# A Defect-Correction Mixed Finite Element Method for Stationary Conduction-Convection Problems

Authors: Zhiyong Si; Yinnian He
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

A defect-correction mixed finite element method (MFEM) for
solving the stationary conduction-convection problems in two-dimension is
given. In this method, we solve the nonlinear equations with an added artificial viscosity term on a grid and correct this solution on the same grid using
a linearized defect-correction technique. The stability is given and the error
analysis inL2 and H1-norm of u, T and the L2-norm of p are derived. The
theory analysis shows that our method is stable and has a good precision.
Some numerical results are also given, which show that the defect-correction
MFEM is highly efficient for the stationary conduction-convection problems.


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## Body

## 1. Introduction

In this paper, we consider the stationary conduction-convection problems in two dimension whose coupled equations governing viscous incompressible flow and heat transfer for the incompressible fluid are Boussinesq approximations to the stationary Navier-Stokes equations.(𝒫) Find (u,p,T)∈X×M×W such that(1.1)-νΔu+(u⋅∇)u+∇p=λjT,x∈Ω,div⁡u=0,x∈Ω,-ΔT+λu⋅∇T=0,x∈Ω,u=0,T=T0,x∈∂Ω,
where Ω is a bounded domain in ℝ2 assumed to have a Lipschitz continuous boundary ∂Ω. 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.As we know the conduction-convection problem contains the velocity vector field, the pressure field and the temperature field, so finding the numerical solution of conduction-convection problems is very difficult. The conduction-convection problems is an important system of equations in atmospheric dynamics and dissipative nonlinear system of equations, so lots of works are devoted to this problem [1–6]. There are also some works devoted to the nonstationary conduction-convection problems [7–10]. In [8], Luo et al. gave an optimizing reduced PLSMFE for the nonstationary conduction-convection problems. They combined PLSMEF method with POD to deal with the problems. In [11], an analysis of conduction natural convection conjugate heat transfer in the gap between concentric cylinders under solar irradiation was studied. In [12], a Newton iterative mixed finite element method for the stationary conduction-convection problems was shown by Si et al. In [13], Si and He gave a coupled Newton iterative mixed finite element method for the stationary conduction-convection problems.The defect-correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Due to its good efficiency, there are many works devoted to this method, for example, [14–28]. In [18], a method making it possible to apply the idea of iterated defect correction to finite element methods was given. A method for solving the time-dependent Navier-Stokes equations, aiming at higher Reynolds' number, was presented in [23]. In [27], an accurate approximations for self-adjoint elliptic eigenvalues was presented. In [28], Stetter exposed the common structural principle of all these techniques and exhibit the principal modes of its implementation in a discretization context.In this paper we present a defect-correction MFEM for the stationary conduction convection problems. In this method, we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct this solution on the same grid using a linearized defect-correction technique. Actually, the defect-correction MFEM incorporates the artificial viscosity term as a stabilizing factor, making both the nonlinear system easier to resolve and the linearized system easier to precondition. The stability and error analysis of the coupled the defect-correction MFEM show that this method is stable and has a good precision. Some numerical experiments show that our analysis is proper and our method is effective. And it can be used for solving the convection-conduction problems with much small viscosity.This paper is organized as follows. In Section2, the functional settings and some assumptions are given. Section 3 is devoted to the defect-correction MFEM. Section 4 gives the stability analysis. Section 5 presents the error analysis. In Section 6, some numerical results and the numerical analysis to validate the effectiveness of the method are laid out.


