A new finite element variational multiscale (VMS) method based on two local Gauss integrations is proposed and analyzed for the stationary conduction-convection problems. The valuable feature of our method is that the action of stabilization operators can be performed locally at the element level with minimal additional cost. The theory analysis shows that our method is stable and has a good precision. Finally, the numerical test agrees completely with the theoretical expectations and the “ exact solution,” which show that our method is highly efficient for the stationary conduction-convection problems.
1. Introduction
The conduction-convection problems constitute an important system of equations in atmospheric dynamics and dissipative nonlinear system of equations. Many authors have worked on these problems [1–8]. The governing equations couple viscous incompressible flow and heat transfer process [9], where the incompressible fluid is the Boussinesq approximation to the nonstationary Navier-Stokes equations. Christon et al. [10] summarized some relevant results for the fluid dynamics of thermally driven cavity. A multigrid (MG) technique was applied for the conduction-convection problems [11, 12]. Luo et al. [13] combined proper orthogonal decomposition (POD) with the Petrov-Galerkin least squares mixed finite element (PLSMFE) method for the problems. In [14], a Newton iterative mixed finite element method for the stationary conduction-convection problems was shown by Si et al. In [15], Si and He gave a defect-correction mixed finite element method for the stationary conduction-convection problems. In [3], an analysis of conduction natural convection conjugate heat transfer in the gap between concentric cylinders under solar irradiation was carried out. In [16], Boland and Layton gave an error analysis for finite element methods for steady natural convection problems. Variational multiscale (VMS) method which defines the large scales in a different way, namely, by a projection into appropriate subspaces, see Guermond [17], Hughes et al. [18–20] and Layton [21], and other literatures on VMS methods [22–24]. The new finite element VMS strategy requires edge-based data structure and a subdivision of grids into patches. It does not require a specification of mesh-dependent parameters and edge-based data structure, and it is completely local at the element level. Consequently, the new VMS method under consideration can be integrated in existing codes with very little additional coding effort.
For the conduction-convection problems, we establish such system that Ω be a bounded domain in Rd(d=2or3), with Lipschitz-continuous boundary ∂Ω. In this paper, we consider the stationary conduction-convection problem as follows:
(1.1)-2ν∇·D(u)+(u·∇u)+∇p=λjT,x∈Ω,∇·u=0,x∈Ω,-ΔT+λu·∇T=0,x∈Ω,u=0,T=T0,x∈∂Ω,
where D(u)=(∇u+∇uT)/2 is the velocity deformation tensor, (u,p,T)∈X×M×W,Ω⊂Rd is a bounded convex domain. u=(u1(x),u2(x))T represents the velocity vector, p(x) the pressure, T(x) the temperature, λ>0 the Grashoff number, j=(0,1)T the two-dimensional vector and ν>0 the viscosity.
The study is organized as follows. In the next section, the finite element VMS method is given. In Section 3, we give the stability. The error analysis is given in Section 4. In Section 5, we show some numerical test. The last but not least is the conclusion given in Section 6.
2. Finite Element VMS Method
Here, we introduce some notations
(2.1)X=H01(Ω)d,M=L02(Ω)={φ∈L2(Ω);∫Ωφdx=0},W=H1(Ω).
For h>0, finite-dimension subspace (Xh,Mh,Wh)⊂(X,M,W) is introduced which is associated with Ωe, a triangulation of Ω into triangles or quadrilaterals, assumed to be regular in the usual sense. In this study, the finite-element subspaces of personal preference are defined by setting the continuous piecewise (bi)linear velocity and pressure subspace, let τh be the regular triangulations or quadrilaterals of the domain Ω and define the mesh parameter h=maxΩe∈τh{diam(Ωe)},(2.2)Xh={v∈X:v|Ωe∈Rl(Ωe)d∀Ωe∈τh},Mh={q∈M:q|Ωe∈Rl(Ωe)∀Ωe∈τh},Wh={ϕ∈M:ϕ|Ωe∈Rl(Ωe)∀Ωe∈τh},
where W0h=Wh∩H01, l≥1 is integers. Rl(Ωe)=Pl(Ωe) if Ωe is triangular and Rl(Ωe)=Ql(Ωe) if Ωe is quadrilateral. Here (Xh,Mh) does not satisfy the discrete Ladyzhenskaya-Babuška-Brezzi (LBB) condition
(2.3)supvh∈Xhd(vh,ph)∥∇vh∥0≥β∥ph∥0,∀ph∈Mh.
