Abstract
This paper considers the two-level iterative finite element methods for the steady natural convection equations under some uniqueness conditions with the Simple-, Oseen- and Newton-type corrections. Firstly, the stability and convergence of the one-level iterative finite element methods are analyzed under some restrictions on physical parameters. Secondly, under the strong uniqueness condition, we develop the two-level finite element method with Simple, Oseen and Newton iterations of m times on the coarse mesh \(\tau _H\) with mesh size H, and then, the considered problem is linearized in three correction schemes with the Simple, Oseen and Newton corrections one time on the fine grid \(\tau _h\) with mesh size \(h\ll H\) based on the obtained iterative solutions. From the theoretical point of view, the results obtained by the two-level iterative methods have the same precision as those obtained by the one-level method which mesh sizes satisfy \(h={\mathcal {O}}(H^2)\) and the iterative steps are greater than some constants. Thirdly, the stability and convergence of one-level Oseen iterative scheme with respect to the mesh size and the iterative time m are provided under a weak uniqueness condition. Finally, some numerical experiments are designed to confirm the established theoretical findings and verify the performance of the proposed numerical schemes.
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1 Introduction
Assume \(\Omega \) is a bounded domain in \({\mathbb {R}}^2\) with Lipschitz continuous boundary \(\partial \Omega \). Consider the following stationary natural convection equations in dimensionless form (see Allard and Ghiaus 2016; Ball 1956; Huang et al. 2015; Luo 2006)
where u, p, T are the velocity, the pressure and the temperature, respectively, f is the forcing function, \(\mathrm{{Pr,Ra}},\kappa \) represent the Prandtl number, Rayleigh number and the thermal conductivity parameter, which are some parameters related to physics, \(j=\left( \begin{array}{c} 0 \\ 1 \\ \end{array} \right) \) is the acceleration of gravity, n is the outward unit normal to the \(\Gamma _T\).
The natural convection problem is a classical model in the atmospheric dynamics, which can be used to describe many physical phenomenon, such as the natural ventilation in industry (Allard and Ghiaus 2016), the katabatic winds (Ball 1956), the buoyancy-driven flows (Chassignet et al. 2012), the dense gas dispersion (Touma et al. 1995). At the same time, it is also widely used in the field of industrial engineering, such as the design of double glazing. From the expression of problem (1.1), we can see that the natural convection problem contains three variables; it not only inherits all difficulties of the Navier–Stokes equations, but also has more complexity since three variables are coupled together by the nonlinear terms. Finding the analytical solutions of problem (1.1) becomes an impossible work (Jiu et al. 2012; Wu 2012). Therefore, designing the efficient numerical scheme becomes an important way to research the behaviors of solutions and many works have been done in recent years, for example, the mixed finite element method (FEM) (Boland and Layton 1990), the projection method (Cibik and Kaya 2011), the variational multiscale method (John et al. 2006; Zhang et al. 2014) and the defect-correction method (Si et al. 2011, 2012). For more recent developments for the natural convection equations, we can refer to above-mentioned papers and the references therein.
The nonlinear equations describe the nature phenomena better than the linear equations, such as the soliton (Hirota 2004), the dissipative system (Hooft 1999) and the quantum chaos (Katsuhiro 1995). However, the nonlinear term needs to be linearized when one implements on computer because the nature of nonlinearity is not well understood. Generally speaking, there are three linearizations for the nonlinear term; they are the Simple, Newton and Oseen iterations. For the incompressible flow problems, three iterations for the steady Navier–Stokes equations were analyzed in He and Li (2009) and Niu et al. (2018); the combinations of the iterative schemes with two-level method (He et al. 2012; He and Wang 2008; He 2015; Layton and Leferink 1995; Si et al. 2011), with the defect-correction method (Huang et al. 2013; Su et al. 2014), with the variational multiscale method (Du et al. 2015; Zhang et al. 2014), with the spectral method (Ge et al. 2022; Zhou et al. 2022, 2020) were also considered. Later, the iterative methods were extended to more complex incompressible flow, such as the incompressible magneto-hydrodynamics (MHD) flow (Dong et al. 2014; Tao and Zhang 2015; Yang et al. 2018; Zhang et al. 2014, 2018), the natural convection equations (Du et al. 2015; Huang et al. 2015; Luo et al. 2003; Si et al. 2011; Su et al. 2014; Wang et al. 2018; Zhang et al. 2014), the thermally coupled MHD problem (Yang and Zhang 2020; Zhang et al. 2016). Two-level method is an efficient way to solve the PDEs numerically; this method has been widely used to solve various problems since it was developed by Xu (1994, 1996), such as for the nonlinear elliptic/parabolic problems (Bi et al. 2018; Chen and Liu 2015; Gong et al. 2021; John et al. 2006), for the incompressible Navier–Stokes equations (He and Wang 2008; He et al. 2012; Huang et al. 2013; Layton and Leferink 1995), for the incompressible MHD flow (Layton et al. 1997; Tao and Zhang 2015; Zhang et al. 2015), for the integral-differential equations (Chen et al. 2019; Jiang et al. 2023), for the fractional PDEs (Liu et al. 2015; Chen et al. 2020; Qiu et al. 2020; Liu et al. 2021) and so on. For more recent works about the iterative method, we can refer to above-mentioned papers and the references therein.