## 2. Functional Setting for the Conduction Convection Problems

In this section, we aim to describe some of the notations and results which will be frequently used in this paper. The Sobolev spaces used in this context are standard [29]. For the mathematical setting of the conduction-convection problems and MFEM of conduction-convection problems (1.1), we introduce the Hilbert spaces (2.1)X=H01(Ω)2,W=H1(Ω),M=L02(Ω)≐{φ∈L2(Ω);∫Ωφdx=0}.ℑh is the uniformly regular family of triangulation of Ω¯, indexed by a parameter h=maxK∈ℑh{hK;hK=diam(K)}. We introduce the finite element subspace Xh⊂X, Mh⊂M, Wh⊂W as follows (2.2)Xh={vh∈X∩C0(Ω¯)2;vh|K∈Pl(K)2,∀K∈Ih},Mh={qh∈M∩C0(Ω¯);qh|K∈Pk(K),∀K∈Ih},Wh={ϕh∈W∩C0(Ω¯);ϕh|K∈Pl(K),∀K∈Ih},
where Pℓ(K) is the space of piecewise polynomials of degree ℓ on K, and ℓ⩾1, k⩾1, l⩾1 are three integers. W0h=Wh∩H01(Ω), and (Xh,Mh) satisfies the discrete LBB condition(2.3)sup⁡vh∈Xhd(φh,vh)‖∇vh‖0⩾β‖φh‖0,∀φh∈Mh,
where d(φ,v)=(φ,divv).With the above notations, the Galerkin mixed variation and the mixed FEM problem for the conduction-convection problems(𝒫) are defined, respectively, as follows.(𝒫′) Find (u,p,T)∈X×M×W such that(2.4)νa(u,v)-d(p,v)+d(φ,u)+b(u,u,v)=λ(jT,v),∀v∈X,φ∈M,a¯(T,ψ)+λb¯(u,T,ψ)=0,∀ψ∈W0.(𝒫′′) Find (uh,ph,Th)∈Xh×Mh×Wh such that(2.5)νa(uh,vh)-d(ph,vh)+d(φh,uh)+b(uh,uh,vh)=λ(jTh,vh),∀vh∈Xh,φh∈Mh,a¯(Th,ψh)+λb¯(uh,Th,ψh)=0,∀ψh∈W0h,
where a(u,v)=(∇u,∇v), d(φ,v)=(φ,divv), a¯(T,ψ)=(∇T,∇ψ), and (2.6)b(u,v,w)=12[∫Ω∑i,k=12ui∂vk∂xiwkdx-∑i,k=12ui∂wk∂xivkdx],∀u,v,w∈X,b¯(u,T,ψ)=12[∫Ω∑i=12ui∂T∂xiψdx-∑i2ui∂ψ∂xiTdx],∀u∈X,T,ψ∈W.The following assumptions and results are recalled (see [7, 29–31]).(A1)
There exists a constant C0 which only depends on Ω, such that
(i)
∥u∥0≤C0∥∇u∥0, ∥u∥0,4≤C0∥∇u∥0, for all u∈H01(Ω)2(orH01(Ω)),(ii)
∥u∥0,4≤C0∥u∥1,for all u∈H1(Ω)2,(iii)
∥u∥0,4≤2∥∇u∥01/2∥u∥01/2,for allu∈H01(Ω)2(orH01(Ω)).(A2)
Assuming ∂Ω∈Ck,α(k⩾0,α>0), then, for T0∈Ck,α(∂Ω), there exists an extension in C0k,α(ℝ2) (denote T0 also), such that
(2.7)‖T0‖k,q≤ɛ,k⩾0,1≤q≤∞,
where ɛ is an arbitrary positive constant.(A3)
b(·,·,·) and b¯(·,·,·) have the following properties.
(i)
For allu∈X, v,w∈X(orT,ψ∈H01(Ω)), there holds that
(2.8)b(u,v,v)=0,b(u,v,w)=-b(u,w,v),(2.9)b¯(u,T,T)=0,b¯(u,T,ψ)=-b¯(u,ψ,T).(ii)
For allu∈X, v∈H1(Ω)2(orT∈H1(Ω)), for allw∈X(orψ∈H01(Ω)), there holds that
(2.10)|b(u,v,w)|≤N‖∇u‖0‖∇v‖0‖∇w‖0,(2.11)|b¯(u,T,ψ)|≤N¯‖∇u‖0‖∇T‖0‖∇ψ‖0,
where
(2.12)N=sup⁡u,v,w⁡|b(u,v,w)|(‖∇u‖0‖∇v‖0‖∇w‖0),N¯=sup⁡u,T,φ⁡|b¯(u,T,φ)|(‖∇u‖0‖∇T‖0‖∇φ‖0).We recall the following existence, uniqueness and regularity result of(𝒫′) (see [7, Chapter 4]).Theorem 2.1 (see [7]).
Under the assumption of (A1)~(A3), letting A≡2ν-1λ(3C0+1)∥T0∥1, B≡2∥∇T0∥0+2(C02λ)-1A, there exist 0<δ1, δ2≤1 such that
(2.13)ν-1NA≤1-δ1,δ1-1ν-1C02λ2BN¯≤1-δ2.
Then, there exists a unique solution (u,p,T)∈X×M×W for (𝒫′), and
(2.14)‖∇u‖0≤A,‖∇T‖0≤B.Some estimates of the trilinear formb are given in the following lemma and the proof can be found in [30, 32–34].Lemma 2.2.
The trilinear formb satisfies the following estimate:
(2.15)|b(uh,vh,w)|+|b(vh,uh,w)|+|b(w,uh,vh)|≤C0|log⁡h|1/2‖∇vh‖0‖∇uh‖0‖w‖0,
for all uh,vh∈Vh, w∈L2(Ω)2.Lemma 2.3.
Suppose that (A1)~(A3) are valid and ɛ is a positive constant, such that
(2.16)32C02λ2N¯ɛ3ν<1,‖∇T0‖0≤ɛ4,‖T0‖0≤C0ɛ4,
then (𝒫′′) has a unique solution (uh,ph,Th)∈Xh×Mh×Wh, such that T|∂Ω=T0 and
(2.17)‖∇uh‖0≤5C02λɛ3ν,‖∇Th‖0≤ɛ.Proof.
The proof of the existence and the uniqueness of the solution has been given by Luo [7]. Let Th=ωh+T0, ψh=ωh in (2.5), we can get
(2.18)a¯(ωh,ωh)=-λb¯(uh,T0,ωh)-a¯(T0,ωh).
Using (2.11) and (2.16), we deduce
(2.19)‖∇ωh‖0≤‖∇T0‖0+λN¯ɛ‖∇uh‖0.
Letting vh=uh, φh=ph in the first equation of (2.5), we get
(2.20)ν‖∇uh‖02=|λ(jTh,uh)|≤λC0‖Th‖0‖∇uh‖0.
By (2.16), we can obtian
(2.21)‖∇uh‖0≤ν-1λC0‖Th‖0≤ν-1λC0(‖ωh‖0+‖T0‖0)≤ν-1λC02‖∇ωh‖0+ν-1λC0‖T0‖0≤ν-1λC0‖T0‖0+ν-1λC02‖∇T0‖0+ν-1λ2C02N¯ɛ‖∇uh‖0.
Using (2.16) again, we get
(2.22)‖∇uh‖0≤5C02λɛ3ν.
By (2.19), we deduce
(2.23)‖∇Th‖0≤‖∇ωh‖0+‖∇T0‖0≤2‖∇T0‖0+λN¯ɛ‖∇uh‖0≤2‖∇T0‖0+5C02λ2N¯ɛ23ν≤ɛ2+ɛ2=ɛ.We introduce the Laplace operator(2.24)Au=-Δu,∀u∈D(A)=H2(Ω)2∩X.Lemma 2.4 (see [35, 36]).
For allu,w∈X, v∈D(A) there holds that
(2.25)|b(u,v,w)|+|b(v,u,w)|+|b(w,u,v)|≤C‖Av‖0‖w‖0‖∇u‖0.