Now, in order to stabilize the convective term appropriately for the higher Reynolds number and avoid the extra storage, we supply finite element VMS method that the local stabilization form of the difference between a consistent and an underintegrated mass matrices based on two local Gauss integrations at element level as the stabilize term
(2.4)G(ph,qh)=ϵd(ak(ph,qh)-a1(ph,qh)).
Here,
(2.5)ak(ph,qh)=pGTMkqG,a1(ph,qh)=pGTM1qG,pGT=[p1,p2…,pN]T,qG=[q1,q2,…,qN],Mij=(ϕi,ϕj),ph=∑i=1Npiϕi,pi=ph(xi),∀ph∈Mh,i=1,2,…,N,Mk=(Mijk)N×N,M1=(Mij1)N×N,
the stabilization parameter ϵd(ϵd=o(h)) in this scheme acts only on the small scales, ϕi is the basis function of the velocity on the domain Ω such that its value is one at node xi and zero at other nodes, and N is the dimension of Mh. The symmetric and positive matrices Mijk, k≥2 and Mij1 are the stiffness matrices computed by using k-order and 1-order Gauss integrations at element level, respectively. pi and qi, i=1,2,…,N are the values of ph and qh at the node xi. In detail, the stabilized term can be rewritten as
(2.6)G(ph,qh)=ϵd∑Ωe∈τh{∫Ωe,kphqhdx-∫Ωe,1phqhdx},∀ph,qh∈Mh,G(p,q)=(p-∐hp,q-∐hq).L2-projection operator ∐h:L2(Ω)→R0 with the following properties [25]:
(2.7)(p,qh)=(∐hp,qh),∀p∈M,qh∈R0;∥∐hp∥0≤c∥p∥0,∀p∈M;∥p-∐hp∥0≤ch∥p∥1,∀p∈H1(Ω)∩M.
Lemma 2.1 (see [<xref ref-type="bibr" rid="B26">26</xref>]).
Let (Xh,Mh) be defined as above, then there exists a positive constant β independent of h, such that
(2.8)|B((u,p);(v,q))|≤c(∥u∥1+∥p∥0)(∥v∥1+∥q∥0)(u,p),(v,q)∈(X,M),β(∥uh∥1+∥vph∥0)≤sup(vh,qh)∈(Xh,Mh)|B((uh,ph);(vh,qh))|∥v∥1+∥q∥0,∀(uh,ph)∈(Xh,Mh),|G(p,q)|≤C∥(I-IIh)p∥0∥(I-IIh)q∥0,∀p,q∈M.
Using the above notations, the VMS variational formulation of problems (1.1) reads as follows.
Find (A1)(uh,ph,Th)∈Xh×Mh×Wh such that
(2.9)a(uh,vh)-d(ph,vh)+d(qh,uh)+b(uh,uh,vh)+G(ph,qh)=λ(jTh,vh),∀vh∈Xh,φh∈Mh;a-(Th,ψh)+λb-(uh,Th,ψh)=0,∀ψh∈W0h.
Given (A2)(uhn-1,Thn-1), find (uhn,phn,Thn)∈Xh×Mh×Wh such that
(2.10)a(uhn,vh)-d(phn,vh)+d(qh,uhn)+b(uhn,uhn-1,vh)+b(uhn-1,uhn,vh)+G(phn,qh)=b(uhn-1,uhn-1,vh)+λ(jThn,vh),∀vh∈Xh,φh∈Mh;a-(Thn,ψh)+λb-(uhn-1,Thn,ψh)=0,∀ψh∈W0h,
where a(u,v)=ν(∇u,∇v),a-(T,ψ)=(∇T,∇ψ)d(q,v)=(q,divv), and
(2.11)b(u,v,w)=((u·∇v),w)+12((divu)v,w)=12((u·∇)v,w)-12((u·∇)w,v),b-(u,T,ψ)=((u·∇T),w)+12((divu)T,ψ)=12((u·∇)T,ψ)-12((u·∇)ψ,T).