In this paper, we design a two-level iterative finite element scheme for the natural convection equations (1.1) and analyze the stability and convergence of numerical solutions under some uniqueness conditions. We firstly assume that \((\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}<\tfrac{1}{4}\). In fact, if \((\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}<\tfrac{1}{4}\) cannot hold, we can choose a positive constant \(\alpha \) such that \(\alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}<\tfrac{1}{4}\). And set \( \sigma =\mathrm{{Pr}}^{-1}\mathrm{{Ra}}\kappa ^{-1}\Vert f\Vert _{-1}(N+\mathrm{{Pr}} {{\overline{N}}}\kappa ^{-1}) \) as a parameter, then problem (1.1) admits a unique solution under \(0<\sigma <1\). Secondly, three iterative FEMs for the natural convection equations are developed and the corresponding theoretical findings of iterative solutions \((u_H^m,p_H^m,T_H^m)\) are established under different uniqueness condition \(0<\sigma \le \tfrac{1}{4}, 0<\sigma \le \tfrac{1}{3}, 0<\sigma \le 1\), the \(H^2\)-stability of numerical solutions in iterative schemes are also provided. Finally, the two-level iterative FEMs are designed and analyzed for the natural convection equations. The two-level methods contain three iterative schemes related to the Stokes; the linearized Navier–Stokes and the Oseen equations are solved on the coarse mesh \(\tau _H\). And then, on the fine mesh, we use the iterative solutions \((u_H^m,p_H^m,T_H^m)\) as the known terms to linearize the nonlinear terms for three correction problems. According to the different unique conditions applicable to the three iteration methods, the parameters \(\sigma \) are divided again, \((0,\frac{1}{4}],(\frac{1}{4},\frac{1}{3}],(\frac{1}{3},1-\alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}]\). Three corrections are presented to find the numerical solution \((u_{mh},p_{mh},T_{mh})\) on the fine mesh \(\tau _h\) with the iterative solution \((u_H^m,p_H^m,T_H^m)\) obtained on the coarse mesh. If \(\sigma \in (0,\frac{1}{4}]\), both of the results obtained by the three iterative methods can be used as the known terms of the correction problems, \(\sigma \in (\frac{1}{4},\frac{1}{3}]\), Simple iterative solution cannot used for the correction problems, and \(\sigma \in (\frac{1}{3},1-\alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}]\), only Oseen iterative solution can be used for the three correction problems. From the established theoretical point of view, the results obtained by the two-level iterative methods have the same precision as those obtained by the one-level method as mesh sizes satisfy \(h={\mathcal {O}}(H^2)\) and iterative steps are greater than some constants.
This paper is organized as follows: in Sect. 2, some mathematical preliminaries and basic results of problem (1.1) are recalled. Section 3 presents three iterative FEMs and provides the stability and convergence of these numerical schemes. Two-level iterative FEMs for problem (1.1) are considered under the condition \(0<\sigma <1-\alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}\) in Sects. 4, 5 and 6, respectively, and the stability and convergence results are also provided in these corresponding sections. Section 7 analyzes one-level Oseen iterative finite element method for the natural convection equations with \(1-\alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}<\sigma <1\). Finally, some numerical results are given to identity with the established theoretical findings and show the performances of the considered numerical schemes.
2 Preliminaries
2.1 Function spaces and basic results
For the mathematical setting of problem (1.1), the following Sobolev spaces are used
The standard norms and semi-norms of above spaces are adopted, we take the spaces \(W^{m,p}(\Omega )\) with norm \(\Vert \cdot \Vert _{m,p},m,p\ge 0\). If one chooses \(m=0,p=2\), the space \(W^{0,2}(\Omega )\) denotes as \(L^2(\Omega )\) with the inner product \((\cdot ,\cdot )\) and norm \(\Vert \cdot \Vert _0\). The spaces X and W are endowed with the product \((\nabla \psi ,\nabla \varphi )\). We write \(W^{m,2}(\Omega )\) as \(H^m(\Omega )\) with norm \(\Vert \cdot \Vert _m\) and \(m\ne 0\).
Denote the Stokes operator by \(A=-P\Delta \) (see He 2015; He and Li 2009; Luo 2006; Temam 1984), where P is the \(L^2\)-orthogonal projection from Y onto H or Z onto Z, \(\Delta \) is the Laplace operator.
Next, we make some regularity assumptions that provided in He (2015), He and Li (2009), Luo (2006) and Temam (1984).
(A1). Assume that the boundary of \(\Omega \) is smooth so that (v, q) of the steady Stokes problem
for the prescribed \(g\in L^2(\Omega )^2\) satisfies
where \(C_i>0\ (i=1,2,3,4,5)\) is a constant depending on \(\Omega \).
(A2). Assume that \(\Omega \) is smooth such that there exists a unique solution \(\psi \in W\) of the following elliptic problem
for any prescribed \(\varphi \in L^2(\Omega )\). Furthermore, the solution \(\psi =A^{-1}\varphi \) satisfies
The validity of Assumptions (A1)–(A2) is known under the condition of \(\partial \Omega \in C^2\) or \(\Omega \) is a convex polygon (see He 2015; Luo 2006; Temam 1984).
Define the bilinear forms \(a(\cdot ,\cdot ), {{\overline{a}}}(\cdot ,\cdot )\) and \( d(\cdot ,\cdot )\) by
and the trilinear forms \( b(\cdot ,\cdot ,\cdot )\) on \(X \times X \times X\) and \({\overline{b}}(\cdot ,\cdot ,\cdot ) \) on \(X\times W \times W\) by
The following properties of \( b(\cdot , \cdot ,\cdot )\) and \( {{\overline{b}}}(\cdot , \cdot ,\cdot )\) can be found in He (2015), He and Li (2009), Luo (2006) and Temam (1984) and their references.
with
With above notations, the weak form of problem (1.1) reads as follows: for all \((v,q,\phi ) \in X \times M \times W\), find \((u,p,T) \in X \times M \times W\), such that
Theorem 2.1
(See Boland and Layton 1990; Luo 2006; Wu 2012) Under the Assumptions (A1)–(A2) and suppose that the physical parameters \(\mathrm{{Pr,Ra}},\kappa ,f \) satisfy
where
Then, problem (2.4) admits a unique solution (u, p, T). Furthermore, it holds
where \(c>0\) is a general constant, depends on \(C_i(i=1,...5)\) and the domain \(\Omega \).
