initial sel-dom time nodes and build the SMFEROE model including seldom unknowns for the D unsteady conduction-convection problem by means of the POD basis. Section  offers the demonstration of the existence and uniqueness and the stability as well as the convergence of the SMFEROE solutions and the algorithm process for the SMFEROE model. In Section , some numerical simulations are supplied to validate the correctness and dependability of the SMFEROE model. Section  generalizes the main conclusions.

Review the fully discrete SMFE model
The following arisen Sobolev spaces as well as their norms are well known (see []).
The weak form for the D unsteady conduction-convection problem is stated as follows.
, then Problem II has a unique solution that satisfies For the integer N > , let k = T/N represent the time step, h = {K} represent the quasiuniformity triangle partition of (see [, ]), P  (K) denote the linear polynomial space on K , and (u n h , p n h , Q n h ) be the SMFE solutions of (u(t), p, Q) at the time nodes t n = nk ( ≤ n ≤ N ). Then the SMFE model including the parameter-free and two local Gauss integrals can be stated as follows.
, ε is a positive parameter-free real, K,j g(x, y) dx dy (j = , ) represent the Gauss integrals on K that are exact for i degree polynomial g(x, y) = p h q h (j = , ), and R h is the Ritz projection from W onto W h (see []).
Note that, ∀q h ∈ M h , the function p h ∈ M h should be piecewise constant as j = . IfŴ h ⊂ L  ( ) is the piecewise constant space on h and the operator h : L  ( ) →Ŵ h is defined as follows, ∀p ∈ L  ( ), then the bilinear functional D(·, ·) can be denoted by Furthermore, the operator h satisfies the following inequalities (see [, , ]): where C >  in this context denotes the constant independent of h and k that is possibly not the same at different places.
The following conclusions of the existence and uniqueness and the stability as well as the convergence of the SMFE solutions to Problem III have been deduced in [].

Theorem  Under the conditions of Theorem , the SMFE model has only a set of solutions
where (u, p, T) represents the generalized solution of Problem II.   Lemma  Suppose that the rank ofÃ is l, λ  ≥ λ  ≥ · · · ≥ λ l >  are the positive eigenvalues ofÃ, and ψ  , ψ  , . . . , ψ l are the corresponding orthonormal eigenvectors. Then the POD bases are denoted by

Constitute the POD basis and build the SMFEROE model
and satisfy the following formula: Let Then it is easily known from functional analysis principles (see, e.g., []) that there are three extensions P h : Moreover, there are the following results (see [, , ]): In addition, there are the following conclusions (see, e.g., [, -]).
are the initial L solutions of Problem III. Furthermore, the projections P h , Z h , and R h hold, respectively, the following properties: where (u, p, Q) ∈ H  ( )  × H m ( ) × H  ( ) represents the generalized solution for the D unsteady conduction-convection problem.
Thus, based on X d × M d × W d , the SMFEROE formulation for the D unsteady conduction-convection problem is set up as follows.

Problem IV
where (u n h , p n h , Q n h ) ∈ X h × M h × W h (n = , , . . . , L) are the initial L SMFE solutions for Problem III.
Remark  It is easily known that Problem III at each time node contains N h (here N h represents the number of vertices of triangles in h , see []) unknowns, but Problem IV at the same time node only has d (d l ≤ L N N h ) unknowns. For the real-world engineering issues, the number N h of vertices of triangles in h exceeds thousands or even millions; whereas d only is the number of the initial seldom eigenvalues and is quite small (for instance, in Section , d = , but N h =  ×  ×   ). Therefore, Problem IV is the SMFEROE model for the D unsteady conduction-convection problem. Especially, Problem IV only uses the initial few known L solutions of Problem III to seek other (N -L) solutions and does not have reduplicated calculations. In other words, the initial L PODbased SMFEROE solutions are gained by means of projecting the initial L SMFE solutions into POD basis, while other (N -L) SMFEROE solutions are gained by means of extrapolation and iterating equations (), (), and (). Therefore, it is thoroughly different from the now available reduced-order models (see, e.g., [-, , ]).