There exists a constant C which only depends on Ω, such that
Assuming ∂Ω∈Ck,α(k≥0,α>0), then, for T0∈Ck,α(∂Ω), there exists an extension T0 in C0k,α(Rd), such that
(2.12)∥T0∥k,q≤ε,k≥0,1≤q≤∞,
where ε is an arbitrary positive constant.
b(·,·,·)andb-(·,·,·) have the following properties.
For all u∈X,v,w∈X,T·φ∈H01(Ω), there holds that
(2.13)b(u,v,w)=-b(u,w,v),b-(u,T,ψ)=-b-(u,ψ,T).
For all u∈X,v∈H1(Ω)d,T∈H1(Ω),forallw∈X(orφ∈H01(Ω)), there holds that
(2.14)|b(u,v,w)|≤N∥∇u∥0∥∇v∥0∥∇w∥0,|b-(u,T,φ)|≤N-∥∇u∥0∥∇T∥0∥∇φ∥0,
where N=supu,v,w|b(u,v,w)|/∥∇u∥0∥∇v∥0∥∇w∥0,N-=supu,v,w|b-(u,T,φ)|/∥∇u∥0∥∇T∥0∥∇φ∥0.3. Stability AnalysisLemma 3.1.
The trilinear form b satisfies the following estimate:
(3.1)|b(uh,vh,w)|+|b(vh,uh,w)|+|b(w,uh,vh)|≤C|logh|1/2∥∇uh∥0∥∇vh∥0∥w∥0.
Theorem 3.2.
Suppose that (B1)-(B3) are valid and ε is a positive constant number, such that
(3.2)64C2Nλε3ν2<1,16C2λ2N-ε3ν<1,∥∇T0∥0≤ε4,∥T0∥0≤Cε4.
Then (uhm,Thm) defined by (A2) satisfies
(3.3)∥∇uhm∥0≤8C2λε3ν,∥∇Thm∥0≤ε.
Proof.
We prove this theorem by the inductive method. For m=1, (3.3) holds obviously. Assuming that (3.3) holds for m=n-1, we want to prove that it holds for m=n. We estimate ∥Δuhn∥ firstly. Letting vh=uhn,qh=0 in the first equation of (2.10) and using (2.13), we get
(3.4)a(uhn,uhn)+b(uhn,uhn-1,uhn)=b(uhn-1,uhn-1,uhn)+λ(jThn,uhn).
Setting Thn-1=khn-1+T0 and using (2.14), we have
(3.5)ν∥∇uhn∥0≤N∥∇uhn∥0∥∇uhn-1∥0+N∥∇uhn-1∥02+C2λ∥∇khn-1∥0+Cλ∥∇T0∥0.
Letting Thn=khn+T0,ψ=khn in the second equation of (2.10), we can obtain
(3.6)a-(khn,khn)=-λb-(uhn,T0,khn)-a-(T0,khn).
Using (2.12), (2.14), and (3.2), we get
(3.7)∥∇khn-1∥0≤λN-∥∇uhn-1∥0∥∇T0∥0+∥∇T0∥0≤λN-ε4∥∇uhn-1∥0+∥∇T0∥0≤3ε8≤3ε4,(ν-N∥∇uhn-1∥0)∥∇uhn∥0≤N∥∇uhn-1∥02+C2λε≤C2λε+64C4N9ν2λ2ε2≤4C2λε3.
Using (3.2), we have ν-N∥∇uhn-1∥0≥7ν/8. Then,
(3.8)∥∇uhn∥0≤8C2λε3ν.