2.2 The stability and convergence of Galerkin finite element method
Set \({\tau _{\mu }}\) be a family of shape-regular triangulations of \(\Omega \) with the mesh size \(\mu >0\), the mesh sizes h and \(H(h\ll H)\) tend to 0. We take the fine grid partition \(\tau _h\) as a mesh refinement generated from the coarse grid \(\tau _H\). Based on the regular partitions \(\tau _h\) and \(\tau _H\), the conforming finite element spaces \((X_h,M_h,W_h)\) and \((X_H, M_H,W_H)\subset (X_h,M_h,W_h)\subset (X,M,W)\) can be constructed. For the finite element spaces \(X_h\) and \(M_h\), the following restriction is required (see Luo 2006; Temam 1984).
(A3). There exists a constant \(\beta _1>0\) such that
There are several mixed finite element spaces satisfying the assumption (A3) when \(\Omega \) is a convex polygonal domain. For example, we can choose the velocity, pressure and temperature finite element spaces as follows:
where
\(P_1(K)\) is the space of linear polynomials on K and \({\hat{b}}\) is the bubble function.
Define the discrete space \(V_{\mu }\) by
Introduce two \(L^2\)-orthogonal projectors \(P_{1\mu }:L^2(\Omega )^2\rightarrow V_{\mu }\) and \(P_{2\mu }:L^2(\Omega )\rightarrow W_{\mu }\) by \((P_{1\mu }u_\mu ,v_\mu )=(u_\mu ,v_\mu ), \forall u_\mu \in L^2(\Omega )^2, v_\mu \in V_\mu \) and \((P_{2\mu }T_\mu ,\phi _\mu )=(T_\mu ,\phi _\mu ), \forall T_\mu \in L^2(\Omega ), \phi _\mu \in W_\mu \). And define the discrete Stokes operators by \(A_{1\mu }=-P_{1\mu }\Delta _{\mu }\) and \(A_{2\mu }=-P_{2\mu }\Delta _{\mu }\), where \(\Delta _{\mu }\) is given by (see He 2015; He and Li 2009; Sermane and Temam 1983).
The discrete norms are defined by \(\Vert v_{\mu }\Vert _{k,\mu }=\Vert A^{\tfrac{k}{2}}_{1\mu }v_{\mu }\Vert _0\) and \(\Vert T_{\mu }\Vert _{k,\mu }=\Vert A^{\tfrac{k}{2}}_{2\mu }T_{\mu }\Vert _0\) with \(k\in {\mathbb {R}}\).
The standard Galerkin finite element method for problem (2.4) reads as: Find \((u_{\mu },p_{\mu },T_{\mu })\in X_{\mu }\times M_{\mu }\times W_{\mu }\), for all \((v_{\mu },q_{\mu },\phi _{\mu })\in X_{\mu }\times M_{\mu }\times W_{\mu }\), such that
Theorem 2.2
(See Luo 2006; Si et al. 2011; Zhang et al. 2015) Under the Assumptions (A1)–(A4) and the uniqueness condition (2.5), set \((u,p,T) \in H^2(\Omega )^2 \cap X\times H^1(\Omega ) \cap M\times H^2(\Omega )\cap W\). Then, the solution \((u_{\mu },p_{\mu },T_{\mu })\) of problem (2.7) satisfies
Moreover, if \(\mu \le (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}(1-\sigma )\) holds, we have
where \(C_r=\Vert u\Vert _2 + \Vert p\Vert _1 + \Vert T\Vert _2\).
Proof
The inequalities (2.8) and (2.10) have been proved in Luo (2006), Si et al. (2011) and Zhang et al. (2015), respectively. We now give the proof of (2.9). Taking \(v_\mu =A_{1\mu }u_\mu , q_\mu =0,\phi _\mu =A_{2\mu }T_\mu \) in (2.7), using the definition of the Stokes operator, the \(L^2\)-orthogonal projectors \(P_{1\mu },P_{2\mu }\) and the Cauchy inequality, ones gets
With the help of (2.8), we complete the proof. \(\square \)
The physical domain with its boundary conditions
The streamlines of velocity obtained by two-level iterative FEM with \(Ra=10^3\), TLSSC,TLSNC,TLSOC (the first line), TLNSC,TLNNC,TLNOC (the second line) and TLOSC,TLONC,TLOOC (the third line)
3 Iterative finite element methods
When one considers the nonlinear problem, the iterative method should be chosen to discrete the nonlinear terms in implementation. Generally speaking, there are three treatments for the nonlinear terms, they are the Simple, Newton and Oseen iterations, both the theoretical findings and numerical results about these iterative schemes have been reported in recent years. For example, He et al. considered the performances of three iterative method for the Navier–Stokes equations in He (2015), He and Li (2009), He and Wang (2008) and He et al. (2012) and then extended to the incompressible MHD flow (Dong et al. 2014; Tao and Zhang 2015; Zhang et al. 2014), the natural convection problem (Huang et al. 2015; Si et al. 2011; Su et al. 2014; Wang et al. 2018) and the thermally coupled MHD equations (Yang and Zhang 2020; Zhang et al. 2016). In order to improve the computational efficiency, two-level iterative FEMs for the incompressible flow have also been considered, such as He and Wang (2008), Huang et al. (2013) and Layton and Leferink (1995) for the Navier–Stokes equations and Tao and Zhang (2015) and Zhang et al. (2015) for the MHD flows and the references therein. Here, we consider the Simple, Newton and Oseen iterative FEMs for the natural convection under different unique conditions. These iterative FEMs can be listed as follows:
Iterative methods 1. (The Simple iterative method). For all \(n=1,2,\ldots \), and \((v_{\mu },q_{\mu },\phi _{\mu }) \in X_{\mu } \times M_{\mu } \times W_{\mu }\), find the iterative solution \((u_{\mu }^n,p_{\mu }^n,T_{\mu }^n) \in X_{\mu } \times M_{\mu } \times W_{\mu }\), such that
Iterative methods 2. (The Newton iterative method). For all \(n=1,2,\ldots \), and \((v_{\mu },q_{\mu },\phi _{\mu }) \in X_{\mu } \times M_{\mu } \times W_{\mu }\), find the iterative solution \((u_{\mu }^n,p_{\mu }^n,T_{\mu }^n) \in X_{\mu } \times M_{\mu } \times W_{\mu }\), such that
Iterative methods 3. (The Oseen iterative method). For all \(n=1,2,\ldots \), and \((v_{\mu },q_{\mu },\phi _{\mu }) \in X_{\mu } \times M_{\mu } \times W_{\mu }\), find the iterative solution \((u_{\mu }^n,p_{\mu }^n,T_{\mu }^n) \in X_{\mu } \times M_{\mu } \times W_{\mu }\), such that
The initial guess \((u^0_{\mu },p^0_{\mu },T^0_{\mu }) \in X_{\mu } \times M_{\mu } \times W_{\mu }\) is defined as follows:
Set
Combining (3.1), (3.2), (3.3) and (3.4) with (2.7), we obtain the following error equations.
Error equations of iterative method 1:
Error equations of iterative method 2:
Error equations of iterative method 3:
And
Theorem 3.1
Under the Assumptions (A1)-(A3) and the conditions of Theorem 2.1, for all \(m\ge 0\), if \(0<\sigma \le \frac{1}{4}\) holds, the solution \((u_h^m,p_h^m,T_h^m)\) of iterative method 1 satisfies:
If \( 0 < \sigma \le \frac{1}{3}\) holds, the solution \((u_h^m,p_h^m,T_h^m)\) of iterative method 2 satisfies:
If \( 0 < \sigma \le 1\) holds, the solution \((u_h^m,p_h^m,T_h^m)\) of iterative method 3 satisfies:
Proof
The results (3.10), (3.12), (3.13), (3.15), (3.16), (3.17) and (3.19) have been provided by Huang in Huang et al. (2015). Now, we consider (3.11), (3.14) and (3.18).
Taking \(v_h=A_{1h}u_h^0,q_h=0\) and \(\phi _h=A_{2h}T_h^0\) in (3.4), we derive that
thus, (3.11) holds for \(m=0\), assuming that (3.11) holds for \(m=n-1\), we want to prove that it holds for \(m=n\). Taking \(v_h=A_{1h}u_h^n, q_h=0, \phi _h=A_{2h}T_h^n\) in (3.1) and using (2.2), (2.3), we get
If \(\Vert A_{2h}T_h^n\Vert _0\le \Vert A_{2h}T_h^{n-1}\Vert _0, \Vert A_{1h}u_h^n\Vert _0\le \Vert A_{1h}u_h^{n-1}\Vert _0\), then the induction assumption means \(\Vert A_{2h}T_h^n\Vert \le c\kappa ^{-1}\Vert f\Vert _0\) and \(\Vert A_{1h}u_h^n\Vert _0\le c\mathrm{{Ra}}\kappa ^{-1}\Vert f\Vert _0\). On the other hand, we assume \(\Vert A_{2h}T_h^{n-1}\Vert _0\le \Vert A_{2h}T_h^n\Vert _0\) and \(\Vert A_{1h}u_h^{n-1}\Vert _0\le \Vert A_{1h}u_h^n\Vert _0\), then from (3.21), we deduce that
Then, the inequality (3.11) holds for \(m=n\) by (3.10). In the same way, we obtain (3.14) and (3.18). \(\square \)
Theorem 3.2
Under the conditions of Theorem 3.1, if \(0<\sigma \le \frac{1}{4}\) holds, the errors between the exact solution (u, p, T) and the numerical solution \((u_{\mu }^m,p_{\mu }^m,T_{\mu }^m)\) of Iterative method 1 satisfy:
If \(0<\sigma \le \frac{1}{3}\) holds, the errors between the exact solution (u, p, T) and the numerical solution \((u_{\mu }^m,p_{\mu }^m,T_{\mu }^m)\) of Iterative method 2 satisfy:
If \(0<\sigma <1\) holds, the error between the exact solution (u, p, T) and the numerical solution \((u_{\mu }^m,p_{\mu }^m,T_{\mu }^m)\) of Iterative method 3 satisfies:
The streamlines of velocity obtained by two-level iterative FEM with \(\mathrm{{Ra}}=10^4\), TLNSC,TLNNC,TLNOC (the first line) and TLOSC,TLONC,TLOOC (the second line)
The streamlines of velocity obtained by TLOSC,TLONC,TLOOC schemes with \(\mathrm{{Ra}}=10^5\)
The temperature contours of TLOSC,TLONC,TLOOC schemes with \(\mathrm{{Ra}}=10^3\) (the first line), \(\mathrm{{Ra}}=10^4\) (the second line) and \(\mathrm{{Ra}}=10^5\) (the third line)
4 Two-level iterative finite element methods with \(0<\sigma \le \frac{1}{4}\)
Based on Theorem 3.1, we know that the iterative methods 1, 2 and 3 are stable and convergent with \(0<\sigma \le \frac{1}{4}\). Two-level iterative FEMs for the natural convection equations (2.4) can be described as follows:
Step 1. Find the solution \((u^m_H,p^m_H,T^m_H)\in X_H \times M_H \times W_H\) defined by the Iterative methods 1, 2 and 3 on the global coarse grid \(\tau _H\) with mesh size H, respectively.