## 3. The Defect-Correction Method

The aim of this section is to give a method for solving the nonlinear system (2.5) on a coarser mesh than one uses when employing the standard FEM; the coarse-mesh solution is corrected using the same grid in our method. The defect-correction method in which we consider incorporates an artificial viscosity parameter σh as a stabilizing factor in the solution algorithm. For a fixed grid parameter h the method requires the solution of one nonlinear system and a few linear correction steps. It is described in the following paragraphs. We consider the following problems which is identical to (2.5) except for an artificial viscosity term.(𝒫*) Find (uh0,ph0,Th0)∈Xh×Mh×Wh such that(3.1)(ν+σh)a(uh0,vh)-d(ph0,vh)+d(φh,uh0)+b(uh0,uh0,vh)=λ(jTh0,vh),∀vh∈Xh,φh∈Mh,(1+σh)a¯(Th0,ψh)+λb¯(uh0,Th0,ψh)=0,∀ψh∈W0h.
We define the residual or named defect R(uh0,ph0,Th0), Q(uh0,ph0,Th0) for the momentum systems as follows:(3.2)(R(uh0,ph0,Th0),vh)=λ(jTh0,vh)-νa(uh0,vh)+d(ph0,vh)-d(φh,uh0)-b(uh0,uh0,vh),(Q(uh0,ph0,Th0),ψh)=-a¯(Th0,ψh)-λb¯(uh0,Th0,ψh).
Define the correction (ɛh0,ϱh0,τh0) satisfying the following linear problem:(3.3)(ν+σh)a(ɛh0,vh)-d(ϱh0,vh)+d(φh,ɛh0)+b(ɛh0,uh0,vh)+b(uh0,ɛh0,vh)=(R(uh0,ph0,Th0),vh),∀vh∈Xh,φh∈Mh,(1+σh)a¯(τh0,ψh)+λb¯(uh0,τh0,ψh)+λb¯(ɛh0,Th0,ψh)=(Q(uh0,ph0,Th0),ψh),∀ψh∈W0h.
Define uh1=uh0+ɛ0, ph1=ph0+ϱh0, Th1=Th0+τh0, which are hoped to be better solutions of the problems. In order to obtain the equations for (uh1,ph1,Th1), we use the residual equation (3.2) to rewrite the linear problems (3.3); we obtain(3.4)(P†){(ν+σh)a(uh1,vh)-d(ph1,vh)+d(φh,uh1)+b(uh0,uh1,vh)+b(uh1,uh0,vh)=λ(jTh1,vh)+σha(uh0,vh)+b(uh0,uh0,vh),∀vh∈Xh,φh∈Mh,(1+σh)a¯(Th1,ψh)+λb¯(uh1,Th0,ψh)+λb¯(uh0,Th1,ψh)=σha¯(Th0,ψh)+λb¯(uh0,Th0,ψh),∀ψh∈W0h.
In general, this method can be described as follows.Step 1.
Solve the nonlinear systems (3.1) for (uh0,ph0,Th0).Step 2.
Fori=1,2,…,m, solve the linear equations
(3.5)(P‡){(ν+σh)a(uhi,vh)-d(phi,vh)+d(φh,uhi)+b(uhi-1,uhi,vh)+b(uhi,uhi-1,vh)=(Thi,vh)+σha(uhi-1,vh)+b(uhi-1,uhi-1,vh),∀vh∈Xh,φh∈Mh,(1+σh)a¯(Thi,ψh)+λb¯(uhi,Thi-1,ψh)+λb¯(uhi-1,Thi,ψh)=σha¯(Thi-1,ψh)+λb¯(uhi-1,Thi-1,ψh),∀ψh∈W0h.
For each i the residual is given by
(3.6)(R(uhi,phi,Thi),vh)=λ(jThi,vh)-νa(uhi,vh)+d(phi,vh)-d(φh,uhi)-b(uhi,uhi,vh),(Q(uhi,phi,Thi),ψh)=-a¯(Thi,ψh)-λb¯(uhi,Thi,ψh).
The correction (ɛhi,ϱhi,τhi) is given by
(3.7)(ν+σh)a(ɛhi,vh)-d(ϱhi,vh)+d(φh,ɛhi)+b(ɛhi,uhi,vh)+b(uhi,ɛhi,vh)=(R(uhi,phi,Thi),vh),∀vh∈Xh,φh∈Mh,(1+σh)a¯(τhi,ψh)+λb¯(uhi,τi,ψh)+λb¯(ɛhi,Thi,ψh)=(Q(uhi,phi,Thi),ψh),∀ψh∈W0h.Remark 3.1.
From the numerical experiments, we see that one or two correction steps is adequate. And this is as same as [24].