Combining (2.12), (2.14), (3.2), and (3.6), we arrive at
(3.9)∥∇khn∥0≤λN-∥∇uhn∥0∥∇T0∥0+∥∇T0∥0≤3ε4,(3.10)∥∇Thn∥0≤∥∇khn∥0+∥∇T0∥0≤ε.
Therefore, we finish the proof.
4. Error Analysis
In this section, we establish the H1-bound of the error uhn-u,Thn-T and L2-bounds of the error phn-p. Setting (en,μn,ηn)=(uhn-uh,phn-ph,Thn-Th). Firstly, we give some Lemmas.
Lemma 4.1.
In [4], If B1-B3 hold, (u,p, T)∈Hm+1(Ω)×Hm(Ω)×Hm+1(Ω) and (uh,ph,Th)∈Xh×Mh×Wh are the solution of problem (A1) and (A2), respectively, then there holds that
(4.1)∥∇(u-uh)∥0+∥p-ph∥0+∥∇(T-Th)∥0≤Chm(∥u∥m+1+∥p∥m+∥T∥m+1).
Lemma 4.2.
Under the assumptions of Theorem 3.2, (A2) has a unique solution (uh,ph,Th)∈Xh×Mh×Wh, such that T|∂Ω=T0 and
(4.2)∥∇uh∥0≤8C2λε3ν,∥∇Th∥0≤ε.
The detail proof we can see [4, 13, 14].
Theorem 4.3.
Under the assumption of Theorem 3.2, there holds
(4.3)∥∇en∥0≤C2λε2n-33ν,∥∇ηn∥0≤ε2n+1,∥μn∥0≤β-1{νε2+4C2λε3,n=1(ν+2Nε)C2λε2n-33ν+N(C2λε2n-43ν)2+C2λε2n,n≥2.
Proof.
Subtracting (2.10) from (2.9), we get the following error equations, namely (en,μn,ηn) satisfies
(4.4)a(en,vh)-d(μn,vh)+d(qh,en)+b(en,uhn-1,vh)+b(uhn-1,en,vh)+G(μn,qh)=b(en-1,en-1,vh)+λ(jηn,vh),(4.5)a-(ηn,ψh)+λb-(en,Thn,ψh)+λb-(uhn-1,ηn,qh)=0.
Here, let ψh=ηn, in (4.5), then we have
(4.6)a-(ηn,ηn)+λb-(en,Thn,ηn)=0.
By using (2.14), we get
(4.7)∥∇ηn∥0≤λN-ε∥∇en∥0.
In (4.4), we take vh=en∈Xh,qh=μn, then
(4.8)a(en,en)+b(en,uhn-1,en)+b(uhn-1,en,en)+G(μn,μn)=b(en-1,en-1,en)+λ(jηn,en).
Using (2.13) and (2.14), we have
(4.9)ν∥∇en∥0+G(μn,μn)≤N∥∇en∥0∥∇uhn-1∥0+N∥∇en-1∥02+C2λ∥∇ηn∥0,
then, we obtain
(4.10)(ν-N∥∇uhn-1∥0)∥∇en∥0≤N∥∇en-1∥02+C2λ∥∇ηn∥0.
By using ν-N∥∇uhn-1∥0≥7ν/8. Equations (3.3) and (4.2), we get
(4.11)78ν∥∇en∥0≤N∥∇en-1∥02+C2λ∥∇ηn∥0≤(N∥∇uhn-1∥0+N∥∇uh∥0+C2λ2N-ε)∥∇en-1∥0≤(16NC2λε3ν+C2λ2N-ε)∥∇en-1∥0=7ν16∥∇en-1∥0,∥∇en∥0≤12∥∇en-1∥0.
From the inductive method, we know, for n=1, subtracting (2.10) from (2.9), we can get
(4.12)a(e1,vh)-d(μ1,vh)+d(qh,e1)+b(uh,uh,vh)+G(μ1,qh)=λ(jTn,vh).