Step 2. Find the solution \((u_{mh},p_{mh},T_{mh})\in X_h \times M_h \times W_h\) defined by the following Simple, Newton and Oseen corrections on fine grid \(\tau _h\) with mesh size h.
Correction 1. (Simple-type correction). Find a fine grid solution \((u_{mh},p_{mh},T_{mh})\in X_h \times M_h \times W_h\) provided by the following Simple-type correction:
Correction 2. (Newton-type correction). Find a fine grid solution \((u_{mh},p_{mh},T_{mh})\in X_h \times M_h \times W_h\) provided by the following Newton-type correction:
Correction 3. (Oseen-type correction). Find a fine grid solution \((u_{mh},p_{mh},T_{mh})\in X_h \times M_h \times W_h\) provided by the following Oseen-type correction:
Denote
Combining (4.1), (4.2) and (4.3) with (2.7), the following error equations for the Corrections 1, 2 and 3 hold: Find \((e_{mh},\eta _{mh},\xi _{mh}) \in X_h \times M_h \times W_h\), for all \((v_h,q_h,\phi _h)\in X_h \times M_h \times W_h\), such that
Error equations based on the Simple-type correction.
Error equations based on the Newton-type correction.
Error equations based on the Oseen-type correction.
Theorem 4.1
Under the conditions of Theorem 3.1 and suppose that \(0< \sigma \le \frac{1}{4}\) holds, the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the iterative methods 1, 2 and 3 with the Correction 1 satisfies:
where \(C_R=\mathrm{{Ra}}\kappa ^{-2}\Vert f\Vert _{-1},{\tilde{C}}=C_6(1-\sigma )^{-1}(\tfrac{\Vert f\Vert _0}{\Vert f\Vert _{-1}})^{\tfrac{1}{2}}\).
Proof
Taking \(\phi _h=T_{mh}\) in the second equation of (4.1), we have
Thanks to (3.10), it yields
Choosing \(v_h=u_{mh},q_h=p_{mh}\) in the first equation of (4.1), we have
By using (3.10) and (4.8), one gets
Next, setting \(v_h=A_{1h}u_{mh},q_h=0\) and \(\phi _h=A_{2h}T_{mh}\) in (4.1), we obtain
which combining (3.10) and (3.11) imply the bounds of \(\Vert A_{1h}u_{mh}\Vert _0\) and \(\Vert A_{2h}T_{mh}\Vert _0\).
On the other hand, taking \(\phi _h=\xi _{mh}\) in the second equation of (4.5) and using (2.3), one finds
then, the following inequality holds by using (2.10) and some simple calculations
Taking \(v_h=e_{mh}, q_h=\eta _{mh}\) in the first equation of (4.5), we obtain
Setting \(q_h=0\) in (4.5) and using the condition (2.6), we obtain
We complete the proof by combining (4.10), (4.11) and (4.12) with the fact that \(1-\sigma \ge \tfrac{3}{4}\).
Theorem 4.2
Under the conditions of Theorem 4.1, the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 1, 2 and 3 with the Correction 2 satisfies:
where \({\tilde{C}}=(C_4+2C_5P_rR_a\kappa ^{-1})(1-\sigma )^{-\tfrac{1}{2}}, C_p=C^2_4N^{-1}(1-\sigma )^{-1}\).
Proof
Taking \(\phi _h=T_{mh}\) in the second equation of (4.2), we have
thanks to (3.10), we have
Setting \(v_h=u_{mh}, q_h=p_{mh}\) in the first equation of (4.2), we get
using (3.10), (4.13), noting \(\sigma =\mathrm{{Pr}}^{-1}\mathrm{{Ra}}\kappa ^{-1}\Vert f\Vert _{-1}(N+\mathrm{{Pr}} {{\overline{N}}}\kappa ^{-1})\) and \(0<\sigma \le \frac{1}{4}\), we obtain
As a consequence, we have
Taking \(v_h=A_{1h}u_{mh},q_h=0\) and \(\phi _h=A_{2h}T_{mh}\) in (4.2), using (2.2), (2.3), it holds
Combining (3.10), (3.11) and (4.13), we obtain the bounds of \(u_{mh}\) and \(T_{mh}\) in \(H^2\)-norm.
It is easy to check that the following identities hold
Taking \(\phi _h=\xi _{mh}\) in the problem (4.6) and employing the Cauchy inequality, we have
Setting \(v_h=e_{mh}, q_h=\eta _{mh}\) in (4.6) and using the Cauchy inequality, we obtain
combining (2.10) and noting the parameter \(0<\sigma <\tfrac{1}{4}\), we arrive at
Furthermore, choosing \(q_h=0\) in (4.6) and using the discrete inf-sup condition (2.6), we have
We finish the proof by combining (4.17), (4.18) with (4.19). \(\square \)
Theorem 4.3
Under the conditions of Theorem 4.1, the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 1, 2 and 3 with the Correction 3 satisfies:
where \({\tilde{C}}=C_6(1-\sigma )^{-1}(\tfrac{\Vert f\Vert _0}{\Vert f\Vert _{-1}})^{\tfrac{1}{2}}\).