## 4. Stability Analysis

In this section, we give the stability analysis. It is given by the following theorems.Theorem 4.1.
Under the assumptions of Lemma2.3, then (uh0,Th0) defined by (𝒫*) satisfies
(4.1)‖∇uh0‖0≤5C02λɛ3(ν+σh),‖∇Th0‖0≤ɛ.
Moreover, if
(4.2)25C02Nλɛ3(ν+σh)2<1,(𝒫*) admits a unique solution.Proof.
We define the set(4.3)BM={ṽh∈Xh;‖∇ṽh‖0≤5C02λɛ3(ν+σh)}.
Letũh be in ℬM. Then(4.4)(1+σh)a¯(Th0,ψh)+λb¯(ũh,Th0,ψh)=0,∀ψh∈W0h
has a unique solution Th0∈Wh such that Th|∂Ω=T0. For a given Th0, we consider the following problem:
(4.5)(ν+σh)a(uh0*,vh)-d(ph0*,vh)+d(φh,uh0*)+b(uh0*,uh0*,vh)=λ(jTh0,vh),∀vh∈Xh,φh∈Mh.
By the theory of the Navier-Stokes equations, we get (4.5) has a unique solution (uh0*,ph0*)∈Xh×Mh (see [31]). It means that (4.4) and (4.5) give a unique uh0*∈Xh for a given ũh∈Xh, we denote
(4.6)uh0*=lhũh.
SettingTh0=ωh0+T0, ψh=ωh0 in (4.4) and using (2.9), we can obtain(4.7)(1+σh)a¯(ωh0,ωh0)=-λb¯(ũh,T0,ωh0)-(1+σh)a¯(T0,ωh0).
Using (2.7), (2.11), and (2.16), we can get
(4.8)(1+σh)‖∇ωh0‖0≤λN¯‖∇ũh‖0‖∇T0‖0+(1+σh)‖∇T0‖0,‖∇ωh0‖0≤λN¯ɛ4‖∇ũh‖0+‖∇T0‖0.
Using the triangle inequality, we have
(4.9)‖∇Th0‖0≤‖∇T0‖0+‖∇ωh0‖0≤λN¯ɛ4‖∇ũh‖0+2‖∇T0‖0≤5C02N¯λɛ212(ν+σh)+ɛ2≤ɛ.
Lettingvh=uh0*, φh=ph0 in (4.5) and using (2.8), we get(4.10)(ν+σh)a(uh0*,uh0*)=λ(jTh0,uh0*).
Letting Th0=ωh0+T0 and using (2.9), we have
(4.11)(ν+σh)‖∇uh0*‖0≤C02λ‖∇ωh0‖0+C0λ‖T0‖0≤C02λ2N¯‖∇ũh‖0‖∇T0‖0+C0λ(1+C0)‖∇T0‖0≤C02λɛ.
Namely,
(4.12)‖∇uh0*‖0≤5C02λɛ3(ν+σh).
Hence, we proved that ℓh maps ℬM to ℬM. It follows from Brouwer's fixed-point theorem that there exits a solution to system (𝒫*).
To prove the uniqueness, assume that(uh01,ph01,Th01),(uh02,ph02,Th02)∈Xh×Mh×Wh, and Th01|∂Ω=Th02|∂Ω=T0 are two solutions of (𝒫*). Then, we obtain that(4.13)(ν+σh)a(uh01-uh02,vh)-d(ph01-ph02,vh)+d(φh,uh01-uh02)+b(uh01-uh02,uh01,vh)+b(uh02,uh01-uh02,vh)=λ(j(Th01-Th02),vh),∀vh∈Xh,φh∈Mh,(1+σh)a¯(Th01-Th02,ψh)+λb¯(uh02,Th01-Th02,ψh)+λb¯(uh01-uh02,Th01,ψh)=0,∀ψh∈W0h.
Let vh=uh01-uh02, φh=ph01-ph02 in the first equation of (4.13), we can get
(4.14)(ν+σh)‖∇(uh01-uh02)‖0≤N‖∇(uh01-uh02)‖0‖∇uh01‖0+C02λ‖∇(Th01-Th02)‖0.
Setting ψh=Th01-Th02 in the second equation of (4.13), we obtain
(4.15)(1+σh)‖∇(Th01-Th02)‖0≤λN¯‖∇(uh01-uh02)‖0‖∇Th01‖0.
By (4.14) and (4.15), we deduce
(4.16)(ν+σh)‖∇(uh01-uh02)‖0≤N‖∇uh01‖0‖∇(uh01-uh02)‖0+C02λ2N¯‖∇Th01‖0‖∇(uh01-uh02)‖0≤5C02Nλɛ3(ν+σh)‖∇(uh01-uh02)‖0+C02λN¯ɛ4ν‖∇(uh01-uh02)‖0.
Using (4.2), we obtain
(4.17)‖∇(uh01-uh02)‖0≤25‖∇(uh01-uh02)‖0.
Namely,
(4.18)‖∇(uh01-uh02)‖0=0.
By (4.15), we see that ∥∇(Th01-Th02)∥0=0. Therefore, it follows that (𝒫*) has a unique solution.
Then, we give the prove of (4.1) without using (4.2). Letting vh=uh0, φh=ph0 in the first equation of (3.1) and using (2.8), we get(4.19)(ν+σh)a(uh0,uh0)=λ(jTh0,uh0).
Letting Th0=ωh0+T0, we have
(4.20)(ν+σh)‖∇uh0‖0≤C02λ‖∇ωh0‖0+C0λ‖T0‖0.
Letting Th0=ωh0+T0, ψh=ωh0 in the second equation of (3.1) and using (2.9), we can obtain
(4.21)(1+σh)a¯(ωh0,ωh0)=-λb¯(uh0,T0,ωh0)-(1+σh)a¯(T0,ωh0).
Using (2.7), (2.11), and (2.16), we can get
(4.22)(1+σh)‖∇ωh0‖0≤λN¯‖∇uh0‖0‖∇T0‖0+(1+σh)‖∇T0‖0,‖∇ωh0‖0≤λN¯ɛ4‖∇uh0‖0+‖∇T0‖0.
By (4.20) and (4.22), we can deduce
(4.23)‖∇uh0‖0≤(ν+σh)-1λC02‖∇ωh0‖0+(ν+σh)-1C0λ‖T0‖0≤(ν+σh)-1(λC0‖T0‖0+C02λ‖∇T0‖0+C02λ2N¯ɛ4‖∇uh0‖0).
Using (2.16), we get
(4.24)‖∇uh0‖0≤5C02λɛ3(ν+σh).
Using (2.7), (2.11), (2.16), and (4.20), we can get
(4.25)‖∇ωh0‖0≤λN¯‖∇uh0‖0‖∇T0‖0+‖∇T0‖0≤λN¯ɛ4‖∇uh0‖0+‖∇T0‖0≤3ɛ4,‖∇Th0‖0≤‖∇ωh0‖0+‖∇T0‖0≤ɛ.
Therefore, we finish the proof.Theorem 4.2.
Under the assumptions of Lemma2.3, and
(4.26)25C02Nλɛ3(ν+σh)2<1,(uh1,Th1) defined by (3.4) satisfies
(4.27)‖∇uh1‖0≤δ,‖∇Th1‖0≤5ɛ6+λN¯δɛ+σhɛ,
where δ≐(103C02λɛ/48+σh(5C02λɛ/3ν))/(7/10)(ν+σh).Proof.
Lettingvh=uh1, φh=ph1 in the first equation of (3.4) and using (2.8), we get
(4.28)(ν+σh)a(uh1,uh1)+b(uh1,uh0,uh1)=b(uh0,uh0,uh1)+σha(uh0,uh1)+λ(jTh0,uh1).
Letting Th0=ωh0+T0 and using (2.10), we have
(4.29)(ν+σh)‖∇uh1‖0≤N‖∇uh1‖0‖∇uh0‖0+σh‖∇uh0‖0+N‖∇uh0‖02+C02λ‖∇ωh1‖0+C0λ‖T0‖0.
Let Th1=ωh1+T0, ψh=ωh1 in the second equation of (3.4), we can obtain
(4.30)(1+σh)a¯(ωh1,ωh1)=-λb¯(uh0,T0,ωh1)-λb¯(uh1,Th0,ωh1)+λb¯(uh0,Th0,ωh1)+σha¯(Th0,ωh1)-(1+σh)a¯(T0,ωh1).
Using (2.11) and (2.16), we can get
(4.31)(1+σh)‖∇ωh1‖0≤λN¯‖∇uh0‖0‖∇T0‖0+λN¯‖∇uh1‖0‖∇Th0‖0+λN¯‖∇uh0‖0‖∇Th0‖0+σh‖∇Th0‖0+(1+σh)‖∇T0‖0,(4.32)‖∇ωh1‖0≤λN¯‖∇uh0‖0‖∇T0‖0+λN¯‖∇uh1‖0‖∇Th0‖0+λN¯‖∇uh0‖0‖∇Th0‖0+σh‖∇Th0‖0+‖∇T0‖0.
Using (4.29), we get
(4.33)(ν+σh)‖∇uh1‖0≤N‖∇uh1‖0‖∇uh0‖0+σh‖∇uh0‖0+N‖∇uh0‖02+C02λ(λN¯‖∇uh0‖0‖∇T0‖0+λN¯‖∇uh1‖0‖∇Th0‖0+λN¯‖∇uh0‖0‖∇Th0‖0+‖∇Th0‖0+‖∇T0‖0)+C0λ‖T0‖0.(ν+σh-N‖∇uh0‖0-C02λ2N¯‖∇Th0‖0)‖∇uh1‖0≤N‖∇uh0‖02+σh‖∇uh0‖0+C02λ2N¯‖∇uh0‖0‖∇T0‖0+C02λ2N¯‖∇uh0‖0‖∇Th0‖0+C02λ‖∇Th0‖0+C02λ‖∇T0‖0+C0λ‖T0‖0.
Using (2.16), (4.26), and Theorem 4.2, we can obtain
(4.34)710(ν+σh)‖∇uh1‖0≤25NC04λ2ɛ29(ν+σh)2+σh5C02λɛ3(ν+σh)+10N¯C04λ3ɛ23(ν+σh)+3C02λ2ɛ22≤103C02λɛ48+σh5C02λɛ3(ν+σh).
Namely,
(4.35)‖∇uh1‖0≤103C02λɛ/48+σh(5C02λɛ/3ν)(7/10)(ν+σh)≐δ.
Using (2.16), (4.31), and (4.35), we can get(4.36)‖∇ωh1‖0≤10N¯C02λ2ɛ23(ν+σh)+λN¯δɛ+σhɛ+ɛ4≤ɛ3+ɛ4+λN¯δɛ+σhɛ=7ɛ12+λN¯δɛ+σhɛ.
Using the triangle inequality, we can get
(4.37)‖∇Th1‖0≤‖∇ωh1‖0+‖∇T0‖0≤5ɛ6+λN¯δɛ+σhɛ.
Therefore, we finish the proof.