Letting vh=e1,qh=μ1 in (4.12) and using (2.14), we have
(4.13)∥∇e1∥0+G(μ1,μ1)≤ν-1N∥∇uh∥02+ν-1C2λ∥∇Th∥0≤64C4λ2Nε29ν3+C2λεν≤4C2λε3ν,
then
(4.14)∥∇e1∥0≤4C2λε3ν.
By (4.7), we have
(4.15)∥∇η1∥0≤λN-ε∥∇en∥0≤λN-4C2λε23ν≤ε4.
Letting qh=0 in (4.12), (2.14), and (3.9), using Lemma 2.1, we get
(4.16)β∥μ1∥0≤ν∥∇e1∥0+N∥∇uh∥02+Cλ∥Th∥0≤νε2+4C2λε3ν.
Assuming that (4.3) is true for n=k-1, using (4.7) and (4.11), we know that both of them are valid for n=k. Using (4.7) holds for n=k, we let qh=0 in (4.4) and using Lemma 2.1, (4.5), and (3.3), we have
(4.17)β∥μn∥0≤(ν+2Nε)∥∇en∥0+N∥∇en-1∥02+C2λ∥∇ηn-1∥0≤(ν+2Nε)C2λε2n-33ν+N(C2λε2n-43ν)2+C2λε2n.
Theorem 4.4.
Under the assumptions of Theorem 4.3, then there holds that
(4.18)limn→∞(∥uhn-uhn-1∥0+∥∇(uhn-uhn-1)∥0)=0,∥∇en∥0+∥μn∥0+∥∇ηn∥0≤F|logh|1/2∥∇(uhn-uhn-1)∥0∥uhn-uhn-1∥0+Hε2n+1,
where F and H are two positive constants.
Proof.
By using (B1) and triangle inequality, we have
(4.19)∥uhn-uhn-1∥0+∥∇(uhn-uhn-1)∥0≤(C+1)(∥∇en∥0+∥∇en-1∥0).
Using Theorem 4.3, letting n→∞, we obtain (4.18). Taking vh=en,qh=μn in (4.4) and using (2.14), we get
(4.20)a(en,en)+b(en,uhn-1,en)+G(μn,μn)=-b(uhn-uhn-1,uhn-uhn-1,en)+λ(jηn,en).
By (2.14) and Lemma 3.1, we deduce
(4.21)(ν-N∥uhn-1∥0)∥∇en∥0+G(μn,μn)≤F|logh|1/2∥∇(uhn-uhn-1)∥0∥uhn-uhn-1∥0+F2λ∥∇ηn∥0.
Combining (3.3) and (4.7), we obtain
(4.22)(ν-8Nε3ν)∥∇en∥0≤F|logh|1/2∥∇(uhn-uhn-1)∥0∥uhn-uhn-1∥0+F2λ2N-ε∥∇en-1∥0.
Using (3.2), we get
(4.23)∥∇en∥0≤F|logh|1/2∥∇(uhn-uhn-1)∥0∥uhn-uhn-1∥0+Hε2n+1.
Combining (3.2), (4.7), and (4.17), we get
(4.24)∥∇ηn∥0≤F|logh|1/2∥∇(uhn-uhn-1)∥0∥uhn-uhn-1∥0+Hε2n+1,∥∇μn∥0≤F|logh|1/2∥∇(uhn-uhn-1)∥0∥uhn-uhn-1∥0+Hε2n+1.
Here, we complete the proof.
Theorem 4.5.
Under the assumptions of Theorem 4.3, the following inequality:
(4.25)∥∇(u-uhn)∥0+∥p-phn∥0+∥∇(T-Thn)∥0≤F1hm(∥u∥m+1+∥p∥m+∥T∥m+1)+F|logh|1/2∥∇(uhn-uhn-1)∥0∥uhn-uhn-1∥0+Hε2n+1,
holds, where F1 and H are the positive constants.
Proof.
By Lemma 4.1, Theorem 4.4, and the triangle inequality, this theorem is obviously true.
5. Numerical Test
This section presents the numerical results that complement the theoretical analysis.