Proof
Choosing \(v_h=u_{mh}, q_h=p_{mh}\) and \(\phi _h=T_{mh}\) in (4.3), we have
Setting \(v_h=A_{1h}u_{mh},q_h=0\) and \(\phi _h=A_{2h}T_{mh}\) in (4.3), one gets
Using (3.10) and (4.20), we obtain the bounds of \(u_{mh}\) and \(T_{mh}\) in \(H^2\)-norm.
Moreover, setting \(v_h=e_{mh}, q_h=\eta _{mh}\) and \(\phi _h=\xi _{mh}\) in (4.7), we obtain
Finally, setting \(q_h=0\) in (4.7) and using the discrete inf-sup condition (2.6), we have
We complete the proof by using inequalities (4.20)–(4.24).
Theorem 4.4
Under the conditions of Theorem 4.1, the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 1, 2 and 3 with the Corrections 1 satisfies:
The numerical solutions \((u_{mh},p_{mh},T_{mh})\) provided by the Iterative methods 1, 2 and 3 with the Correction 2 satisfy:
The numerical solutions \((u_{mh},p_{mh},T_{mh})\) provided by the Iterative methods 1, 2 and 3 with the Correction 3 satisfy:
Remark 4.1
For \(0<\sigma \le \frac{1}{4}\), based on Theorems 4.1, 4.2, 4.3 and 4.4, we know that the fine grid solution \((u_{mh},p_{mh},T_{mh})\) based on the Iterative methods \(i\ (i=1,2,3)\) with the Corrections \(j\ (j=1,2,3)\) is stable and convergent. Furthermore, the combination of the Iterative method 2 and the Correction 2 is the better choice than others because this combination has the second-order convergence on the iterative step m.
Remark 4.2
Under the assumptions of Theorems 3.1 and 4.4, we choose m such that
for the Iterative method 1, and
for the Iterative method 2, and
for the Iterative method 3. Namely, the iterative number m should be satisfied:
5 Two-level iterative finite element methods with \(\frac{1}{4}<\sigma \le \frac{1}{3}\)
For the case of \(\frac{1}{4}<\sigma \le \frac{1}{3}\), from Theorem 4.1, we know that the Iterative methods 2 and 3 are stable and convergent. Therefore, we consider the two-level iterative methods in which the solution \((u^m_H,p^m_H,T^m_H)\in X_H \times M_H \times W_H\) of the Iterative methods 2 and 3 are obtained on coarse grid \(\tau _H\), and the correction solutions \((u_{mh},p_{mh},T_{mh})\) of the Simple, Newton and Oseen corrections are given on the fine grid \(\tau _{h}\). The two-level iterative finite element methods can be designed as follows.
Step 1. Find the numerical solution \((u^m_H,p^m_H,T^m_H)\in X_H \times M_H \times W_H\) defined by the Iterative methods 2 and 3 on the coarse grid \(\tau _H\), respectively.
Step 2. Find the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by (4.1), (4.2) and (4.3) on the fine grid \(\tau _h\), respectively.
Theorem 5.1
Under the conditions of Theorem 4.1 and suppose that \(\frac{1}{4}<\sigma \le \frac{1}{3}\) holds, the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 2 and 3 with the Correction 1 satisfies:
where \(C_R=\mathrm{{Ra}}\kappa ^{-2}\Vert f\Vert _{-1},{\tilde{C}}=C_6(1-\sigma )^{-1}(\tfrac{\Vert f\Vert _0}{\Vert f\Vert _{-1}})^{\tfrac{1}{2}}\).
Proof
Combining (4.10), (4.11), (4.12) with the fact \(1+\frac{16}{9}\sigma \le \frac{5}{3},\ 1-\sigma \ge \tfrac{2}{3}\), we finish the proof. \(\square \)
Theorem 5.2
Under the conditions of Theorem 4.1, the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 2 and 3 with the Correction 2 satisfies:
where \({\tilde{C}}=(C_4+2C_5P_rR_a\kappa ^{-1})(1-\sigma )^{-\tfrac{1}{2}}, C_p=C^2_4N^{-1}(1-\sigma )^{-1}\).
Proof
We complete the proof by combining Theorem 3.1, the fact \(\frac{1}{4}<\sigma \le \frac{1}{3}\) with (4.13)–(4.19). \(\square \)
Theorem 5.3
Under the conditions of Theorem 4.1, the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 2 and 3 with the Correction 3 satisfies:
where \({\tilde{C}}=C_6(1-\sigma )^{-1}(\tfrac{\Vert f\Vert _0}{\Vert f\Vert _{-1}})^{\tfrac{1}{2}}\).
Proof
Combining (4.20)–(4.25) and using the fact \(1-\sigma \ge \tfrac{2}{3}\), we finish the proof. \(\square \)
Theorem 5.4
Under the conditions of Theorem 4.1, the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 2 and 3 with the Corrections 1 satisfies:
The numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 2 and 3 with the Correction 2 satisfies:
The numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 2 and 3 with the Correction 3 satisfies:
Remark 5.1
For \(\frac{1}{4}\le \sigma \le \frac{1}{3}\), from Theorems 5.1, 5.2, 5.3 and 5.4, we know that the fine grid solution \((u_{mh},p_{mh},T_{mh})\) based on the Iterative methods \(i\ (i=2,3)\) with the Corrections \(j\ (j=1,2,3)\) is stable and convergent. Furthermore, the combination of the Iterative method 2 and the Correction 2 is the better choice than others because the combination has the second order convergence on the iterative step m.