## 5. Error Analysis

In this section, we establish theH1 and L2-bounds of the error u-uhi, T-Thi, i=0,1 and L2-bound of the error p-phi, i=0,1. In order to obtain the error estimates, we define the Galerkin projection (Rh,Qh)=(Rh(u,p),Qh(u,p)):(X,M)→(Xh,Mh), such that(5.1)a(Rh-u,vh)-d(Qh-p,vh)+d(qh,Rh-u)=0,∀(u,p)∈(X,M),(vh,qh)∈(Xh,Mh).Lemma 5.1 (see [37, 38]).
The Galerkin projection(Rh,Qh) satisfies
(5.2)‖Rh-u‖0+h(‖∇(Rh-u)‖0+‖Qh-p‖0)≤Chr+1(ν‖u‖r+1+‖p‖r),r=1,2.Lemma 5.2 (see [7]).
There exitsr̃h:W→Wh for all ψ∈W holds that
(5.3)(∇(ψ-r̃hψ),∇ψh)=0,∀ψh∈Wh,(5.4)∫Ω(ψ-r̃hψ)dx=0,‖∇r̃hψ‖0≤‖∇ψ‖0.
When ψ∈Wk,q(Ω)(1≤q≤∞), there holds
(5.5)‖ψ-r̃hψ‖-s,q≤Chk+s|ψ|k,q,-1≤s≤m,0≤k≤r+1.
There exits r¯h:W0→W0h for all ψ∈W0 holds that
(5.6)(∇(ψ-r¯hψ),∇ψh)=0,∀ψh∈W0h,‖∇r¯hψ‖0≤‖∇ψ‖0.
When ψ∈Wr,q(Ω)(1≤q≤∞), there holds
(5.7)‖ψ-r¯hψ‖-s,q≤Chk+s|ψ|r,q,-1≤s≤r,0≤k≤r+1.Lemma 5.3 (see [7]).
If (A1)~(A3) hold and (u,p,T)∈Hr+1(Ω)×Hr(Ω)×Hr+1(Ω) and (uh,Ph,Th) are the solution of problem (𝒫′) and (𝒫′′), respectively, then there holds that
(5.8)‖∇(u-uh)‖0+‖p-ph‖0+‖∇(T-Th)‖0≤Chr(‖u‖r+1+‖p‖r+‖T‖r+1).Lemma 5.4.
Under the assumptions of Lemma2.3, (uh,ph,Th) is the solution of (3.1), (uh0,ph0,Th0) defined by (3.4), then there hold
(5.9)‖∇(uh-uh0)‖0≤50σC02λh21ν+10C02λσɛh7ν,‖∇(Th-Th0)‖0≤2σhɛ+σhɛ14ν,β‖ph-ph0‖0≤95σhC02λɛ21ν+19σhC02λɛ7+σhC02λ+σhC02λ14ν.Proof.
Subtracting (3.1) from (2.5) we get the error equations, namely (uh-uh0,ph-ph0,Th-Th0) satisfy
(5.10)νa(uh-uh0,vh)-σha(uh0,vh)+d(φh,uh-uh0)-d(ph-ph0,vh)+b(uh0,uh-uh0,vh)+b(uh-uh0,uh,vh)=λ(j(Th-Th0),vh),∀vh∈Xh,φh∈Mh,a¯(Th-Th0,ψh)-σha¯(Th0,ψh)+λb¯(uh-uh0,Th,ψh)+b¯(uh0,Th-Th0,ψh)=0,∀ψh∈W0h.
Letting vh=uh-uh0,φh=ph-Ph0 in the first equation of (5.10) and using (2.11), (2.8), and (A1), we can get
(5.11)ν‖∇(uh-uh0)‖0≤σh‖∇uh0‖0+N‖∇(uh-uh0)‖0‖∇uh‖0+C02λ‖∇(Th-Th0)‖0.
Hence, we deduce
(5.12)(ν-N‖∇uh‖0)‖∇(uh-uh0)‖0≤σh‖∇uh0‖0+C02λ‖∇(Th-Th0)‖0.
Letting ψh=Th-Th0 in the second equation of (5.10) and using (2.9), we obtain
(5.13)a¯(Th-Th0,Th-Th0)+σha¯(Th0,Th-Th0)+λb¯(uh-uh0,Th,Th-Th0)=0.
Using (2.11), we can get
(5.14)‖∇(Th-Th0)‖0≤σh‖∇Th0‖0+λN¯‖∇(uh-uh0)‖0‖∇Th‖0.
By (5.12), we deduce
(5.15)(ν-N‖∇uh‖0)‖∇(uh-uh0)‖0≤σh‖∇uh0‖0+C02λσh‖∇Th0‖0+C02λ2N¯‖∇(uh-uh0)‖0‖∇Th‖0.
Using (4.1), we can obtain
(5.16)(ν-N‖∇uh‖0-3ν32)‖∇(uh-uh0)‖0≤σh‖∇uh0‖0+C02λσh‖∇Th0‖0≤5σhC02λɛ3(ν+σh)+C02λσhɛ.
By using (2.16) and (2.17), there holds
(5.17)ν-N‖∇uh‖0-3ν32≥7ν10.
Therefore, we can deduce
(5.18)‖∇(uh-uh0)‖0≤50σhC02λɛ21ν(ν+σh)+10C02λσhɛ7ν.
By (5.14) and (5.18), we can have
(5.19)‖∇(Th-Th0)‖0≤σhɛ+λN¯ɛ(50σhC02λɛ21ν(ν+σh)+10C02λσhɛ7ν)≤2σhɛ+σhɛ14ν.
Letting φh=0, vh=uh-uh0 in the first equation of (5.10) and using (2.3), we have
(5.20)β‖ph-ph0‖0≤ν‖∇(uh-uh0)‖0+σh‖∇uh0‖0+N‖∇(uh-uh0)‖0‖∇uh‖0+C02λ‖∇(Th-Th0)‖0≤50σhC02λɛ21(ν+σh)+10σhC02λɛ7+5σhC02λɛ3ν+10σhC02λɛ21ν+2σhC02λɛ7+σhC02λɛ+σhC02λ+σhC02λ14ν≤95σhC02λɛ21(ν+σh)+19σhC02λɛ7+σhC02λ+σhC02λ14ν.
Hence, we finish the proof.Theorem 5.5.
Under the assumptions of Lemmas2.3 and 5.3, the following inequality
(5.21)‖∇(u-uh0)‖0+‖p-ph0‖0+‖∇(T-Th0)‖0≤Chr(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch,