5.1. Convergence Analysis
In our experiment, Ω=[0,1]×[0,1] is the unit square in R2. Let T0=0 on left and lower boundary of the cavity, ∂T/∂n=0 on upper boundary of the cavity, and T0=4y(1-y) on right boundary of the cavity (see Figure 1). Physics model of the cavity flows: t=0, that is, n=0 initial values on boundary. In general, we cannot know the exact solution of the stationary conduction-convection equations. In order to get the exact solution, we design the procedure as follows. Firstly, solving the stationary conduction-convection equations by using the P2-P1-P2 finite element pair, which holds stability, on the finer mesh, we take the solution as the exact solution. Secondly, the absolute error is obtained by comparing the exact solution and the finite element solutions with VMS methods. Finally, we can easily obtain errors and convergence rates.
From (a) to (c): physics model of the cavity flows, vertical midlines for Re = 2000, h = 1/100, horizontal midlines for Re = 2000, h = 1/100.
5.2. Driven Cavity
In this experiment, Ω=[0,1]×[0,1] is the unit square in R2. Let T0=0 on left and lower boundary of the cavity, ∂T/∂n=0 on upper boundary of the cavity, and T0=4y(1-y) on right boundary of the cavity (see Figure 1). Physics model of the cavity flows: t=0, that is, n=0 initial values on boundary. Solving the stationary conduction-convection equations by using the P2-P1-P2 finite element pair, which holds stability results, on the finer mesh, we take the solution as the exact solution. From Figures 1 and 2, we know that the solution of finite element VMS using P1-P1-P1 element agree completely with the “exact solution.” In Figure 3, we choose Re=2000, divide the cavity into M×N=100×100, from left to right shows the numerical streamline, the numerical isobar, and the numerical isotherms. In Figure 4, we choose Re=3000, divide the cavity into M×N=100×100, from left to right shows the numerical streamline, the numerical isobar, and the numerical isotherms.
From (a) to (b): vertical midlines for Re = 3000, h = 1/100, horizontal midlines for Re = 3000, h = 1/100.
For Re = 2000, h = 1/100, from (a) to (c): velocity streamlines, the pressure level lines, numerical isotherms.
For Re = 3000, h = 1/100, from (a) to (c): velocity streamlines, the pressure level lines, numerical isotherms.
Remark 5.1.
Our VMS finite element method based on two local Gauss integrations and εd=0.1h is suitable for the Sobolev space. Throughout the paper, our analysis and numerical tests are all carried out for the P1-P1-P1 element (see Tables 1 and 2).
VMS: P1-P1-P1 element.
1/h
∥u-uh∥0
∥u-uh∥1
∥T-Th∥0
∥T-Th∥1
∥p-ph∥0
10
0.000194122
0.00493006
0.00740049
0.277241
0.00506075
20
4.91998e-005
0.00252269
0.00208824
0.153561
0.00308312
40
1.21288e-005
0.00126459
0.0005746
0.0838877
0.00180539
60
5.35135e-006
0.000842093
0.000266991
0.0583954
0.00131444
80
2.98429e-006
0.000630808
0.000154418
0.0457979
0.00105007
VMS: P1-P1-P1 element.
1/h
uL2 rate
uH1 rate
TL2 rate
TH1 rate
pL2 rate
10
/
/
/
/
/
20
1.9802
0.9666
1.8253
0.8523
0.7150
40
2.0202
0.9963
1.8617
0.8723
0.7721
60
2.0180
1.0028
1.8903
0.8934
0.7827
80
2.0300
1.0042
1.9033
0.8447
0.7806
6. Conclusion
In this paper, we studied a finite element VMS algorithm based on two local Gauss integrations to solve the stationary conduction-convection problem. From Figures 1 and 2, we see that the solution of VMS using P1-P1-P1 and εd=0.1h agrees completely with the “exact solution,” which shows that our method is highly efficient for the stationary conduction-convection problems. Numerical tests tell us that VMS finite element method based on two local Gauss integrations is very effective.
Acknowledgments
The project is supported by NSF of China (10971164) and the Research Foundation of Xianyang Normal University (06xsyk265).
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