The existence and uniqueness and the stability as well as the convergence of SMFEROE solutions and the algorithm process for the SMFEROE model 4.1 The existence and uniqueness and the stability as well as the convergence of the SMFEROE solutions
The existence and uniqueness and the stability as well as the convergence of the solutions for the SMFEROE formulation of the D unsteady conduction-convection problem have the following main conclusions.

Theorem  Under the conditions of Theorem , Problem IV has only a set of solutions
Proof When  ≤ n ≤ L, from (), we immediately gain unique (u n d , , and using () and Hölder's and Cauchy's inequalities, we obtain It follows from () that Taking a square root for () and utilizing ( n By choosing ϕ d = Q n d in () and by making use of () and Hölder 's and Cauchy's inequalities, we obtain Summing () from L+ to n yields By extracting a square root for (), making use of ( n i= a  i ) / ≥ n i= |a i |/ √ n and () when n = L, and then simplifying, we obtain By noting that · - ≤ C ·  and by using () when n = L, from () and (), we obtain Combining () with () yields that () holds when L +  ≤ n ≤ N . If p n d = , () is distinctly correct.
When  ≤ n ≤ L, with Lemma  and (), we immediately obtain (). When L +  ≤ n ≤ N , by subtracting Problem IV from Problem III choosing ψ h = ψ d , q h = q d , and ϕ h = ϕ d , we acquire Let e n = P d u n h -u n d , f n = u n h -P d u n h , η n = Z d p n hp n d , and ξ n = p n h -Z d p n h . First, from (), (), and (), we obtain e n   + kμ ∇e n   = P d u n h -u n d , e n + kA P d u n h -u n d , e n = -f n , e n + u n h -u n d , e n + kA u n h -u n d , e n = f n- -f n , e n + kB p n hp n d , e n -kA  u n h , u n h , e n By extraction of a square root to () and making use of ( n i= a  i ) / ≥ n n= |a i |/ √ n, we gain e n  + k Moreover, from Lemma  as well as Theorem , we acquire Combining () and () with () and using Lemma  and () when n = L yield First, by making use of () and Lemma , we acquire And then, when N  μ - ∇u n d  ≤ / (n = , , . . . , N ), with Lemma , (), and Hölder's and Cauchy's inequalities, we have Combining () with () and using Lemma , Theorems  and , the same technique as () yield that Summing () from L +  to n yields that By extraction of a square root to () and making use of ( n i= a  i ) / ≥ n i= |a i |/ √ n and (), we acquire With the triangle inequality of norm, (), and Lemma , we acquire By combining () with () and making use of Lemma , we acquire Combining () with () yields (). When η n = , () is distinctly correct. Thus, the argument of Theorem  is accomplished.
By combining Theorem  with Theorem , we immediately acquire the following conclusion.  Figure 1 The computational domain and the initial boundary values.   Further, by comparing the SMFE model with the SMFEROE model with  POD bases implementing the numerical simulations when t = , Pr = , and Re = ,, we find that the SMFE model includes  ×  ×   unknowns on every time node and the elapsed time is about  minutes, but the SMFEROE model with  POD bases only has  ×  unknowns at the same time node and the corresponding elapsed time is no more than  seconds, i.e., the elapsed time of the SMFE model is  times more than that of the SM-FEROE model with  POD bases. Thus, the SMFEROE model can immensely decrease the elapsed time and ease the computational load so that it could immensely ease the truncated error amassing in the calculation procedure. This implies that the SMFEROE model is effective and dependable for solving the D unsteady conduction-convection problem.

Conclusions
In this article, we have established the SMFEROE model for the D unsteady conductionconvection problem by means of the POD technique. We first extract the initial seldom L (L N ) SMFE solutions for the D unsteady conduction-convection problem and formulate the snapshots. Next, we constitute the POD basis by the snapshots by means of the POD technique. And then, the subspaces generated with the initial seldom POD basis substitute the MFE subspaces in the SMFE model in order to establish the SMFEROE model for the D unsteady conduction-convection problem. Finally, we analyze the existence and uniqueness and the stability as well as the convergence of the SMFEROE solutions for the D unsteady conduction-convection problem and supply the algorithm process for the SMFEROE model. Comparing the numerical simulation results of the SMFEROE solutions with the SMFE solutions validates the dependability and correctness of the SM-FEROE model.