holds, where C is a positive constant numbers.Proof.
By Lemmas5.3, 5.4, and the triangle inequality this theorem is obviously true.Lemma 5.6.
For allu∈H2(Ω)∩X, ω∈W0, ψ∈H2(Ω)∩W0, there hold that
(5.22)|b¯(u-Rh,ω,ψ)|≤C‖u-Rh‖0‖Aω‖0‖∇ψ‖0,(5.23)|b¯(u,T-r̃hT,ψ)|≤C‖Au‖0‖T-r̃hT‖0‖∇ψ‖0.Proof.
Lettingω¯=(ω,0)T, we have
(5.24)b¯(u-Rh,ω,ψ)=b(u-Rh,ω¯,ψ).
Using (2.25), we can deduce (5.22). Because T-r̃hT∈W0, (5.23) holds.Theorem 5.7.
Under the assumptions of Lemmas2.3 and 5.3, the following inequality:
(5.25)‖u-uh0‖0+‖T-Th0‖0≤Chr+1(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch,
holds, where C is a positive constant.Proof.
Subtracting (3.1) from (2.4) we get the error equations, namely,
(5.26)νa(u-uh0,vh)-σha(uh0,vh)+b(u-uh0,uh0,vh)+b(u,u-uh0,vh)-d(p-ph0,vh)+d(φh,u-uh0)=λ(j(T-Th0),vh),∀vh∈Xh,φh∈Mh,a¯(T-Th0,ψh)-σha¯(Th0,ψh)+λb¯(u-uh0,T,ψh)+λb¯(uh0,T-Th0,ψh)=0,∀ψh∈W0h.
Letting eh0=Rh-uh0, ηh0=Qh-ph0, ξh0=r̃hT-Th0 and using (5.1) and (5.3), we can get
(5.27)νa(eh0,vh)-σha(uh0,vh)+b(u-uh0,uh0,vh)+b(u,u-uh0,vh)-d(ηh0,vh)+d(φh,eh0)=λ(j(T-Th0),vh),∀vh∈Xh,φh∈Mh,a¯(ξh,ψh)-σha¯(Th0,ψh)+λb¯(u-uh0,T,ψh)+λb¯(uh0,T-Th0,ψh)=0,∀ψh∈W0h.
Taking vh=eh0, φh=ηh0 in the first equation of (5.27), we obtain
(5.28)νa(eh0,eh0)-σha(uh0,eh0)+b(eh0,uh0,eh0)+b(u-Rh,uh0,eh0)+b(u,u-Rh,eh0)=λ(j(T-Th0),eh0),∀vh∈Xh,φh∈Mh.
Using (2.10) and (A1), we deduce
(5.29)(ν-N‖∇uh0‖0)‖∇eh0‖02≤|b(u-Rh,u,eh0)|+|b(uh0,u-Rh,eh0)|+|λ(j(T-Th0),eh0)|+|σha(uh0,eh0)|≤N(‖∇u‖0+‖∇uh0‖0)‖∇(u-Rh)‖0‖∇eh0‖0+C02λ‖∇(T-Th0)‖0‖∇eh0‖0+σh‖∇uh0‖0‖∇eh0‖0.
Using Theorem 2.1, (2.16), (4.1), and (5.2), we can obtain
(5.30)‖∇eh0‖0≤Chr+1(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch.
Taking ψh=ξh0 in the second equation of (5.27) and using (2.9) we have
(5.31)a¯(ξh0,ξh0)-σha¯(Th0,ξh0)+λb¯(uh0,T-r̃hT,ξh0)+λb¯(u-Rh,T,ξh0)+λb¯(eh0,T,ξh0)=0.
By (2.9), we have
(5.32)b¯(uh0,T-r̃hT,ξh0)+b¯(u-Rh,T,ξh0)+b¯(eh0,T,ξh0)=-b¯(eh0,T-r̃hT,ξh0)-b¯(u-Rh,T-r̃hT,ξh0)+b¯(u,T-r̃hT,ξh0)+b¯(u-Rh,T,ξh0)+b¯(eh0,T-r̃hT,ξh0)+b¯(eh0,ξh0,ξh0)+b¯(eh0,Th0,ξh0).
Letting T=ω+T0, ω∈W0 and using Lemma 5.6, we can get
(5.33)|b¯(uh0,T-r̃hT,ξh0)+b¯(u-Rh,T,ξh0)+b¯(eh,T,ξh0)|≤N¯‖∇eh0‖0‖∇(T-r̃hT)‖0‖∇ξh0‖0+N¯‖∇(u-Rh)‖0‖∇(T-r̃hT)‖0‖∇ξh0‖0+N¯‖∇eh0‖0‖∇(T-r̃hT)‖0‖∇ξh0‖0+N¯‖∇eh0‖0‖∇Th0‖0‖∇ξh0‖0+C‖Au‖0‖T-r̃hT‖0‖∇ξh0‖0+C‖u-Rh‖0‖Aω‖0‖∇ξh0‖0+C‖∇(u-Rh)‖0‖∇T0‖0‖∇ξh0‖0.
By assumption (A2), letting ɛ<h and using Lemma 5.1 and (2.16), (4.26), and (5.33), we can deduce
(5.34)‖∇ξh0‖0≤Chr+1(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch.
Hence, we have
(5.35)‖T-Th0‖0≤‖T-r̃hT‖0+‖ξh0‖0≤‖T-r̃hT‖0+C0‖∇ξh0‖0≤Chr+1(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch.
By (2.10) and (2.25), we can deduce(5.36)|b(u-Rh,u,eh0)|+|b(uh0,u-Rh,eh0)|≤|b(u-Rh,u,eh0)|+|b(u,u-Rh,eh0)|+|b(u-Rh,u-Rh,eh0)|+|b(eh,u-Rh,eh0)|≤C‖Au‖0‖u-Rh‖0‖∇eh0‖0+N(‖∇(u-Rh)‖0+‖∇eh0‖0)‖∇(u-Rh)‖0‖∇eh0‖0≤Ch2‖∇eh0‖0.
Using (5.29), we can obtain
(5.37)(ν-N‖∇uh0‖0)‖∇eh0‖02≤Chr+1(‖u‖r+1+‖p‖r+‖T‖r+1)‖∇eh0‖0+C0λ‖T-Th0‖0‖∇eh0‖0+σh‖∇uh0‖0‖∇eh0‖0.
By using (2.16) and (4.1), there holds
(5.38)ν-N‖∇uh0‖0≥4ν5.
Hence, we can deduce from (5.37)