Remark 5.2
Under the conditions of Theorems 3.1 and 5.4, we set m to satisfy:
for the Iterative method 2, and
for the Iterative method 3. Namely, the iterative step m should be chosen as:
6 Two-level iterative finite element methods with \(\frac{1}{3}<\sigma \le 1-\alpha (\frac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\frac{1}{2}}\)
From Theorem 3.1, we know that only the Iterative method 3 is stable and convergent with \(\frac{1}{3}<\sigma \le 1-\alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}\); thus, we consider the two-level iterative finite element methods in which the iterative solution \((u^m_H,p^m_H,T^m_H)\) obtained by the Iterative method 3 on the coarse grid \(\tau _H\) and the correction solutions \((u_{mh},p_{mh},T_{mh})\) of the Simple-, Newton- and Oseen-type corrections are solved on the fine grid \(\tau _{h}\). The two-level iterative finite element methods can be described as follows.
Step 1. Find the numerical solution \((u^m_H,p^m_H,T^m_H)\in X_H \times M_H \times W_H\) defined by the Iterative methods 3 on the coarse grid with the iterative step m satisfies \(m\ge \frac{\ln \frac{\kappa C_rh}{Ra\Vert f\Vert _{-1}}}{\ln \sigma }\).
Step 2. Find the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the corrections (4.1), (4.2) and (4.3) on the fine grid.
Theorem 6.1
Under the conditions of Theorem 3.1 with \(\frac{1}{3}<\sigma \le 1-\alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}\), the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 3 with the Correction 1 satisfies:
where \(C_R=\mathrm{{Ra}}\kappa ^{-2}\Vert f\Vert _{-1},{\tilde{C}}=C_6(1-\sigma )^{-1}(\tfrac{\Vert f\Vert _0}{\Vert f\Vert _{-1}})^{\tfrac{1}{2}}\).
Proof
Combining (4.8)–(4.12) with \(1+\sigma \le 2\) and \(1-\sigma \ge \alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}\), we finish the proof. \(\square \)
Theorem 6.2
Under the conditions of Theorem 6.1, the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 3 with the Correction 2 satisfies:
where \({\tilde{C}}=(C_4+2C_5P_rR_a\kappa ^{-1})(1-\sigma )^{-\tfrac{1}{2}}, C_p=C^2_4N^{-1}(1-\sigma )^{-1}\).
Proof
Noting the fact that \(1-\sigma \ge \alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}\) and \(1+\sigma \ge 2\), we complete the proof. \(\square \)
Theorem 6.3
Under the conditions of Theorem 6.1, the numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 3 with the Correction 3 satisfies:
where \({\tilde{C}}=C_6(1-\sigma )^{-1}(\tfrac{\Vert f\Vert _0}{\Vert f\Vert _{-1}})^{\tfrac{1}{2}}\).
Proof
Combining (4.20)–(4.25) and using the fact \(1-\sigma \ge \alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}\), we finish the proof. \(\square \)
Theorem 6.4
The numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 3 with the Correction 1 satisfies:
The numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 3 with the Correction 1 satisfies:
The numerical solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative methods 3 with the Correction 1 satisfies:
Remark 6.1
For \(\frac{1}{3}<\sigma \le 1-\alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}\), from Theorems 6.1, 6.2, 6.3 and 6.4, we know that the fine grid solution \((u_{mh},p_{mh},T_{mh})\) obtained by the Iterative method 3 with the Corrections \(j\ (j=1,2,3)\) is stable and convergent. Furthermore, we can see that the combination of the Iterative method 3 and Correction 2 is a better choice for natural convection equations since this combination is of the second-order convergence on the iterative step m.
7 One-level iterative finite element method with \(1-\alpha (\frac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\frac{1}{2}}< \sigma < 1\)
From Theorem 3.1, we know that iterative finite element method 3 is the unique choice under the weak unique condition \(1-\alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}< 1\). In this section, we consider the stability and convergence of one-level iterative finite element solution \((u_{mh},p_{mh},T_{mh})\) on fine grid \(\tau _h\).
Step 1. Find the iterative solution \((u_{mh},p_{mh},T_{mh})\) defined by Iterative method 3 on the fine grid \(\tau _h\).
Theorem 7.1
Under the conditions of Theorem 3.1 and \(1-\alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}<\sigma < 1\), the iterative solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative method 3 satisfies:
Proof
We complete the proof by combining Theorem 3.1 and the fact \(1-\alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}<\sigma < 1\). \(\square \)
Theorem 7.2
Under the conditions of Theorem 7.1, the iterative solution \((u_{mh},p_{mh},T_{mh})\) defined by the Iterative method 3 on the fine mesh \(\tau _h\) satisfies:
Proof
Thanks to the triangular inequality, Theorems 3.1 and 7.1, we finish the proof. \(\square \)
8 Numerical examples
In this section, we present some numerical tests to identify with the established theoretical findings and show the performances of the two-level iterative FEMs for the natural convection equations (1.1). The mesh consists of the triangular elements that obtained by dividing \(\Omega \) into subsquares of equal size and then drawing the diagonal in each sub-square. The mixed stable finite element pair \(P_1b\)-\(P_1\)-\(P_1b\) is used to approximate the velocity-pressure and temperature; the iterative tolerance \(10^{-5}\) is adopted in all numerical implementations.
8.1 An analytical solution: convergence validation
In this test, the natural convection equations are defined on a convex domain \(\Omega =[0,1]^2\) and the parameter \(\mathrm{{Pr}}=\mathrm{{Ra}}=\kappa =1\) is chosen. The boundary and initial conditions and body force f are selected by the following exact solutions
where \(u_1\) and \(u_2\) are the components of u. In order to show the performances of the two-level iterative FEMs (4.1), (4.2) and (4.3), we also provide the numerical results of Galerkin FEM (2.7).