(5.39)‖∇eh0‖0≤Chr+1(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch.
Therefore, we can deduce
(5.40)∥u-uh0∥0≤‖u-Rh‖0+‖eh0‖0≤‖u-Rh‖0+C0‖∇eh0‖0≤Chr+1(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch.Theorem 5.8.
Under the assumptions of Lemmas2.3 and 5.3, then there holds
(5.41)‖∇(u-uh1)‖0+‖∇(T-Th1)‖0≤Chr(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch2,‖u-uh1‖0+‖T-Th1‖0+h‖p-ph1‖0≤Chr+1(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch2,
where C is a positive constant.Proof.
Subtracting (3.4) from (2.4) we get the error equations, namely,
(5.42)νa(u-uh1,vh)-σha(uh1,vh)+b(u,u,vh)-b(uh1,uh0,vh)-b(uh0,uh1,vh)-d(p-ph1,vh)+d(φh,u-uh1)=λ(j(T-Th1),vh)-b(uh0,uh0,vh)-σha(uh0,vh),∀vh∈Xh,φh∈Mh,a¯(T-Th1,ψh)-σha¯(Th1,ψh)+λb¯(u,T,ψh)-λb¯(uh1,Th0,ψh)-λb¯(uh0,Th1,ψh)=-σha¯(Th0,ψh)-λb¯(uh0,Th0,ψh),∀ψh∈W0h.
Lettingeh1=Rh-uh1, ηh1=Qh-ph1, ξh1=r̃hT-Th1, using (5.1) and (5.3) and adding and subtracting appropriate terms in the above expression yields(5.43)(ν+σh)a(eh1,vh)+b(uh0,u-uh1,vh)+b(u-uh1,uh0,vh)-d(ηh1,vh)+d(φh,eh1)=λ(j(T-Th0),vh)+σha(Rh-uh0,vh)-b(u-uh0,u-uh0,vh),∀vh∈Xh,φh∈Mh,(1+σh)a¯(ξh1,ψh)+λb¯(uh0,T-Th1,ψh)-λb¯(u-uh1,Th0,ψh)=σha¯(r̃hT-Th0,ψh)-λb¯(u-uh0,T-Th0,ψh),∀ψh∈W0h.
Letting vh=eh1, φh=ηh1 in the first equation of (5.43), we can deduce
(5.44)(ν+σh)a(eh1,eh1)+b(uh0,u-Rh,eh1)+b(u-Rh,uh0,eh1)+b(eh1,uh0,eh1)=λ(j(T-Th0),eh1)+σha(Rh-uh0,eh1)-b(u-uh0,u-uh0,eh1)
By (2.10) and (2.25), we can deduce
(5.45)(ν+σh-N‖∇uh0‖0)‖∇eh1‖0≤C‖Auh0‖0‖u-Rh‖0+σh‖∇(Rh-uh0)‖0+N‖∇(u-uh0)‖02+λC0‖T-Th0‖0.
Using (2.16), (4.1), (4.26), (5.21), and (5.2), we can obtain
(5.46)‖∇eh1‖0≤Chr+1(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch2.
Using (5.2) and triangle inequality, we can have
(5.47)‖∇(u-uh1)‖0≤‖∇(u-Rh)‖0+‖∇e1‖0≤Chr(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch2,‖u-uh1‖0≤‖u-Rh‖0+‖e1‖0≤‖u-Rh‖0+C0‖∇e1‖0≤Chr+1(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch2.
Lettingψh=ξh1 in the second equation of (5.43) and using (2.8), we can deduce(5.48)(1+σh)a¯(ξh1,ξh1)+λb¯(uh0,T-r̃hT,ξh1)-λb¯(u-uh1,Th0,ξh1)=σha¯(T-Th0,ξh1)-λb¯(u-uh0,T-Th0,ξh1).
Letting Th0=ωh0+T0 and using (2.11), (5.22), and (5.23), we have
(5.49)(1+σh)‖∇ξh1‖0≤Cλ‖Auh0‖0‖T-r̃hT‖0+Cλ‖u-uh1‖0‖Aωh0‖0+N¯λ‖∇(u-uh1)‖0‖∇T0‖0+σh‖∇(T-Th0)‖0+λN¯‖∇(u-uh0)‖0‖∇(T-Th0)‖0.
Using (5.5), (5.21), (5.47), we can obtain
(5.50)‖∇ξh1‖0≤Chr+1(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch2.
Using (5.2) and triangle inequality, we can have
(5.51)‖∇(T-Th1)‖0≤‖∇(T-r̃hT)‖0+‖∇ξh1‖0≤Chr(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch2,‖T-Th1‖0≤‖T-r̃hT‖0+‖ξh1‖0≤‖T-r̃hT‖0+C0‖∇ξh1‖0≤Chr+1(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch2.
Takingφh=0, vh=Rh-uh1 in the first equation of (5.43) and using (2.3), we have(5.52)β‖ηh1‖0≤(ν+σh)‖∇eh1‖0+σh‖∇(u-uh0)‖0+2N‖∇uh0‖‖∇(u-uh1)‖0+C0λ‖T-Th0‖0+N‖∇(u-uh0)‖02.
By (4.1), (5.21), and (5.47), we can deduce
(5.53)‖ηh1‖0≤Chr(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch2.
Using (5.2) and triangle inequality, we can have
(5.54)‖p-ph1‖0≤‖p-Qh‖0+‖ηh1‖0≤Chr(‖u‖r+1+‖p‖r+‖T‖r+1)+Ch2.


## 6. Numerical Experiments

In this section, we present some numerical examples with a physical model of square cavity stationary flow. We choose differentν for comparison. The side length of the square cavity and the boundary conditions are given by Figure 1. From Figure 1, we can see that the T=0 on left and lower boundaries, ∂T/∂n=0 on upper boundary, and T=4y(1-y) on right boundary of the cavity. We use P2-P1-P2 finite element here.Figure 1
Physics model of the cavity flows.Firstly, we chooseν=1/2000, σ=0.4 and divide the cavity into M×N=40×40, that is, h=2/40. Figure 2 gives the numerical isotherms (a) and the numerical isobar (b). Figure 3 gives the numerical streamline. From the numerical results, we can see that our method is stable and has a good precision.The numerical Isotherms (a) and the numerical Isobar (b) forν=1/2000 by the defect-correction MFEM with h=2/40, σ=0.4.
(a)(b)Figure 3
The numerical streamline forν=1/2000 by the defect-correction MFEM with h=2/40, σ=0.4.Secondly, we chooseν=1/5000, σ=0.4 to show our our method suiting for solving the conduction convection problems with small viscosity. It is well known that it is more and more difficult to solve the problem by numerical method as ν changing smaller and smaller. Hence, we divide the cavity into M×N=100×100, namely h=2/100. Figure 4 gives the numerical isotherms (a) and the numerical isobar (b), and Figure 5 shows the numerical streamline. At last, we choose ν=1/6000, σ=0.4. Figure 6 gives the numerical isotherms (a) and the numerical isobar (b), and Figure 7 shows the numerical streamline.The numerical Isotherms (a) and the numerical Isobar (b) forν=1/5000 by the defect-correction MFEM with h=2/100, σ=0.4.
(a)(b)Figure 5
The numerical streamline forν=1/5000 by the defect-correction MFEM with h=2/100, σ=0.4.The numerical Isotherms (a) and the numerical Isobar (b) forν=1/6000 by the defect-correction MFEM with h=2/100, σ=0.4.
(a)(b)Figure 7
The numerical streamline forν=1/6000 by the defect-correction MFEM with h=2/100, σ=0.4.Just as Remark3.1, we only use one correction step in our numerical experiments. From the numerical, we can see that when ν=0.5×10-3 the numerical streamline is very regular. The pressure is small near the wall. But the numerical streamline changes more and more immethodical with ν changing smaller and smaller. And the pressure changes bigger near the wall. In conclusion, the defect-correction MFEM is highly efficient for the stationary conduction-convection problems and it can be used for solving the convection-conduction problems with much small viscosity.



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*Source: 370192-2011-02-06.xml*