Firstly, we present the errors and the CPU time of numerical scheme (2.7) in Table . It is obvious that the errors of numerical solution \((u_h,p_h,T_h)\) become smaller and smaller as the mesh is refined, the convergence order of the velocity, temperature in \(H^1\) norm is 1, which confirm Theorem 2.2 well. The theoretical convergence order of pressure in \(L^2\)-norm is 1, numerical results show some superconvergences, the reason may lie in the smoothness of exact solutions.
Secondly, we present the errors of two-level iterative finite element methods in Tables , and . In order to simplify the notations, we use TLSSC, TLSOC and TLSNC to denote two-level Simple iteration with Simple, Oseen and Newton corrections, respectively. Similarly, TLNSC, TLNNC, TLNOC, TLOSC, TLONC and TLOOC are the two-level Newton iteration with Simple, Newton and Oseen corrections, two-level Oseen iteration with Simple, Newton and Oseen corrections, respectively. Compared with Table 1, we can see that the accuracy of two-level iterative FEMs with different corrections is comparable with that of Galerkin FEM (2.7) with the mesh size h takes \(\frac{1}{9}, \frac{1}{16},...,\frac{1}{100}\) and \(H=h^{\frac{1}{2}}\).
Finally, we compare the CPU time among four numerical schemes. From Tables 1, 2, 3 and 4, we know that the two-level methods with different iterations not only keep good accuracy but also speed less time than standard Galerkin FEM. Furthermore, we compare the results obtained from these two-level iterative FEMs with different corrections. From Tables 2, 3 and 4, we can see that two-level Newton iterative scheme has the best accuracy, while two-level Simple iterative scheme spends the least computational time.
8.2 Thermal-driven cavity problem
The problem of thermal-driven cavity is used as a suitable benchmark for testing the natural convection problem. The simplicity of geometry and clear boundary conditions makes this test attractive. The domain consists of a square cavity with differentially heated vertical walls where right and left walls are kept at \(T_r\) and \(T_l\), respectively, with \(T_l>T_r\). The remaining walls are insulated, and there is no heat transfer through them. The boundary conditions are no-slip boundary conditions for the velocity at four walls \((u=0)\) and Dirichlet boundary conditions for the temperature at vertical walls. As the horizontal walls are adiabatic, we employ \(\frac{\partial T}{\partial n}=0\). Figure shows the physical domain of the thermal-driven cavity flow problem. In this test, we take \(\kappa =1,f=0,T_r=0\) and \(T_l=1\). While we consider the air as the cavity filling fluid in our model, we take the fixed value \(Pr=0.71\). We do the computations with Rayleigh number varying from \(10^3\) to \(10^5\). The performances of two-level iterative FEMs are compared with the famous benchmark solutions of de Vahl Davis (1983), Cibik and Kaya (2011), Manzari (1999), Massarotti et al. (1998) and Wan et al. (2001). We also present the results of standard Galekrkin FEM (2.7) where we keep the same mesh size \(h=\frac{1}{64}\) for the two-level iterative methods.
We firstly present the peak values of the vertical velocity at \(y=0.5\) and the horizontal velocity at \(x=0.5\) obtained by two-level iterative FEMs. Table summarizes the maximum vertical velocity values at the mid-height and mid-width with different Rayleigh numbers. For comparison, the velocity values obtained by Manzari (1999), Massarotti et al. (1998), de Vahl Davis (1983) and Wan et al. (2001) are presented. From Table 5, we can see that the results of two-level iterative methods are in excellent agreement with the benchmark data even at coarser grid \(H=\frac{1}{8}\) as the Rayleigh number increases. Furthermore, we also compare the CPU time of two-level iterative methods with the standard Galerkin FEM in Table 5. It is obvious that the two-level methods take less time than the standard Galerkin finite element method.
Finally, we present the streamlines and isotherms of the natural convection problem obtained by the two-level iterative methods with different Rayleigh number in Figs. , , and . It is clear from these figures that three iterative schemes on the coarse mesh with different corrections on fine mesh run well with \(Ra=10^3\). However, as the Rayleigh number increases, the Simple and Newton iterative methods fail successively since the restriction of strong uniqueness condition \(0<\sigma \le 1-\alpha (\tfrac{\Vert f\Vert _{-1}}{\Vert f\Vert _0})^{\tfrac{1}{2}}\). Moreover, from Figs. 2, 3 and 4, we see that the circular vortex of streamlines at the cavity center begins to deform into an ellipse and then break up into two vortices tending to approach to the corners differentially as Rayleigh number increases. For the temperature isolines, the parallel behavior is distorted as these lines seem to have a flat behavior in the central part of the region as the Rayleigh number increases. Near the sides of the cavity, isolines tend to be vertical only. The temperature slopes with \(Ra=10^5\) at the corners of the differentially heated sides are more immersed than the case of the lower Rayleigh number. These graphics are comparable with the results provided by Cibik and Kaya (2011); Manzari (1999); Massarotti et al. (1998); de Vahl Davis (1983); Wan et al. (2001), but less CPU time is required, it shows that the two-level iterative methods have good computational performances.
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Communicated by Frederic Valentin.
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This work was supported by National Natural Science Foundation of China (nos. 11971152, 12271468), Shandong Province Natural Science Foundation (nos. ZR2021ZD03, ZR2021MA010 ) and the Natural Science Foundation of Henan Province (no. 202300410167)
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Zhang, H., Chen, C. & Zhang, T. Two-level iterative finite element methods for the stationary natural convection equations with different viscosities based on three corrections. Comp. Appl. Math. 42, 11 (2023). https://doi.org/10.1007/s40314-022-02147-z
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DOI: https://doi.org/10.1007/s40314-022-02147-z